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Main Content

propagationModel

Create RF propagation model

Syntax

``pm = propagationModel(modelname)``
``pm = propagationModel(___,Name,Value)``

Description

````pm = propagationModel(modelname)` creates an RF propagation model for the specified model.```

example

````pm = propagationModel(___,Name,Value)` updates the model using one or more name-value pairs. For example, ```pm = propagationModel('rain','RainRate',96)``` creates a rain propagation model with a rain rate of 96 mm/h. Enclose each property name in quotes.```

Examples

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Specify transmitter and receiver sites.

```tx = txsite('Name','MathWorks Apple Hill',... 'Latitude',42.3001, ... 'Longitude',-71.3504, ... 'TransmitterFrequency', 2.5e9); rx = rxsite('Name','Fenway Park',... 'Latitude',42.3467, ... 'Longitude',-71.0972);```

Create the propagation model for a heavy rainfall rate.

`pm = propagationModel('rain','RainRate',50)`
```pm = Rain with properties: RainRate: 50 Tilt: 0 ```

Calculate the signal strength at the receiver using the rain propagation model.

`ss = sigstrength(rx,tx,pm)`
```ss = -87.1559 ```

Create a transmitter site.

`tx = txsite`
```tx = txsite with properties: Name: 'Site 1' Latitude: 42.3001 Longitude: -71.3504 Antenna: 'isotropic' AntennaAngle: 0 AntennaHeight: 10 SystemLoss: 0 TransmitterFrequency: 1.9000e+09 TransmitterPower: 10 ```

Create a Longley-Rice propagation model using the `propagationModel` function.

`pm = propagationModel('longley-rice','TimeVariabilityTolerance',0.7)`
```pm = LongleyRice with properties: AntennaPolarization: 'horizontal' GroundConductivity: 0.0050 GroundPermittivity: 15 AtmosphericRefractivity: 301 ClimateZone: 'continental-temperate' TimeVariabilityTolerance: 0.7000 SituationVariabilityTolerance: 0.5000 ```

Find the coverage of the transmitter site using the defined propagation model.

`coverage(tx,'PropagationModel',pm)`

Input Arguments

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Type of propagation model specified as one of these:

• `'freespace'` — Free space propagation model.

• `'rain'` — Rain propagation model. For more information, see [3].

• `'gas'` — Gas propagation model. For more information, see [7].

• `'fog'` — Fog propagation model. For more information, see [2].

• `'close-in'` — Close-in propagation model typically used in urban macro-cell scenarios. For more information, see [1].

Note

The close-in model implements a statistical path loss model and can be configured for different scenarios. The default values correspond to an urban macro-cell scenario in a non-line-of-sight (NLOS) environment.

• `'longley-rice'` — Longley-Rice propagation model. This model is also known as Irregular Terrain Model (ITM). You can use this model to calculate point-to-point path loss between sites over an irregular terrain, including buildings. Path loss is calculated from free-space loss, terrain diffraction, ground reflection, refraction through atmosphere, tropospheric scatter, and atmospheric absorption. For more information and list of limitations, see [4].

Note

The Longley-Rice model implements the point-to-point mode of the model, which uses terrain data to predict the loss between two points.

• `'tirem'` — Terrain Integrated Rough Earth Model™ (TIREM™). You can use this model to calculate point-to-point path loss between sites over an irregular terrain, including buildings. Path loss is calculated from free-space loss, terrain diffraction, ground reflection, refraction through atmosphere, tropospheric scatter, and atmospheric absorption. This model needs access to an external TIREM library. The actual model is valid from 1 MHZ to 1000 GHz. But with Antenna Toolbox™ elements and arrays the frequency range is limited to 200 GHz.

• `'raytracing'` — A multipath propagation model that uses ray tracing analysis to compute propagation paths and corresponding path losses. Path loss is calculated from free-space loss, reflection loss due to material, and antenna polarization loss. You can perform ray tracing analysis using the image method (the default) or the shooting and bouncing rays (SBR) method. Specify a method using the `'Method'` property. Both methods include surface reflections but do not include effects from refraction, diffraction, or scattering. Both ray tracing methods are valid for a frequency range of 100 MHz to 100 GHz. For information about differences between the image and SBR methods, see Choose a Propagation Model. Use the `raytrace` function to plot the propagation paths between the sites.

You can use these functions on RF propagation models:

• `range` — Calculate the range of the radio wave under different propagation scenarios. The `range` function does not support the `'longley-rice'`, `'tirem'`, or `'raytracing'` propagation models.

• `pathloss` — Calculate the path loss of radio wave propagation between the transmitter and receiver sites under different propagation scenarios.

• `add` — Add propagation models.

Dependencies

To specify `'tirem'`, requires the Antenna Toolbox.

Data Types: `char`

Name-Value Pair Arguments

Specify optional comma-separated pairs of `Name,Value` arguments. `Name` is the argument name and `Value` is the corresponding value. `Name` must appear inside quotes. You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.

Example: `'RainRate',50` sets the rate of rainfall in the rain propagation model to 50.
Rain

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Rain rate, specified as a nonnegative scalar in millimeters per hour (mm/h).

Dependencies

To specify `'RainRate'`, you must specify `'rain'` propagation model.

Data Types: `double`

Polarization tilt angle of the signal, specified as a scalar in degrees.

Dependencies

To specify `'Tilt'`, you must specify `'rain'` propagation model.

Data Types: `double`

Gas

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Air temperature, specified as a scalar in Celsius (C).

Dependencies

To specify `'Temperature'`, you must specify `'gas'` propagation model.

Data Types: `double`

Dry air pressure, specified as a scalar in pascals (Pa).

Dependencies

To specify `'AirPressure'`, you must specify `'gas'` propagation model.

Data Types: `double`

Water vapor density, specified as a scalar in grams per cubic meter (g/m3).

Dependencies

To specify `'WaterDensity'`, you must specify `'gas'` propagation model.

Data Types: `double`

Fog

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Air temperature, specified as a scalar in Celsius (C).

Dependencies

To specify `'Temperature'`, you must specify `'fog'` propagation model.

Data Types: `double`

Liquid water density, specified as a scalar in grams per cubic meter (g/m3).

Dependencies

To specify `'WaterDensity'`, you must specify `'fog'` propagation model.

Data Types: `double`

Close-In

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Free-space reference distance, specified as a scalar in meters.

Dependencies

To specify `'ReferenceDistance'`, you must specify the `'close-in'` propagation model.

Data Types: `double`

Path loss exponent, specified as a scalar.

Dependencies

To specify `'PathLossExponent'`, you must specify `'close-in'` propagation model.

Data Types: `double`

Standard deviation of the zero-mean Gaussian random variable, specified as a scalar in decibels (dB).

Dependencies

To specify `'Sigma'`, you must specify `'close-in'` propagation model.

Data Types: `double`

Number of data points of zero-mean Gaussian random variable, specified as an integer.

Dependencies

To specify `'NumDataPoints'`, you must specify `'close-in'` propagation model.

Data Types: `double`

Note

The close-in model is valid for distances greater than or equal to the `'ReferenceDistance'` property. If a distance less than the `'ReferenceDistance'` is used, path loss is `0`.

Longley-Rice

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Polarization of transmitter and receiver antennas, specified as `'horizontal'` or `'vertical'`. Both antennas are assumed to have the same polarization. This value is used to calculate path loss due to ground reflection.

Dependencies

To specify `'AntennaPolarization'`, you must specify `'longley-rice'` propagation model.

Data Types: `char` | `string`

Conductivity of the ground, specified as a scalar in Siemens per meter (S/m). This value is used to calculate path loss due to ground reflection. The default value corresponds to average ground.

Dependencies

To specify `'GroundConductivity'`, you must specify `'longley-rice'` propagation model.

Data Types: `double`

Relative permittivity of the ground, specified as a scalar. Relative permittivity is expressed as a ratio of absolute material permittivity to the permittivity of vacuum. This value is used to calculate the path loss due to ground reflection. The default value corresponds to average ground.

Dependencies

To specify `'GroundPermittivity'`, you must specify `'longley-rice'` propagation model.

Data Types: `double`

Atmospheric refractivity near the ground, specified as a scalar in N-units. This value is used to calculate the path loss due to refraction through the atmosphere and tropospheric scatter. The default value corresponds to average atmospheric conditions.

Dependencies

To specify `'AtmosphericRefractivity'`, you must specify `'longley-rice'` propagation model.

Data Types: `double`

Radio climate zone. This value is used to calculate the variability due to changing atmospheric conditions. The default value corresponds to average atmospheric conditions in a particular climate zone.

Dependencies

To specify `'ClimateZone'`, you must specify `'longley-rice'` propagation model.

Data Types: `char` | `string`

Time variability tolerance level of the path loss, specified as a scalar between [0.001, 0.999]. Time variability occurs due to changing atmospheric conditions. This value gives the required system reliability or the fraction of time during which the actual path loss is expected to be less than or equal to model prediction. For more information, see [5].

Dependencies

To specify `'TimeVariabilityTolerance'`, you must specify `'longley-rice'` propagation model.

Data Types: `double`

Situation variability tolerance level of the path loss, specified as a scalar in between [0.001, 0.999]. Situation variability occurs due to uncontrolled or hidden random variables. This value gives the required system confidence or the fraction of similar situations for which the actual path loss is expected to be less than or equal to the model prediction. For more information, see [5].

Dependencies

To specify `'SituationVariabilityTolerance'`, you must specify `'longley-rice'` propagation model.

Data Types: `double`

TIREM

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Polarization of transmitter and receiver antennas, specified as `'horizontal'` or `'vertical'`. Both antennas are assumed to have the same polarization. This value is used to calculate path loss due to ground reflection.

Dependencies

To specify `'AntennaPolarization'`, you must specify `'tirem'` propagation model.

Data Types: `char` | `string`

Conductivity of the ground, specified as a numeric scalar in Siemens per meter (S/m) in the range of 0.0005 to 100. This value is used to calculate path loss due to ground reflection. The default value corresponds to average ground.

Dependencies

To specify `'GroundConductivity'`, you must specify `'tirem'` propagation model.

Data Types: `double`

Relative permittivity of the ground, specified as a numeric scalar in the range of 1 to 100. Relative permittivity is expressed as a ratio of absolute material permittivity to the permittivity of vacuum. This value is used to calculate the path loss due to ground reflection. The default value corresponds to average ground.

Dependencies

To specify `'GroundPermittivity'`, you must specify `'tirem'` propagation model.

Data Types: `double`

Atmospheric refractivity near the ground, specified as a numeric scalar in N-units in the range of 250 to 400. This value is used to calculate the path loss due to refraction through the atmosphere and tropospheric scatter. The default value corresponds to average atmospheric conditions.

Dependencies

To specify `'AtmosphericRefractivity'`, you must specify `'tirem'` propagation model.

Data Types: `double`

Absolute air humidity near ground,specified as a numeric scalar in `g/m^3` units in the range of 0 to 110. You can use this value to calculate path loss due to atmospheric absorption. The default value corresponds to the absolute humidity of air at 15 degrees Celsius and 70 percent relative humidity.

Dependencies

To specify `'Humidity'`, you must specify `'tirem'` propagation model.

Data Types: `double`

Ray Tracing

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Ray tracing method, specified as one of the following values:

• `'image'` — Use the image method, which supports up to two path reflections and calculates exact propagation paths.

• `'sbr'` — Use the shooting and bouncing rays (SBR) method, which supports up to 10 path reflections and calculates approximate propagation paths. The SBR method is generally faster than the image method.

Specify the maximum number of path reflections by using the `'MaxNumReflections'` property.

For more information about differences between the image and SBR methods, see Choose a Propagation Model.

Dependencies

To specify the ray tracing method, you must specify the `modelname` input argument as `'raytracing'`.

Data Types: `char` | `string`

Angular separation of launched rays, specified as one of the following values:

• `'high'` — Rays have an angular separation in the range [0.9912, 1.1845] measured in degrees, so that the model launches 40,962 rays.

• `'medium'` — Rays have an angular separation in the range [0.4956, 0.5923] measured in degrees, so that the model launches 163,842 rays.

• `'low'` — Rays have an angular separation in the range [0.2478, 0.2961] measured in degrees, so that the model launches 655,362 rays.

Because the model launches more rays, ray tracing analysis with low angular separation can require more time than with high angular separation.

When creating coverage maps using the `coverage` function, you can improve the results by choosing a lower angular separation.

Dependencies

To specify the angular separation of launched rays, you must specify the `modelname` argument as `'raytracing'` and the `'Method'` property as `'sbr'`.

Data Types: `char` | `string`

Maximum number of path reflections to search for using ray tracing, specified as an integer. Supported values depend on the value of the `'Method'` property.

• When `'Method'` is `'image'`, supported values are `0`, `1`, and `2`.

• When `'Method'` is `'sbr'`, supported values are in the range [0,10].

The default value of `1` results in a search for line-of-sight propagation paths and single-reflection propagation paths.

Dependencies

To specify the maximum number of path reflections, you must specify the `modelname` argument as `'raytracing'`.

Data Types: `double`

Coordinate system of the site location, specified as `'geographic'` or `'cartesian'`. If you specify `'geographic'`, material types are defined using `'BuildingMaterial'` or `'TerrainMaterial'` properties. If you specify `'cartesian'`, material types are defined using the `'SurfaceMaterial'` properties.

Data Types: `string` | `char`

Surface material of geographic buildings, specified as one of these: `'perfect-reflector'`, `'concrete'`, `'brick'`, `'wood'`, `'glass'`, `'metal'`, or `'custom'`. The material type is used to calculate reflection loss where propagation paths reflect off of building surfaces. For more information, see ITU Permittivity and Conductivity Values for Common Materials.

When `'BuildingsMaterial'` is set to `'custom'`, the material permittivity and conductivity are specified in the `BuildingsMaterialPermittivity` and `BuildingsMaterialConductivity` properties.

Dependencies

To specify `'BuildingsMaterials'`, you must set `'CoordinateSystem'` to `'geographic'`.

Data Types: `char` | `string`

Relative permittivity of the buildings surface material, specified as a nonnegative scalar. Relative permittivity is expressed as a ratio of absolute material permittivity to the permittivity of vacuum. This value is used to calculate path loss due to reflection. The default value corresponds to concrete at 1.9 GHz.

Dependencies

To specify `'BuildingsMaterialPermittivity'`, you must set `'CoordinateSystem'` to `'geographic'` and `'BuildingsMaterial'` to `'custom'`.

Data Types: `double`

Conductivity of the buildings surface material, specified as a nonnegative scalar in Siemens per meter (S/m). This value is used to calculate path loss due to reflection. The default value corresponds to concrete at 1.9 GHz.

Dependencies

To specify `'BuildingsMaterialConductivity'`, you must set `'CoordinateSystem'` to `'geographic'` and `'BuildingsMaterial'` to `'custom'`.

Data Types: `double`

Surface material of terrain, specified as one of these: `'perfect-reflector'`, `'concrete'`, `'brick'`, `'water'`, `'vegetation'`, `'loam'`, or `'custom'`. The material type is used to calculate reflection loss where propagation paths reflect off of terrain surfaces. For more information, see ITU Permittivity and Conductivity Values for Common Materials.

When `'TerrainMaterial'` is set to `'custom'`, the material permittivity and conductivity are specified in the `'TerrainMaterialPermittivity'` and `'TerrainMaterialConductivity'` properties.

Dependencies

To specify `'TerrainMaterial'`, you must set `'CoordinateSystem'` to `'geographic'`.

Data Types: `char` | `string`

Relative permittivity of the terrain material, specified as a nonnegative scalar. Relative permittivity is expressed as a ratio of absolute material permittivity to the permittivity of vacuum. This value is used to calculate path loss due to reflection. The default value corresponds to concrete at 1.9 GHz.

Dependencies

To specify `'TerrainMaterialPermittivity'`, you must set `'CoordinateSystem'` to `'geographic'` and `'TerrainMaterial'` to `'custom'`.

Data Types: `double`

Conductivity of the terrain material, specified as a nonnegative scalar in Siemens per meter (S/m). This value is used to calculate path loss due to reflection. The default value corresponds to concrete at 1.9 GHz.

Dependencies

To specify `'TerrainMaterialConductivity '`, you must set `'CoordinateSystem'` to `'geographic'` and set `'TerrainMaterial'` to `'custom'`.

Data Types: `double`

Surface material of Cartesian map surface, specified as one of these: `'plasterboard'`,`'perfect-reflector'`, `'ceilingboard'`, `'chipboard'`, `'floorboard'`, `'concrete'`, `'brick'`, `wood`, `'glass'`, `'metal'`, `'water'`, `'vegetation'`, `'loam'`, or `'custom'`. The material type is used to calculate reflection loss where propagation paths reflect off of surfaces. For more information, see ITU Permittivity and Conductivity Values for Common Materials.

When `'SurfaceMaterial'` is set to `'custom'`, the material permittivity and conductivity are specified in the `'SurfaceMaterialPermittivity'` and `'SurfaceMaterialConductivity'` properties.

Dependencies

To specify `'SurfaceMaterial'`, you must set `'CoordinateSystem'` to `'cartesian'`.

Data Types: `char` | `string`

Relative permittivity of the surface material, specified as a nonnegative scalar. Relative permittivity is expressed as a ratio of absolute material permittivity to the permittivity of vacuum. This value is used to calculate path loss due to reflection. The default value corresponds to plaster board at 1.9 GHz.

Dependencies

To specify `'SurfaceMaterialPermittivity'`, you must set `'CoordinateSystem'` to `'cartesian'` and `'SurfaceMaterial'` to `'custom'`.

Data Types: `double`

Conductivity of the surface material, specified as a nonnegative scalar in Siemens per meter (S/m). This value is used to calculate path loss due to reflection. The default value corresponds to plaster board at 1.9 GHz.

Dependencies

To specify `'SurfaceMaterialConductivity '`, you must set `'CoordinateSystem'` to `'cartesian'` and set `'SurfaceMaterial'` to `'custom'`.

Data Types: `double`

More About

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N-Units

The refractive index of air n is related to the dielectric constants of the gas constituents of an air mixture. The numerical value of n is only slightly larger than one. To make the calculation more convenient, you can use N units, which are given by the formula: $N=\left(n-1\right)×{10}^{6}$

ITU Permittivity and Conductivity Values for Common Materials

ITU-R P.2040-1 [8] and ITU-R P.527-5 [9] present methods, equations, and values used to calculate real relative permittivity, conductivity, and complex relative permittivity for the common materials.

Compatibility Considerations

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Behavior changed in R2021a

References

[1] Sun, S.,Rapport, T.S., Thomas, T., Ghosh, A., Nguyen, H., Kovacs, I., Rodriguez, I., Koymen, O.,and Prartyka, A. "Investigation of prediction accuracy, sensitivity, and parameter stability of large-scale propagation path loss models for 5G wireless communications." IEEE Transactions on Vehicular Technology, Vol.65, No 5, pp 2843-2860, May 2016.

[2] ITU-R P.840-6. "Attenuation due to cloud and fog." Radiocommunication Sector of ITU

[3] ITU-R P.838-3. "Specific attenuation model for rain for use in prediction methods." Radiocommunication Sector of ITU

[4] Hufford, George A., Anita G. Longley, and William A.Kissick. "A Guide to the Use of the ITS Irregular Terrain Model in the Area Prediction Mode." NTIA Report 82-100. Pg-7.

[5] SoftWright Homepage https://www.softwright.com/faq/support/longley_rice_variability.html

[6] Seybold, John. Introduction to RF Propagation. Wiley, 2005

[7] ITU-R P.676-11. "Attenuation by atmospheric gases." Radiocommunication Sector of ITU

[8] ITU-R P.2040-1. "Effects of Building Materials and Structures on Radiowave Propagation Above 100MHz." International Telecommunications Union - Radiocommunications Sector (ITU-R). July 2015.

[9] ITU-R P.527-5. "Electrical characteristics of the surface of the Earth." International Telecommunications Union - Radiocommunications Sector (ITU-R). August 2019.

[10] Yun, Zhengqing, and Magdy F. Iskander. “Ray Tracing for Radio Propagation Modeling: Principles and Applications.” IEEE Access 3 (2015): 1089–1100. https://doi.org/10.1109/ACCESS.2015.2453991.

[11] Schaubach, K.R., N.J. Davis, and T.S. Rappaport. “A Ray Tracing Method for Predicting Path Loss and Delay Spread in Microcellular Environments.” In [1992 Proceedings] Vehicular Technology Society 42nd VTS Conference - Frontiers of Technology, 932–35. Denver, CO, USA: IEEE, 1992. https://doi.org/10.1109/VETEC.1992.245274.

See Also

Introduced in R2019b

Communications Toolbox Documentation

Bridging Wireless Communications Design and Testing with MATLAB

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Record types¶

The Record construction is a macro allowing the definition of records as is done in many programming languages. Its syntax is described in the grammar below. In fact, the Record macro is more general than the usual record types, since it allows also for “manifest” expressions. In this sense, the Record construction allows defining “signatures”.

Command RecordStructure record_definition with record_definition*
::=
>? ident_decl : sort? := { ;? }?
::=
#[ ]* name | num?
|
|
:= term
::=
{| ;? |}
::=

Each record_definition defines a record named by ident_decl. The constructor name is given by ident. If the constructor name is not specified, then the default name Build_ident is used, where ident is the record name.

If sort is omitted, the default sort is Type. Notice that the type of an identifier can depend on a previously-given identifier. Thus the order of the fields is important. binder parameters may be applied to the record as a whole or to individual fields.

>?

If provided, the constructor name is automatically declared as a coercion from the class of the last field type to the record name (this may fail if the uniform inheritance condition is not satisfied). See Implicit Coercions.

Notations can be attached to fields using the decl_notations annotation.

Record and Structure are synonyms.

More generally, a record may have explicitly defined (a.k.a. manifest) fields. For instance, we might have: Record ident binder* : sort := { ident1 : type1 ; ident2 := term2 ; ident3 : type3 }. in which case the correctness of type3 may rely on the instance term2 of ident2 and term2 may in turn depend on ident1.

Example

The set of rational numbers may be defined as:

Record Rat : Set := mkRat  { sign : bool  ; top : nat  ; bottom : nat  ; Rat_bottom_cond : 0 <> bottom  ; Rat_irred_cond :      forall x y z:nat, (x * y) = top /\ (x * z) = bottom -> x = 1  }.
Rat is defined sign is defined top is defined bottom is defined Rat_bottom_cond is defined Rat_irred_cond is defined

Note here that the fields Rat_bottom_cond depends on the field bottom and Rat_irred_cond depends on both top and bottom.

Let us now see the work done by the Record macro. First the macro generates a variant type definition with just one constructor: Variant ident binder* : sort := ident0 binder*.

To build an object of type ident, provide the constructor ident0 with the appropriate number of terms filling the fields of the record.

Example

Let us define the rational $$1/2$$:

Theorem one_two_irred : forall x y z:nat, x * y = 1 /\ x * z = 2 -> x = 1.
1 subgoal ============================ forall x y z : nat, x * y = 1 /\ x * z = 2 -> x = 1
one_two_irred is declared
Definition half := mkRat true 1 2 (O_S 1) one_two_irred.
half is defined
Check half.
half : Rat

Alternatively, the following syntax allows creating objects by using named fields, as shown in this grammar. The fields do not have to be in any particular order, nor do they have to be all present if the missing ones can be inferred or prompted for (see Program).

Definition half' :=   {| sign := true;      Rat_bottom_cond := O_S 1;      Rat_irred_cond := one_two_irred |}.
half' is defined

The following settings let you control the display format for types:

Flag Printing Records

If set, use the record syntax (shown above) as the default display format.

You can override the display format for specified types by adding entries to these tables:

Table Printing Record qualid

Specifies a set of qualids which are displayed as records. Use the Add and Remove commands to update the set of qualids.

Table Printing Constructor qualid

Specifies a set of qualids which are displayed as constructors. Use the Add and Remove commands to update the set of qualids.

This syntax can also be used for pattern matching.

Eval compute in (   match half with   | {| sign := true; top := n |} => n   | _ => 0   end).
= 1 : nat

The macro generates also, when it is possible, the projection functions for destructuring an object of type ident. These projection functions are given the names of the corresponding fields. If a field is named _ then no projection is built for it. In our example:

Eval compute in top half.
= 1 : nat
Eval compute in bottom half.
= 2 : nat
Eval compute in Rat_bottom_cond half.
= O_S 1 : 0 <> bottom half

An alternative syntax for projections based on a dot notation is available:

Eval compute in half.(top).
= 1 : nat
Flag Printing Projections

This flag activates the dot notation for printing.

Example

Set Printing Projections.
Check top half.
half.(top) : nat
::=
|
term0 .( @ qualid )

Syntax of Record projections

The corresponding grammar rules are given in the preceding grammar. When qualid denotes a projection, the syntax term0.(qualid) is equivalent to qualid term0, the syntax term0.(qualid arg+) to qualid arg+ term0. and the syntax term0.(@qualid term0+) to @qualid term0+ term0. In each case, term0 is the object projected and the other arguments are the parameters of the inductive type.

Note

Records defined with the Record keyword are not allowed to be recursive (references to the record's name in the type of its field raises an error). To define recursive records, one can use the Inductive and CoInductive keywords, resulting in an inductive or co-inductive record. Definition of mutually inductive or co-inductive records are also allowed, as long as all of the types in the block are records.

Note

Induction schemes are automatically generated for inductive records. Automatic generation of induction schemes for non-recursive records defined with the Record keyword can be activated with the Nonrecursive Elimination Schemes flag (see Generation of induction principles with Scheme).

Warning ident cannot be defined.

It can happen that the definition of a projection is impossible. This message is followed by an explanation of this impossibility. There may be three reasons:

1. The name ident already exists in the environment (see Axiom).

2. The body of ident uses an incorrect elimination for ident (see Fixpoint and Destructors).

3. The type of the projections ident depends on previous projections which themselves could not be defined.

Error Records declared with the keyword Record or Structure cannot be recursive.

The record name ident appears in the type of its fields, but uses the keyword Record. Use the keyword Inductive or CoInductive instead.

Error Cannot handle mutually (co)inductive records.

Records cannot be defined as part of mutually inductive (or co-inductive) definitions, whether with records only or mixed with standard definitions.

During the definition of the one-constructor inductive definition, all the errors of inductive definitions, as described in Section Inductive types, may also occur.

Coercions and records in section Classes as Records of the chapter devoted to coercions.

Primitive Projections¶

Flag Primitive Projections

Turns on the use of primitive projections when defining subsequent records (even through the Inductive and CoInductive commands). Primitive projections extended the Calculus of Inductive Constructions with a new binary term constructor r.(p) representing a primitive projection p applied to a record object r (i.e., primitive projections are always applied). Even if the record type has parameters, these do not appear in the internal representation of applications of the projection, considerably reducing the sizes of terms when manipulating parameterized records and type checking time. On the user level, primitive projections can be used as a replacement for the usual defined ones, although there are a few notable differences.

Flag Printing Primitive Projection Parameters

This compatibility flag reconstructs internally omitted parameters at printing time (even though they are absent in the actual AST manipulated by the kernel).

Primitive Record Types¶

When the Primitive Projections flag is on, definitions of record types change meaning. When a type is declared with primitive projections, its match construct is disabled (see Primitive Projections though). To eliminate the (co-)inductive type, one must use its defined primitive projections.

For compatibility, the parameters still appear to the user when printing terms even though they are absent in the actual AST manipulated by the kernel. This can be changed by unsetting the Printing Primitive Projection Parameters flag.

There are currently two ways to introduce primitive records types:

1. Through the Record command, in which case the type has to be non-recursive. The defined type enjoys eta-conversion definitionally, that is the generalized form of surjective pairing for records: r = Build_R (r.(p$$_{1}$$) … r.(p$$_{n}$$)). Eta-conversion allows to define dependent elimination for these types as well.

2. Through the Inductive and CoInductive commands, when the body of the definition is a record declaration of the form Build_R { p$$_{1}$$ : t$$_{1}$$; … ; p$$_{n}$$ : t$$_{n}$$ }. In this case the types can be recursive and eta-conversion is disallowed. These kind of record types differ from their traditional versions in the sense that dependent elimination is not available for them and only non-dependent case analysis can be defined.

Reduction¶

The basic reduction rule of a primitive projection is p$$_{i}$$ (Build_R t$$_{1}$$t$$_{n}$$) $${\rightarrow_{\iota}}$$ t$$_{i}$$. However, to take the $${\delta}$$ flag into account, projections can be in two states: folded or unfolded. An unfolded primitive projection application obeys the rule above, while the folded version delta-reduces to the unfolded version. This allows to precisely mimic the usual unfolding rules of constants. Projections obey the usual simpl flags of the Arguments command in particular. There is currently no way to input unfolded primitive projections at the user-level, and there is no way to display unfolded projections differently from folded ones.

Compatibility Projections and match¶

To ease compatibility with ordinary record types, each primitive projection is also defined as a ordinary constant taking parameters and an object of the record type as arguments, and whose body is an application of the unfolded primitive projection of the same name. These constants are used when elaborating partial applications of the projection. One can distinguish them from applications of the primitive projection if the Printing Primitive Projection Parameters flag is off: For a primitive projection application, parameters are printed as underscores while for the compatibility projections they are printed as usual.

Additionally, user-written match constructs on primitive records are desugared into substitution of the projections, they cannot be printed back as match constructs.

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# Samplets: Construction and Scattered Data Compression

26 Pages Posted: 22 Mar 2022

See all articles by Helmut Harbrecht

## Helmut Harbrecht

affiliation not provided to SSRN

## Michael D. Multerer

Swiss Finance Institute - USI Lugano

### Abstract

We introduce the concept of samplets by transferring the construction of Tausch-White wavelets to scattered data. This way, we obtain a multiresolution analysis tailored to discrete data which directly enables data compression, feature detection and adaptivity. The cost for constructing the samplet basis and for the fast samplet transform, respectively,is $$\Ocal(N)$$, where $$N$$ is the number of data points.Samplets with vanishing moments can be used to compresskernel matrices, arising, for instance, in kernel based learning or Gaussian process regression. This leads tosparse matrices with only $$\mathcal{O}(N\log N)$$remaining entries. We provide estimates for the compression error and present an algorithm that computes the compressed kernel matrix with computationalcost $$\mathcal{O}(N\log N)$$. The accuracy of the approximation is controlled by the number of vanishing moments.Besides the cost efficient storage of kernel matrices,the sparse representation enables the application of sparsedirect solvers for the numerical solution of corresponding linear systems.In addition to a comprehensive introduction to samplets andtheir properties, we present numerical studies to benchmarkthe approach. Our results demonstrate that samplets mark aconsiderable step in the direction of making large scattered datasets accessible for multiresolution analysis.

Suggested Citation

Harbrecht, Helmut and Multerer, Michael D., Samplets: Construction and Scattered Data Compression. Available at SSRN: https://ssrn.com/abstract=4053305 or http://dx.doi.org/10.2139/ssrn.4053305


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# Tag Info

## Hot answers tagged transfer-function

9

You can apply a so-called all-pass transformation to a discrete-time low-pass prototype filter in order to convert it to other standard filters (such as high-pass, band-pass, and band-stop). This is accomplished by transforming the complex variable $z$ in the transfer function of the prototype filter by a function $G(z)$ which satisfies $|G(e^{j\omega})|=1$, ...

6

Note that in this case you can see that the system is causal only from the given implementation. It's important to understand that you can't see it from the difference equation (if no initial conditions are given), and in general you can't see it from the transfer function either (if no region of convergence is given). The only case for which the expression ...

6

In your original code you defined the sampling interval to be $T_s=0.05$ (idd1 = iddata(ddout, ddin, 0.05);). Yet, according to the data file, the time step between sample points appears to be $T_s=0.003$. Changing $T_s$ from $0.5$ to $0.003$ should compress the step response by a factor of $0.05/0.003=16.67$, which compares favorably to the relation between ...

5

What you are looking for are what we, in the audio space, call comb filters. Comb filters may or may not have a feedback path, just like FIR and IIR filters. In fact, there is a generalized theory about designing comb filters based simply on designing any digital filter. In fact, you can think of every LTI digital filter as being a comb filter where the ...

4

The given transfer function has a specific form. With the denominator polynomial $A(z)=2+bz^{-2}$, you can write the transfer function as $$H(z)=\frac{z^{-2}}{2}\frac{A\left(\frac{1}{z}\right)}{A(z)}\tag{1}$$ Since for $z=e^{j\omega}$ (i.e., on the unit circle, where we evaluate the frequency response), we have $|A(z)|=|A(z^{-1})|$, the magnitude of $H(e^{... 4 Or another way, visually, with the plot of the magnitude response of your filter shown below you see that it goes to zero at$\pi/2$. So what you could do is use the relationship between the angular frequency$\omega$in [radians/sample] and the relative frequency$f$in [cycles/sample] $$\omega = 2\pi f = 2\pi \left(\frac FF_s\right)$$ Then compute$F_s$... 4 Purely by inspection of the block diagram the system is causal, because the output is the sum of the current input sample and stuff that's delayed -- there's no$z$blocks in there to predict the future, just$z^{-1}$block to react to the past. Also by your method of finding the transfer function, the system is causal -- with a$3^{rd}$order numerator and ... 4 This is unfortunately a difficult problem. A microphone doesn't have a single frequency response. It has a different one for each direction of incidence. Especially for second order microphones (dipoles, cardioids, etc.), the frequency response also depends heavily on the distance and radiation impedance of the source. So first you need to decide: what ... 4 The transfer function is$H(s) = \frac{16.94s + 579.5}{s^2 + 507.2s + 1224}$This transfer function has 2 poles, one slow pole at -2.4248 and a fast pole at -504.7752. The function has a slowish zero at -34.2. Good news, your poles and zero are all in the left-half plane. It is much easier to control a system with zeroes and poles in the left-half plane ... 3 For continuous-time systems, you obtain the frequency response by evaluating the transfer function$H(s)$on the imaginary axis$s=j\omega$(assuming stability). In discrete-time, you get the$\mathcal{Z}$-transform instead of the Laplace transform, and the imaginary axis is replaced by the unit circle. So the frequency response is obtained by evaluating the ... 3 That purple array is giving the impulse response over time. You can get it directly from the difference equation. Assume initial rest,$y[-1]=0$, then write out the impulse response for$n=0, 1, 2, .... If you do that you will get: \begin{align} h[0]&=1+0 \\ h[1] &= 0 + 0.5 \\ h[2] &= 0 + 0.25 \\ h[3] &= 0 + 0.125 \\ ... \end{align} 3 Basically the transfer function is given by (Continuous time case): $$H \left( j \omega \right) = K \frac{ \left( j \omega - {q}_{1} \right) \left( j \omega - {q}_{2} \right) }{ \left( j \omega - {p}_{1} \right) \left( j \omega - {p}_{2} \right) }$$ Now, given your frequency data sampled an N $points$ {\left\{ {\omega}_{i} \right\}}_{i = 1}^{N} $we ... 3 However when I look at the closed loop transfer function, I would say that this system is unstable for 𝐺𝐻=−1. In this case the transfer function becomes infinity so a bounded input will result in a unbounded (=infinity) output. This depends on your definition of stability.$GH = -1$is called marginally stable because depending on how you look at it, it ... 3 [EDIT: note that the note you refer to compute the discrete-time Fourier transform, via a continuous argument in frequency. And not a DFT. You are apparently computing a 3-point DFT] What I usually call the size-2 binomial filter is$\beta_1=\frac{1}{2}[1\;1], the 2-point moving average, whose Fourier representation is well-known, or easy to compute. $$... 3 I don't really understand the thing about cosines (as in: how is that helpful?) – a DFT is really just a mapping from a complex vector with N elements to a complex vector with N elements; and your calculation seems to be wrong, and I'm not sure where, but doing two of the three elements of the DFT manually might actually be enough to clear things up. \... 3 The general approach would be to find the zeros of the filter's transfer function. If you do things right it will turn out that there are two complex conjugate zeros on the unit circle at \pm j, i.e., at half the Nyquist frequency. Now you need to choose the sampling frequency such that 80 Hertz corresponds to half the Nyquist frequency. 3 If the OP is actually interested in selecting only one individual frequency from the even or odd harmonics, then a moving average filter (MAF) would be ideal since this can provide a null at every other frequency when the frequencies are harmonically related. The low-pass MAF with all the coefficients as 1 will pass DC at f=0 and provide nulls spaced by 1/... 2 HINT: There's no need to compute the frequency response by evaluating the transfer functions for z=e^{j\omega}. Just compute the poles and zeros and see where they are. Estimate the effect of the poles and zeros on the frequency response by trying to see how they influence the behavior of the transfer function on the unit circle. 2 Explicitly making the imaginary part of your FFT coefficients 0 will change your magnitude response. It is very clear if you consider this:$$ X_k = a_k + jb_k\\ |X_k| = \sqrt{a_k^2 + b_k^2} $$Now, you're explicitly setting new coefficients to be X'_k = a_k, and hence |X'_k| = |a_k|. Taking the IFFT of your new frequency response will obviously not be ... 2 No, at least not the way it's written. U_1(\omega) contains two frequency peaks (10Hz and 0Hz) and U_2(\omega) only one at \omega _2. G(\omega) can change the relative height of the peaks but the overall but not create new ones. A_2 and U_2(\omega) can vary the height and position of the peak but not the number of peaks. 2 HINT: The frequency response has the form$$H(e^{j\omega})=a+be^{-j\omega}+ce^{-2j\omega}+be^{-3j\omega}+ae^{-4j\omega}\tag{1}$$which can be rewritten as$$H(e^{j\omega})=e^{-2j\omega}\big[ae^{2j\omega}+be^{j\omega}+c+be^{-j\omega}+ae^{-2j\omega}\big]\tag{2}$$Now note that the term in brackets is purely real-valued. I trust that you can take it from here. 2 There are a few things I can note about your question. As far as I have always learned, the nyquist stability criterion is taken over the openloop transfer function. if you take the closed loop transfer function, you should count the encirclements of 0 instead (if i recall correctly). The formal definition of stability, as expressed by the Lyapunov's ... 2 A transfer function describes an LTI system. As such, the given system can be described by a transfer function. However, if there are non-zero initial conditions, the system is no longer linear because there's a contribution in the output that does not depend on the input signal but only on the initial conditions. Consequently, the transfer function cannot ... 2 Your filter is a 5-tap delay line FIR filter. The FIR filter's coefficients are [2 6 1 5 1]. If x is your filter's input sequence and you execute the following MATLAB command: h = [2 6 1 5 1]; you can then implement the filter (to produce a y output sequence) using the following MATLAB command: y = filter(h, 1, x); You can see your filter's frequency ... 2$$\begin{align} Y(z) &= E(z) + \frac{1}{1-z^{-1}}(X(z) - z^{-1} Y(z)) \\ &= E(z) + \frac{1}{1-z^{-1}}X(z) - \frac{z^{-1}}{1-z^{-1}} Y(z) \\ \\ Y(z) + \frac{z^{-1}}{1-z^{-1}} Y(z) &= E(z) + \frac{1}{1-z^{-1}}X(z) \\ \left(1 + \frac{z^{-1}}{1-z^{-1}}\right) Y(z) &= E(z) + \frac{1}{1-z^{-1}}X(z) \\ \frac{1}{1-z^{-1}} Y(z) &= E(z) + \frac{1}... 2 The closed loop poles are the roots of the polynomial $$D(s)=s^2+2s+2+K\tag{1}$$ and, according to the root locus plot, they ares_{1,2}=-1\pm 2j$. Consequently, we get $$2+K=|1+2j|^2=5\quad\Longrightarrow\quad K=3\tag{2}$$ With$K=3$we obtain $$H(0)=\frac{K}{2+K}=\frac35\tag{3}$$ which leaves step response$C$as the only option. 2 The system is causal, provided that the recursion is forward; i.e., it's recursed for increasing$k$. Seeing that you are confused about causality tests, let me elaborate on it. Let's put the definition of causality from Oppenheim's Signals & Systems book : A system is causal if the output at any time depends only on values of the input at the present ... 2 You can use a window as an FIR filter (convolve with window in time), such that the zero crossings of the window transform are at$\omega_0 = 2\pi * 2k*200$to filter the even harmonics, and phase shift the window by transform$\frac{\pi}{2}$to filter the odd harmonics. For example, a rectangular window of length$M$will have zero crossings at$\omega = \...

1

You can use the manage/export functionality to convert your Audacity filter preset into a text file. The file contains frequency/value pairs plus some side info. It looks like Audacity builds an 8192 FFT grid and uses a B-spline to interpolate the frequency/value pairs over the entire grid. I assume that just do an inverse FFT plus some circular time ...

1

If the system described by the transfer function $H(s)$ is stable, you can obtain its frequency response by substituting $s=j\omega$, and use the relation that you found: $$S_Y(\omega)=S_X(\omega)\big|H(j\omega)\big|^2\tag{1}$$ where $S_X(\omega)$ and $S_Y(\omega)$ denote the power spectra of the system's input and its output, respectively.

Only top voted, non community-wiki answers of a minimum length are eligible


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## The Vector Cross Product

The vector or cross product of two vectorsandwrittenis also a vector. It is at right angles to bothandand if the angle betweenandisthenThe vectorcan be calculated using one of the following methods:

This can be hard to remember and many people prefer to find the determinant of a matrix, for which the vectors written in ijk form.

Example: Find the vector product ofand


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# Simulation of leaf curl disease dynamics in chili for strategic management options

## Abstract

Leaf curl, a whitefly-borne begomovirus disease, is the cause of frequent epidemic in chili. In the present study, transmission parameters involved in tripartite interaction are estimated to simulate disease dynamics in a population dynamics model framework. Epidemic is characterized by a rapid conversion rate of healthy host population into infectious type. Infection rate as basic reproduction number, R0 = 13.54, has indicated a high rate of virus transmission. Equilibrium population of infectious host and viruliferous vector are observed to be sensitive to the immigration parameter. A small increase in immigration rate of viruliferous vector increased the population of both infectious host and viruliferous vector. Migrant viruliferous vectors, acquisition, and transmission rates as major parameters in the model indicate leaf curl epidemic is predominantly a vector -mediated process. Based on underlying principles of temperature influence on vector population abundance and transmission parameters, spatio-temporal pattern of disease risk predicted is noted to correspond with leaf curl distribution pattern in India. Temperature in the range of 15–35 °C plays an important role in epidemic as both vector population and virus transmission are influenced by temperature. Assessment of leaf curl dynamics would be a useful guide to crop planning and evolution of efficient management strategies.

## Introduction

Begomoviruses (family Geminiviridae) are the most destructive plant viruses that cause diseases like leaf curl, mosaic, yellow mosaic and yellow vein mosaic in numerous crop plants in the tropical and subtropical world. Leaf curl disease in chili is known to cause by several begomoviruses of which Chili leaf curl virus (ChiLCV) is the most predominant in India1,2,3. ChiLCV is efficiently spread by whitefly (Bemisia tabaci) which is abundant year-round in tropical and subtropical climates where wide variety of hosts serves as reservoirs3,4,5,6,7. Leaf curl in chili (ChiLCD) is a common disease but recent frequent epidemic outbreaks witnessed repeatedly in the chili growing areas of Central and Southern India is now a growing concern3,8,9,10. ChiLCD is the most damaging in terms of yield loss across the cropping regions11. In severe cases, even up to 100% losses of marketable fruit have been reported12. In absence of epidemiological information so far no definite management strategy has been evolved. Interaction or contact rate between viruliferous vector and the growing host at initial stages is crucial in epidemic development and thus the period of high contact rate appears critical point of interventions13,14. Simulation of epidemics resulted from complex host–vector interaction is of great interest for on-farm management and assessment of agricultural policies and practices15,16,17. Time of whitefly population abundance, especially the growth of viruliferous flies and its effect on host associated leaf curl risk at regional and global scales is of great interest for precise interventions. Type of intervention that ensures low contact rates and keeps the proportion of infectious host at minimum level, are useful in designing precise management strategies18. Chili being cultivated in a wide variety of agro-ecosystems in India, disease dynamics in relation to tripartite interaction and transmission behavior are useful for evolving management strategies universally applicable to all climates. Understanding the complex host–vector interactions is also necessary to assess emerging disease risk under rapidly changing cropping patterns as well as global temperature rise scenario6,19,20.

Leaf curl epidemic is resulted from a complex tripartite interaction between host, vector, and virus where virus transmission plays a central role15. Population dynamics models have been used to characterize epidemics resulted from tripartite interaction especially for whitefly transmitted begomovirus in cassava21,22,23,24 and tomato25. Environmental factor particularly temperature plays important role in transmission and vector movements17,22,26. Virus transmission, latent period, infectious period, replication, titer, movements and symptoms expression etc. are directly linked to temperature27,28,29. For prediction of whitefly population abundance, temperature-dependent model, explaining developmental rates of each life stage of B. tabaci, is used30,31,32,33. Availability of large scale geo-spatial temperature data facilitates prediction of spatio-temporal distribution of whitefly abundance. To unfold tripartite interaction in the agroecosystem, it is necessary to link the time of whitefly abundance in relation to virus transmission and actual disease risk.

Establishment of disease dynamics in terms of epidemic parameters such as transmission and acquisition rate, prediction of vector abundance time and associated disease risk period may help to develop a testing framework for evolving precise intervention strategies15,34. Further, assessment of temperature influence on transmission parameters and prediction of potential geographical distribution in terms of whitefly abundance is likely to shed light on tripartite interaction which may provide insights for developing a rational management policy.

The current paper has addressed the estimation of transmission parameters for simulation of leaf curl dynamics and explaining tripartite interaction in a population dynamics modeling framework. Leaf curl disease dynamics established was used to identify the most sensitive parameter contributing to the growth of infectious host and viruliferous whitefly population in the field. Temperature influence on transmission parameters and whitefly population abundance was used for approximation of disease risk and compare known leaf curl disease distribution patterns in the chili growing areas of the subcontinent.

## Results

### Leaf curl virus (ChiLCV) transmission in chili

Transmission parameters, estimated under semi-controlled condition, were observed to vary in response to temperature. Transmission or inoculation rate as the number of plant/vector/week was 0.12 at 25 °C as compared to 0.04 and 0.07 at 15 °C and 35 °C, respectively (Fig. 1). The number of vectors used in the inoculation experiment influenced the transmission rate. The transmission rate was nine times higher when plants were inoculated with one viruliferous vector as compared to ten. Therefore, the transmission rate was affected by vector aggregation as well as temperature. Acquisition rate at 25 °C was observed to be 0.70 vector/plant/week which was higher than that at 15 °C and 35 °C where the values were in the range of 0.17 to 0.28. Within 20 min, vectors were observed to acquire virus and by 6 h of feeding access period, almost all the vectors were found to acquire ChiLCV as judged by the positive result in PCR test. Temperature response was typically follow a unimodal pattern as the transmission parameter values were observed to increase with the increase in temperature up to a favorable point. Higher transmission and acquisition efficiency at moderate temperature (25 °C) than at lower (15 °C) and decreased afterword higher ranges of temperatures (35 °C) indicated transmission parameters are governed by optimal and suboptimal ranges of temperature.

It appeared leaf curl disease or incidence distribution would likely be affected by prevalent temperature in time and location as virus transmission is influenced by temperature.

### Leaf curl incidence as proportion of leaf curl infection and virulifreous vector in chili field

In the experimental field, chili plants were noted to have an infection from the very first week of transplantation as a proportion (0.01) of samples tested PCR-positive for ChiLCV (Table 1). Proportion of infection was observed to increase rapidly to 0.52 within the fifth week period and by 10th week reached 0.76 level. Population of viruliferous whitefly (V) from the time of transplantation was 0.65 and went on increasing and reached 0.98 by the end of 12th week. Whitefly population sampled from the field (brinjal, cucurbits, tomato and Leucana leucocephala) was observed to be PCR-positive even at the time of transplantation and proportion of viruliferous vector was estimated to be 0.65.

It appeared that chili planting during August was at very high-risk period when a high proportion of viruliferous vector already existed nearby field or in border plants.

### Population dynamics model for simulation of leaf curl epidemic in chili

For simulation of epidemics, population dynamics models were fitted on leaf curl incidence data observed under field conditions. Transmission parameters estimated under semi-controlled conditions were tuned to match population of healthy host and viruliferous vector. Values of model parameters and variables tuned or adjusted for leaf curl epidemic analysis are given in Table 2. Population of healthy host (S) was predicted fitting host dynamics model (dS/dt = − aVS) and found to correspond with the observed proportion of healthy chili plants in the field (Fig. 2). Transmission or inoculation rate (a = 0.12) was approximately close to simulate healthy host population in the field as indicated by low RMSE. Acquisition rate (b = 0.70) estimated under semi-controlled conditions and used to simulate virulifeorus vector population fitting vector population dynamics model (dV/dt = b[1 − V]I/(Kx + 1 − V) + iv − ev − uV)was in good agreement as indicated by the low RMSE value. Immigration rate (iv) was approximated to be 0.34 flies plant−1 week−1 with high infectivity (0.65) which led to a high infection rate in chili within 2-month period of transplantation. Adjustment of death rate to 0.34 indicated high mortality than the usual rate. Dynamics of vector population were observed to be better explained by the Michaelis–Menten parameter (Kx = 2.8) than the simple kinetic parameter.

Estimation of basic reproduction number (R0) indicated from one infected host about 13.54 plants are infected in the next generation. High transmission rate resulted in quick growth in the leaf curl epidemic.

Fitted population dynamics model on field data based on transmission parameters (a and b estimated) appeared to represent the dynamic tripartite interaction in the field. Thus tripartite interaction captured in the model is likely to represent the dynamics of host and viruliferous vector population that were involved in the epidemic process.

### Critical epidemic parameters in leaf curl disease dynamics-sensitivity analysis

Tripartite interaction captured in a dynamic model was used to assess the behavior of other parameters like immigration and emigration, death and birth rate of whiteflies which are difficult to observe but to be determined analytically. Sensitivity analysis was made to determine how changes in parameter values affected the epidemic process underpinning disease management clues. Changes in parameter values particularly for transmission rate (a), immigration rate (iv) and acquisition rate (b) were noted to have a marked influence on population of host as well as viruliferous vector. Transmission rate parameter (a) was observed to be responsive throughout the crop period indicating the importance of the vector activity in disease spread (Fig. 3A). Equilibrium population of infectious host, as well as viruliferous vector, was sensitive to immigration parameter (iv) as both the population got increased with the increase in parameter value. Increase in infectious host population was more at a lower level (0.05 and 0.1) than at a higher level (0.4) of vector population (Fig. 3B). It indicated immigration of viruliferous-vector is very important at the initial stage of virus transmission than at a later stage of epidemic development when high level of viruliferous population may be there. Measures to check migration of viruliferous vectors or avoiding migrant viruliferous vectors would be important in disease management. Immigration rate was also realized to be responsive for an increase in viruliferous vector population. The sudden jump in viruliferous vector population is likely to be in presence of an infectious host population (I = 0.4) even at a low rate of vector migration (Fig. 3C). Therefore, immigration of viruliferous vector was the most important parameter which acted concurrently in the flare-up of both the infectious population of host and vector. Acquisition rate was sensitive to increase viruliferous vector population at a higher level of infectious host (I = 0.4) as compared to the low level (I = 0.05–0.1) of population (Fig. 3D).

Therefore, tripartite interaction for leaf curl virus in chili was simplified through a schematic diagram maintaining the qualitative behavior of the process that is useful in understanding the dynamics of the transmission process and possible management interventions cues (Fig. 4).

Migrant viruliferous vector population appeared to play an important role in initial infection as well as to flare up both infectious host and virulifeorus vector population in the field for later exponential growth and make epidemic progress.

### Temperature influence on whitefly population abundance

At the experimental site, whitefly population was observed in two distinct seasonal peaks, one during February–May and another in mid July–November period (Fig. 5A). Normalized population count was higher during July–November than February-May period. Temperature profile during Feb–March-mid April and mid July–November was within the threshold range (11–35 °C for whitefly population development) than the rest period. Whitefly developmental rate r(T) estimated based on threshold temperature (11–35 °C) and expressed as weekly temperature index [WTI = sum r(T)]was found to correspond with higher peak of whitefly population count in July–November (mean 22–35 °C) and low peak in February–May (10–32 °C) (Fig. 5B). Accumulated weekly temperature index and accumulated population count was in good agreement (r = 0.97). Therefore, accumulated weekly temperature index as an indicator of environmental suitability represented whitefly population (Fig. 5C).

It was evident that sum of developmental rate based on temperature profile reflected whitefly population abundance. Therefore, based on temperature index, whitefly population can be predicted for large areas as the geo-spatial temperature data is available.

### Prediction of spatio-temporal distribution of whitefly population abundance

Geographical distribution of monthly temperature index as an indicator of whitefly population abundance had shown distinct variation in intensity in time and space (Fig. 6). Monthly temperature index (MTI) or environmental suitability for whitefly abundance was at maximum intensity during July–November period almost throughout the country. In general, Southern peninsular region was observed to be suitable almost throughout the year as compared to other regions. It indicated major part of chili growing areas in the subcontinent found to be at high risk as possibility of vector population during this period is expected to be high. Marked increase in temperature index observed from July and continued till October. Chili planting during this season demands attention to check potential vector activity. During January–June period, Northwestern, Eastern and parts of Central India appeared to be relatively less favorable for whitefly development and likely to be less favorable for disease transmission.

Because of overlapping threshold temperatures of population abundance (11 °C, 22 °C and 35 °C) and transmission ability (measured at 15 °C, 25 °C and 35 °C) spatio-temporal distribution of vector abundance was appropriated as the measure of leaf curl risk. Therefore, spatio-temporal distribution pattern of vector abundance can be used as a reflection of probable disease risk or otherwise disease transmission ability.

Seasonal variation in leaf curl incidence in the major chili growing areas was confirmed based on preliminary survey and geographical distribution pattern for the disease reported8,9. In Northern India the chili crop grown during August-November period is affected by severe leaf curl epidemic, whereas the crop transplanted during December–January receives fewer incidences when whitefly population is comparatively low. Southern region is perpetually affected by severe epidemics as favorable temperature prevails almost throughout the year. It became evident that the period of high vector population and leaf curl incidence are overlapping as whitefly abundance and virus transmission are favored by similar temperature ranges.

## Discussion

Leaf curl disease dynamics has been simulated taking chili-ChiLCV-whitefly interaction into an epidemiological modeling frame work. Real process in tripartite interaction has been simplified based on mathematical models while maintaining the qualitative behavior of the mechanism useful in understanding the kinetics of transmission process. Immigration of viruliferous vectors and transmission parameters are the major parameters in the model noted to have a strong influence on the leaf curl epidemic. Otherwise, tripartite interaction is predominantly a vector mediation process. Current finding provides an important clue for emphasizing prevention of migrant vectors rather than the vector control normally thought for. Based on underlying principles of temperature influence on vector abundance and virus transmission spatio-temporal pattern of disease risk has been predicted. Spatio-temporal patterns for disease risk mapped across wide variants of agro-ecosystems may be used for crop planning if possible to avoid the period of high vector population. Modeling framework assumes susceptible chili cultivar and the viruliferous vectors are present in the agroecosystem. Therefore, parameter estimates are based on susceptible hosts. Adjustment of parameter values for host resistance may be required for general application of the finding. Epidemic analysis framework can be followed for testing and evolution of management strategies. A model framework of tripartite interaction necessary for evaluation of chili germplasms against leaf curl disease is now available.

For leaf curl dynamics in chili, SI model frame is good enough to explain epidemics as E (latently infected) and R (removed or post-infectious) categories are not essential elements in the model. Transmission studies have indicated inoculated plants 2-days onwards become PCR- positive. It appears virus multiplication in infected chili plants is quick and can serve as a potent source within a short period. Weekly observation is taken for estimation of infection proportion which covered enough multiplication time. Therefore, it may be the reason the model could explain dynamics without consideration of E type as latent period got covered in the observation interval. Based on the present finding it is surmised that latent period for leaf curl virus in chilli is shorter as far as transmission capability is concerned. Infected chili plants (maintained in isolation) are observed to remain infectious for more than 5 months as both acquisition and transmission of the virus is possible from those plants. Chili being a crop of 4–5-month duration, SI model serves the purpose for epidemic analysis without inclusion of R type as post-infectious category does not exist.

A high population of viruliferous vector as well as high disease incidence within a short period was an important feature in the epidemic. For such large number, both infectious host and vector are to be borne out of the field. It is possible under a situation where both the population goes increasing simultaneously as they are inter-dependent. Whitefly as persistent-circulative vector, requires virus acquisition from available infectious hosts in the field. Present finding suggests a gradual increase in both the population occurs in the field simultaneously. Basic reproduction number estimated has reflected a higher number of new infections in the next generation. Viruliferous migrant vectors make new infections or infectious hosts and new generation fly acquires virus from the newly infected host. Therefore, migrant viruliferous vector is the driving force to operate the tripartite interaction. We hypothesize a positive feedback mechanism is in work where few migratory viruliferous vectors produce few infectious hosts which in turn facilitates more viruliferous vectors and the process continues till restrictions are imposed in the agro-ecosystem. Positive feedback is a process in which the end products of action cause more of that action to occur in a feedback loop and leads to exponential growth35,36. Therefore, complete avoidance or intervention to prevent viruliferous vector population in the field is very important. As gradual built-up of both infectious host and vector happens at early stages of crop, complete prevention of migrant vectors at the start of chili planting is expected to keep the plants infection-free. Therefore, management of leaf curl is dependent on how best measures are taken to check migrant vector’s entry into the field without interference in intercultural operations. Prevention of infectious vector’s entry from the very beginning or complete avoidance of migrant vectors from the crop-start may be possible by covering the rows with synthetic plant covers that prevents insects but not interfering plant growth. Now-a-days plant covers are available. But the crop requires irrigation, fertilization and intercultural operations. Before going for management trials, there are important issues to resolve. Firstly, how long the plants are to be under complete cover as it hinders intercultural? It requires testing what is the minimum period of cover after which minimum tripartite interaction may happen but not affecting yield and quality. Secondly, what are the cares (say mulching to check weeds) at the time of transplanting to be taken so that plants are protected from vector and at the same time intercultural operations can be allowed without vector’s entry. Thirdly, the cost involvement for the alternatives to be used for vector prevention requires evaluation. Without sorting out these issues management trials are again poised to remain open without any conclusion.

Effect of vector aggregation affects transmission parameters21. Variation in transmission rate has been realized in the current study as the parameter estimates (a) with one whitefly was nine times higher than with ten flies (expected some degree of aggregation). Density of the host also affects vector mobility and therefore virus transmission17. Therefore, inclusion of aggregation parameter for both host and vector would have given a more precise simulation of the dynamics.

Both transmission and acquisition rates are affected by temperature and the unimodal response may possibly be a general phenomenon as far as vector transmissions of viruses are concerned. Similar behaviour of transmission is reported for aphid-transmitted viruses like Banana-bunchy-top in banana27 and leaf roll in potato37 where temperature thresholds (below 10 °C and above 35 °C) affected both the events. Maximum rate of transmission is obtained at 25 °C which is the normal temperature for aphid growth. For chili leaf curl, greater virus transmission ability (transmission and acquisition efficiency) at 25 °C and decreasing effect of temperature at or below 15 °C and at 35 °C or above, suggests vector population growth, as well as virus transmission, is remarkably influenced by temperature. Thermo tolerance and gene expression following heat stress in the whitefly Bemisia tabaci B and Q biotypes has shown low sensitivity in transmission for low or high temperature38,39. Temperature influence on virus transmission abilities and its approximation with vector abundance is important information that facilitated its spatio-temporal distribution. Geographical distribution of begomoviruses has been reported to be analogous to occurrence of whitefly in the world40. It is to be seen whether a similar pattern of temperature influence on the transmission is applicable for other plant viruses. If it is true, then prediction of vector abundance and transmission abilities may be used as a critical input for evolving management strategies against most virus diseases.

Seasonal whitefly population abundance has been predicted based on temperature thresholds for developmental rate. Potential vector growth as a map of potential risk of pest occurrence is an important component for management decisions18. Spatio-temporal distribution of whitefly abundance and virus transmission ability is approximated as leaf curl risk has shown good correspondence with empirical observation as well as available reports on leaf curl incidence3,12.

For the prediction of potential disease risk based on temperature thresholds, characteristic behavior of unimodal developmental response to temperature has been captured by three-parameter beta functions41. The beta function of three parameters is more robust to capture the developmental rate as parameters have meaningful biological interpretation than the two-parameter models30,33,42,43,44.

To sum up, leaf curl dynamics is established and migrant viruliferous vectors are denoted as the initiator of the epidemic process. For efficient management of the disease, emphasis must be on prevention of vector’s entry right from the stage of seedling preparation and transplanting. Temperature plays an important role, particularly in transmission process. Thus leaf curl risk prediction based on temperature may be a useful guide to evolve intervention strategies. The assessment framework may be also applicable to other viruses transmitted by whiteflies.

## Material and methods

### Population dynamical models for simulation of leaf curl epidemic

For population dynamics modeling, host population is divided into healthy (S, virus free), latent (E, carrying virus) and infectious (I, serves as virus source) and post-infectious or removed (R, no longer virus source) type. Similarly, whitefly population is divided into aviruliferous (X, virus free) and viruliferous (V, capable of transmission) type. Due to interaction, host population (S) is converted into latent category (E) once visited and probed by viruliferous whitefly (V) and after a period latent plants are transferred into infectious (I) type. Over-infectious or dead plants are called as post-infectious or removed (R) type.

For leaf curl dynamics, following population dynamics models previously used to explain tripartite interactions for plant viruses, were considered21,23,24,25,26.

$$dS/dt \, = \, {-} aVS$$
(1)
$$dI/dt = \, aVS$$
(2)
$$dX/dt \, = cX + cV \, {-} \, bXI/\left( { \, K_{x} + X} \right){-} uV \, + \, i_{x} {-} e_{x}$$
(3)
$$dV/dt \, = \, bXI/\left( { \, K_{x} + X} \right) \, + \, i_{v} {-} e_{v} {-} \, uV \, = \, b\left( {1{-} \, V} \right)I/\left( { \, K_{x} + 1{-} \, V} \right) \, + \, i_{v} {-} e_{v} {-} uV$$
(4)

For chili, host population was divided only into two categories, healthy S, and infectious I type. Chili plants once infected were observed to remain infectious till the end of the crop season. So post-infectious or removed R type was not included in the model. Similarly, vector population was categorized into aviruliferous X, and viruliferous V type. Since the vector was capable of virus transmission with 20 min inoculation feeding period, latent category was not included in the model.

In tripartite interaction, virus transmission between host and vector is dependent on parameters a (transmission or inoculation rate) and b (acquisition rate). Parameter a determines conversion or transfer rate of healthy host (S) into infectious category (I) which is dependent on the number of healthy host (S) and viruliferous vector population (V). Virus acquisition rate by an aviruliferous vector (X) is defined by parameter b, which is dependent on the number of healthy vector (X) and number of available infectious host (I). Other parameters included in the model are rate of immigration and emigration ofviruliferous (iv and ev respectively) and aviruliferous (ix and ex respectively) vectors, c and u birth and death rate respectively (both aviruliferous and viruliferous), Michaelis–Menten constant (Kx).

Estimated parameters are tuned by fitting models on field data where infections were observed under natural conditions. Setting the model in a state of real situation (calibrated), behaviors of other parameters were assessed. Parameter sensitivity was performed on the equilibrium points of population of host and vector to see the impact of changes in their values make changes in host (S* and I*) and vector (V*) population.

Equilibrium points:

$$S* = 1 - I*$$
(5)
$$I* = \, \left( {u \, e_{v} + u^{2} V - u \, i_{v} } \right)/\left( {b\left( { \, i_{v} - \, e_{v} + \, i_{x} - \, e_{x} - uV} \right)} \right)$$
(6)
$$X* = \, \left( {e_{x} - i_{x} + V\left( {u - c} \right)} \right)/\left( {c - bI} \right)$$
(7)
$$V* = \, \left( {u \, e_{v} - u \, i_{v} - I\left( { \, i_{x} - \, e_{x} } \right) + I\left( { \, i_{v} - \, e_{v} } \right)} \right)/\left( {uI - u^{2} } \right)$$
(8)

The equilibrium points for host and vector population (S*, I* and V*) were derived from simpler equations of tripartite interaction to avoid quadratic roots in system solution.

Temperature influence on major transmission parameters was assessed for prediction of probable geographical distribution of the disease to explain epidemic process and predict disease risk.

### Parameter estimation-transmission (a) and acquisition (b) rates

Transmission or inoculation rate for the ChiLCV was estimated under semi-controlled conditions26. Ten viruliferous whiteflies were allowed to feed on eight chili plants (susceptible cv HPH-1041, Syngenta) grown in earthen pot (dai 10ʺ) and maintained in an insect proof cage. Experiment was performed under three levels of temperature exposure viz., 15 °C, 25 °C and 35 °C. Whiteflies were killed after 7 days by spraying insecticide (@1 ml/L Confidor, Bayer). Leaves from each plant were collected for virus detection through PCR (ChiLCV specific primer pairs) after 1, 2, 3, 4, 5, and 7 days’ post inoculation. Transmission rate (a) was estimated by fitting the dynamical model Eq. (1).

For estimation of acquisition rate (b), ten aviruliferous whiteflies were released on an infectious plant maintained in an insect proof cage. The experiment was performed at three temperatures viz., 15 °C, 25 °C and 35 °C to assess temperature influence on virus acquisition. Vectors were collected at 0, 10, 20 min, 1, 2, 4, 8, 12, 24, and 48 h interval for PCR detection (ChilCV specific primers). Acquisition rate, b was estimated fitting the dynamical model:

$$dV/dt \, = bXI = \, b\left( {1 - V} \right)I$$

X = number of aviruliferous vectors in time; V = number of virulifeorus vector in time; I = number of infectious plants in the exposure in time; b = acquisition rate, X encounters with I.

For fitting dynamic models, R software was used. Precision of the model was tested by comparing simulated values of healthy host and viruliferous vector population with observed values estimated through sampling in the field. The RMSE was used as a measure of the discrepancies between observed values and model simulations.

$$RMSE = \sqrt {\frac{{\sum\nolimits_{i = 1}^{n} {(x_{sim,i} - x_{obs,i} )^{2} } }}{n}}$$

where xsim,I is the simulated value for healthy host and viruliferous vector, xobs,I is the observed value for corresponding populations, and n is the number of observation, with a value of 12 in the present evaluation.

### Assessment of infectious host (I) and whitefly vector (V) populations in experimental field

In a field plot experiment at IARI New Delhi (18.6317°N, 77.2241°8E, 219.7 m), 500 seedlings of susceptible chili (cv HPH-1041) were transplanted in the second week of August 2019 and grown under normal package of practice without any management interventions.

For estimation of host (infectious) and vector population (viruliferous) samples of leaves and flies were taken from the first week of post transplanting. Weekly composite of 200 leaves, taking one leaf from the top of 200 randomly selected plants was prepared. Similarly, a composite of 100 whiteflies collected (using a suction device) from five randomly selected plants. Both leaf and whitefly samples were grouped for testing of ChiLCV through PCR. Group testing was done following the scheme mentioned in the Table S1. Uniform PCR protocol (ChiLCV specific primers) was followed for leaf and whitefly samples45 (Table S1). Proportion of infected leaf or whitefly (P) was estimated using the following formula46,47,48,49:

$$P = 1 - \left( {\left( {n - X} \right)/n} \right)^{1/m}$$

where n = number of groups made for leaf and whitefly samples; m = number of leaves or whiteflies in each group; and X = number of groups tested as PCR-positive.

At transplanting, leaf, as well as whitefly samples from the resident crops (cucurbits, brinjal and tomato) and border tree (Leucana leucocephala), were also collected for testing in similar methods of sampling.

Weekly proportion of infected leaves and flies based on PCR test was considered as the proportion of infectious host (I) and viruliferous vector (V) population in the field. Proportion of infected host and vectors estimated were considered as I and V as once they are infected remain infectious.

### Basic reproduction number (R0) for leaf curl in chili

Basic reproduction number calculated as the total number of infections arising from one newly infected host and whitefly introduced into the healthy host population50. It represents the ratio between subsequent generations at population level. The R0 was calculated as dominant eigen value of the next generation matrix (A) where the time step of the matrix is the generation time51:

$$A=\left(\begin{array}{cc}0& aS\\ bX& 0\end{array}\right)\left(\begin{array}{cc}\tau I& 0\\ 0& \tau Y\end{array}\right)$$

where a = transmission rate, b = acquisition rate, $$\tau I$$= time period an infected host remains infectious, $$\tau Y$$= time period the vector remains viruliferous, S and X = healthy host and non-viruliferous vector population density (number/m2), respectively. R0value calculated from the equation50,52:

$$R_{0}^{2} - \alpha S\tau Y \cdot bX\tau I = 0$$

### Prediction of whitefly population abundance based on temperature influence

Whitefly population was observed during 2016–2019 at IARI New Delhi experimental field where regular planting of chili was followed. Flies from five plants were collected through a suction device at the weekly intervals (Julian week) and the mean count was expressed as a normalized population per week53.

$${\text{Normalized}}\;{\text{population}}\;{\text{count }} = \left( {Mean \, count - Min_{mean \, count} } \right)/ \, (Max_{mean \, count} - \, Min_{mean \, count} )$$

To assess the effect of temperature on developmental rate [r(T)] for whitefly population, non-linear beta function which utilizes cardinal temperatures of whitefly growth and development was used41,54:

$$\begin{gathered} {\text{r}}(T) = \left[ {\left( {T_{{upper}} - T_{{air {\text{-}} h}} } \right)/\left( {T_{{upper}} - T_{{opt}} } \right)} \right]*\left[ {{{\left( {T_{{air {\text{-}} h}} - T_{{lower}} } \right)} \mathord{\left/ {\vphantom {{\left( {T_{{air {\text{-}} h}} - T_{{lower}} } \right)} {\left( {T_{{opt}} - T_{{lower}} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {T_{{opt}} - T_{{lower}} } \right)}}} \right]\wedge {{\left( {T_{{opt}} - T_{{lower}} } \right)} \mathord{\left/ {\vphantom {{\left( {T_{{opt}} - T_{{lower}} } \right)} {\left( {T_{{upper}} - T_{{opt}} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {T_{{upper}} - T_{{opt}} } \right)}} \hfill \\ {\text{if}}\;\;T_{{lower}} \le T \le T_{{upper}} \;\;{\text{and}}\;\;0\;\;{\text{otherwise}} \hfill \\ \end{gathered}$$

where, Tupper (35 °C), Tlower (11 °C) and Topt (22 °C) are the upper, lower and optimum threshold temperatures, respectively, for egg laying and adult development4,31,55,56. Tair-h is the hourly air temperature (°C).

Daily temperature for the year 2016–2019 was collected from the Institute’s Meteorological observatory IARI New Delhi (28.7041° N, 77.1025° E). Daily minimum (Tmin) and maximum (Tmax) temperatures were converted into hourly temperatures (Tair-h) based on the standard formula57, which provides smooth transformation from minimum to maximum air daily temperature:

$$T_{air - h} = \left( {T_{max} + T_{min} } \right)/2 + \left( {T_{max} - T_{min} } \right)/2*cos(0.2618*(h - TimeVar)$$

where TimeVar is the hour of the day corresponding to the time of occurrence of Tmax.

For estimation of temperature influence, hourly temperature (Tair-h) was corrected for upper and lower threshold limits for whitefly growth and development. Hourly estimates of r (T)were calculated and daily r(T) were obtained summing hourly values for the day. Weekly temperature index (WTI) was calculated by summing daily r(T) values for the week:

$$WTI={\sum }_{t1}^{t2}r\left(T\right)$$

where t1 and t2 are the r (T) for the first and last day of the week.

Accumulated temperature index (WTI) for the crop period (August–November) was related to the accumulated population of whitefly count as a measure of environmental suitability for population abundance.

### Prediction of spatio-temporal distribution for whitefly population abundance

For prediction of whitefly population abundance based on environmental suitability, daily minimum and maximum temperature were downloaded from the IMD portal (http://mausam.imd.gov.in). Temperature data for the period 2001–2018 from 85 geo-referenced meteorological stations of important chili growing states of India was considered. For probable distribution of vector population, monthly cumulative temperature index (MTI) was estimated.

$$MTI={\sum }_{t1}^{t2}r\left(T\right)$$

where t1 and t2 are the r (T) for the first and last day of the month.

The cumulative temperature indices (MTI) were plotted using ArcGis10.0 (http://www.arcgis.com). For continuous surface, IDW (inverse distance weightage) interpolation technique was applied and spatial maps were generated. For verification of predicted distribution, a preliminary survey across major chili growing areas conducted for leaf curl incidence.

## References

1. 1.

Chattopadhyay, B. et al. Infectivity of the cloned components of a begomovirus: DNA beta complex causing chilli leaf curl disease in India. Adv. Virol. 153, 533–539 (2008).

2. 2.

Khan, M. S., Raj, S. K. & Singh, R. First report of tomato leaf curl New Delhi virus infecting chilli in India. Plant. Pathol. 55, 289 (2006).

3. 3.

Senanayake, D. M. J. B., Mandal, B., Lodha, S. & Varma, A. First report of Chilli leaf curl virus affecting chilli in India. Plant. Pathol. 56, 343 (2007).

4. 4.

Ellango, R. et al. Distribution of bemisiatabaci genetic groups in India. Environ. Entomol. 44, 1258–1264 (2015).

5. 5.

Mishra, M. D., Raychaudhuri, S. P. & Jha, A. Virus causing leaf curl of chilli (Capsicum annum L.). Indian J. Microbiol. 3, 73–76 (1963).

6. 6.

Navas-Castillo, J., Fiallo-Olive, E. & Sanchez-Campos, S. Emerging virus diseasestransmitted by whiteflies. Annu. Rev. Phytopathol. 49, 219–248 (2011).

7. 7.

Ram, K., Singh, B. P. & Parihar, S. B. S. Population dynamics of whitefly (Genn.) on potato crop in relation to weather factors. Proc. Natl Acad. Sci. India 75, 43–46 (2005).

8. 8.

Chaubey, A. N. & Mishra, R. S. Survey of chilli leaf curl complex disease in eastern part of Uttar Pradesh. Biomed. J. Sci. Tech. Res. https://doi.org/10.26717/BJSTR.2017.01.000589 (2017).

9. 9.

Kumar, R., Kumar, V., Kadiri, S. & Palicherla, S. R. Epidemiology and diagnosis of chilli leaf curl virus in central India, a major chilli growing region. Indian Phytopathol. 69(4s), 61–64 (2016).

10. 10.

Uday, B. O. & Jayanta, T. Occurrence and distribution of chilli leaf curl complex disease in West Bengal. Biomed. J. Sci. Tech. Res. https://doi.org/10.26717/BJSTR.2018.03.000948 (2018).

11. 11.

Nigam, K., Suhail, S., Verma, Y., Singh, V. & Gupta, S. Molecular characterization of begomovirus associated with leaf curl disease in chilli. World J. Pharm. Res. 4, 1579–1592 (2015).

12. 12.

Thakur, H., Jindal, S. K., Sharma, A. & Dhaliwal, M. S. Chilli leaf curl virus disease: A serious threat for chilli cultivation. J. Plant Dis. Prot. 125, 239–249 (2018).

13. 13.

Cohen, S. & Antignus, Y. Tomato yellow leaf curl virus, a whitefly-borne geminivirus of tomatoes. Adv. Dis. Vector Res. 10, 259–288 (1994).

14. 14.

Cohen, S. Epidemiology of whitefly-transmitted viruses. In Whiteflies, their Bionomics, Pest Status and Management (ed. Gerling, D.) 211–225 (Intercept, Andover, 1990).

15. 15.

Jeger, M. J., Bosch, F. V. D., Madden, L. V. & Holt, J. K. A model for analysing plant-virus transmission characteristics and epidemic development. IMA J. Math. Appl. Med. Biol. 15, 1–18 (1998).

16. 16.

Jeger, M. J., Madden, L. V. & Bosch, F. V. D. Plant virus epidemiology: Applications and prospects of mathematical modeling and analysis to improve and disease control. Plant Dis. 102, 837–854 (2018).

17. 17.

Zhang, X. S., Holt, J. J. & Colvin, J. A general model of plant-virus disease infection incorporating vector aggregation. Plant. Pathol. 49, 435–444 (2000).

18. 18.

Lapidot, M. & Polston, J. E. Biology and epidemiology of Bemisia-vectored viruses. In Bemisia: Bionomics and Management of a Global Pest (eds Stansly, P. A. & Naranjo, S. E.) 227–231 (Springer, Berlin, 2010).

19. 19.

Anderson, P. K. et al. Emerging infectious diseases of plants: Pathogen pollution, climate change and agrotechnology drivers. Trends Ecol. Evol. 19, 535–544 (2004).

20. 20.

Jones, R. A. C. Plant virus ecology and epidemiology: Historical perspectives, recent progress and future prospects. Ann. Appl. Biol. 164, 320–347 (2014).

21. 21.

Chan, M. & Jeger, M. J. An analytical model of plant virus disease dynamics with roguing and replanting. J. Appl. Ecol. 3, 413–427 (1994).

22. 22.

Holt, J. & Chancellor, T. C. B. Simulation modelling of the spread of rice tungro virus disease: The potential for management by roguing. J. Appl. Ecol. 33, 927–936 (1996).

23. 23.

Holt, J. K., Jeger, M. J., Thresh, J. M. & Otim-Nape, G. W. An Epidemiological model incorporating vector population dynamics applied to African cassava mosaic virus disease. J. Appl. Ecol. 34, 793 (1997).

24. 24.

Kinene, T., Luboobi, L., Nannyonga, B. & Mwanga, G. D. A mathematical model for the dynamics and cost effectiveness of the current controls of cassava brown streak disease in Uganda. J. Math. Comput. Sci. 5, 567–600 (2015).

25. 25.

Holt, J., Colvin, J. & Muniyappa, V. Identifying control strategies for tomato leaf curl virus disease using an epidemiological model. J. Appl. Ecol. 36, 625–633 (1999).

26. 26.

Nakasuji, R., Miyai, S., Kawamoto, H. & Kiritani, K. Mathematical epidemiology of rice dwarf virus transmitted by green rice leafhoppers: A differential equation model. J. Appl. Ecol. 22, 839–847 (1985).

27. 27.

Anhalt, M. D. & Almeida, R. P. Effect of temperature, vector life stage, and plant access period on transmission of banana bunchy top virus to banana. Phytopathology 98, 743–748 (2008).

28. 28.

Chellappan, P., Vanitharani, R. & Fauquet, C. M. MicroRNA-binding viral protein interferes with Arabidopsis development. Proc. Natl. Acad. Sci. USA 102, 10381–10386 (2005).

29. 29.

Kassanis, B. Therapy of virus infected plants. J. R. Agric. Soc. Engl. 126, 105–114 (1965).

30. 30.

Bonato, O., Amandine, L., Claire, V. & Jacques, F. Modeling temperature-dependent bionomics of Bemisiatabaci (Q-biobiotype). Physiol. Entomol. 32, 50–55 (2007).

31. 31.

Butler, G. D. J., Henneberry, T. J. & Clayton, T. E. Bemisiatabaci (Homoptera; Aleyrodidae): Development, oviposition, and longevity in relation to temperature. Ann. Entomol. Soc. Am. 76, 310–313 (1983).

32. 32.

Gilioli, G., Pasquali, S., Parisic, S. & Winterd, S. Modelling the potential distribution of Bemisiatabaci in Europe in light of the climate change scenario. Pest Manag. Sci. 70, 1611–1623 (2014).

33. 33.

Han, E. J., Choi, B. R. & Lee, J. H. Temperature-dependent development models of Bemisiatabaci (Gennadius) (Hemiptera: Aleyrodidae) Q biotype on three host plants. J. Asia-Pac. Entomol. 16, 5–10 (2013).

34. 34.

Jones, R. A. C. Trends in plant virus epidemiology: Opportunities from new or improved technologies. Virus Res. 186, 3–19 (2014).

35. 35.

Crespi, B. J. Vicious circles: Positive feedback in major evolutionary and ecological transitions. Trends Ecol. Evol. 19, 627–633 (2004).

36. 36.

Thomas, R., Thieffry, D. & Kaufman, M. Dynamical behavior of biological regulatory networks. 1. Biological role of feedback loops and practical use of the concept of the loop-characteristic state. Bull. Math. Biol. 57, 247–276 (1995).

37. 37.

Singh, N. P., McCoy, M. T., Tice, R. R. & Schneider, E. L. A simple technique for quantitation of low levels of DNA damage in individual cells. Exp. Cell Res. 175, 184–191 (1988).

38. 38.

Mahadav, A., Kontsedalov, S., Czosnek, H. & Ghanim, M. Thermotolerance and gene expression following heat stress in the whitefly Bemisiatabaci B and Q biotypes. Insect Biochem. Mol. Biol. 39, 668–676 (2009).

39. 39.

Pusag, J. C., Hemayet, J. S. M., Lee, K. S., Lee, S. & Lee, K. Y. Upregulation of temperature susceptibility in Bemisiatabaci upon acquisition of Tomato yellow leaf curl virus (TYLCV). J. Insect Physiol. 58, 1343–1348 (2012).

40. 40.

Brown, J. K., Frolich, D. R. & Rosell, R. C. The sweet potato or silver leaf whiteflies: Biotypes of B. tabaci or a species complex?. Ann. Rev. Entomol. 40, 511–534 (1995).

41. 41.

Yan, W. & Hunt, L. A. An equation for modelling the temperature response of plants using only the cardinal temperatures. Ann. Bot. 84, 607–614 (1999).

42. 42.

Briere, J. F., Pracros, P., Le Roux, A. Y. & Pierre, J. S. A novel rate model of temperature-dependent development for arthropods. Environ. Entomol. 28, 22–29 (1999).

43. 43.

Shi, P., Feng, G., Sun, Y. & Chen, C. A simple model for describing the effect of temperature on insect developmental rate. J. Asia-Pac. Entomol. 16, 5–10 (2011).

44. 44.

Tsai, J. H. & Wang, K. H. Development and reproduction of Bemisiaa rgentifolii (Homoptera: Aleyrodidae) on five host plants. Environ. Entomol. 25, 810–816 (1996).

45. 45.

Mandal B. Epidemics, transmission, characterisation and diagnosis of begomoviruses inchilli leaf curl in India. In, Asian Solanaceous Round Table- 2017 (ASRT-2): ‘Challenges and Future Trends in R&D for Solanaceous Crops in Asia’. February 23–25, 2017. Venue: Ruangkaw Room, Vajiranusorn Building, the Faculty of Agriculture, Kasetsart University, Bangkok, Thailand. https://apsaseed.org/wp-content/uploads/2017/02/5.-Epidemics-Transmission-Characterisation-and-Diagnosis-of-begomoviruses-in-chilli-leaf-curl-in-India-Bikash-Mandal.pdf (2017).

46. 46.

Hepworth, G. Exact confidence intervals for proportions estimated by group testing. Biometrics 52, 1134–1146 (1996).

47. 47.

Hughes, G. & Gottwald, T. R. Survey methods for assessment of citrus tristeza virus incidence. Phytopathology 88, 715–723 (1998).

48. 48.

Moran, J. R., Wilson, J. M., Garrett, R. G. & Smith, P. R. ELISA indexing of commercial carnations for carnation mottle virus using a urease-antibody conjugate. Plant. Pathol. 34, 467–471 (1985).

49. 49.

Walter, S. D., Stephen, W. H. & Barry, J. B. Estimation of infection rates in populations of organisms using pools of variable size. Am. J. Epidemiol. 112, 124–128 (1980).

50. 50.

Van den, B. F. & Jeger, M. J. The basic reproduction number of vector-borne plant virus epidemics. Virus Res. 241, 196–202 (2017).

51. 51.

Heesterbeek, J. A. P. A brief history of R0 and a recipe for its calculation. Acta Biotheor. 50, 189–204 (2002).

52. 52.

van den Bosch, F., McRoberts, N., van den Berg, F. & Madden, L. V. The basic reproduction number of plant pathogens: Matrix approaches to complex dynamics. Phytopathology 98, 239–249 (2008).

53. 53.

Youn, E. & Jeong, M. K. Class dependent feature scaling method using naïve Bayes classifier for text data mining. Pattern Recogn. Lett. 30, 477–485 (2009).

54. 54.

Magarey, R. D., Sutton, T. B. & Thayer, C. L. A simple generic infection model for foliar fungal plant pathogens. Phytopathology 95, 92–100 (2005).

55. 55.

Horowitz, M., Shimoni, Y., Parnes, S., Gotsman, M. S. & Hasin, Y. Heat acclimation: Cardiac performance of isolated rat heart. J. Appl. Physiol. 60, 9 (1986).

56. 56.

Wang, K. & Tsai, J. H. Temperature effect on development and reproduction of silverleaf whitefly (Homoptera: Aleyrodidae). Ann. Entomol. Soc. Am. 89, 375–384 (1996).

57. 57.

Bregaglio, S., Donatelli, M., Confalonieri, R., Acutis, M. & Orlandini, S. An integrated evaluation of thirteen modelling solutions for the generation of hourly values of air relative humidity. Theor. Appl. Climatol. 102, 429–438 (2010).

## Acknowledgements

Authors are thankful to the Head, Division of Plant Pathology, Joint Director (R) and Director IARI New Delhi for providing support and encouragements in the study. Authors express indebted to DBT for financial support and encouragements provided for the work.

## Funding

Funding was provded by Department of Biotechnology, Ministry of Science and Technology, India BT/PR28199/AGIII/103/990/2018.

## Author information

Authors

### Contributions

P.S., B.M. and A.G. conceived the idea and designed the experiments. B.R. and S.D. carried out the experiment. B.R., S.M.S., and P.S. interpreted the results and P.S. analysed the data. P.S. and B.R. wrote the manuscript. B.M. and A.G. reviewed the manuscript.

### Corresponding author

Correspondence to Parimal Sinha.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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Roy, B., Dubey, S., Ghosh, A. et al. Simulation of leaf curl disease dynamics in chili for strategic management options. Sci Rep 11, 1010 (2021). https://doi.org/10.1038/s41598-020-79937-0

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# Inspired by Huan Le Quang

Find the number of positive integers $$n$$ such that the following is also an integer:

$\frac{ n^2 + 11n + 2 } { n-5} .$

Inspiration

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