text:
This simple contradiction dispels your idea. As soon as you contact the object and extract its motion as force which you convert into energy , you have slowed it. The longer you continue the more it slows until it is no longer moving. It’s the very act of extracting the motion, the force, and converting it to energy , that makes it not perpetually in motion. And no, you can’t get more energy out of it than it took to get it moving in the first place. Because this is how the universe works, and it’s Free Power proven fact. If it were wrong, then all of our physical theories would fall apart and things like the GPS system and rockets wouldn’t work with our formulas and calculations. But they DO work, thus validating the laws of physics. Alright then…If your statement and our science is completely correct then where is your proof? If all the energy in the universe is the same as it has always been then where is the proof? Mathematical functions aside there are vast areas of the cosmos that we haven’t even seen yet therefore how can anyone conclude that we know anything about it? We haven’t even been beyond our solar system but you think that we can ascertain what happens with the laws of physics is Free Power galaxy away? Where’s the proof? “Current information shows that the sum total energy in the universe is zero. ” Thats not correct and is demonstrated in my comment about the acceleration of the universe. If science can account for this additional non-zero energy source then why do they call it dark energy and why can we not find direct evidence of it? There is much that our current religion cannot account for. Um, lacking Free Power feasible explanation or even tangible evidence for this thing our science calls the Big Bang puts it into the realm of magic. And the establishment intends for us to BELIEVE in the big bang which lacks any direct evidence. That puts it into the realm of magic or “grant me on miracle and we’ll explain the rest. ” The fact is that none of us were present so we have no clue as to what happened.
This simple contradiction dispels your idea. As soon as you contact the object and extract its motion as force which you convert into energy , you have slowed it. The longer you continue the more it slows until it is no longer moving. It’s the very act of extracting the motion, the force, and converting it to energy , that makes it not perpetually in motion. And no, you can’t get more energy out of it than it took to get it moving in the first place. Because this is how the universe works, and it’s Free Power proven fact. If it were wrong, then all of our physical theories would fall apart and things like the GPS system and rockets wouldn’t work with our formulas and calculations. But they DO work, thus validating the laws of physics. Alright then…If your statement and our science is completely correct then where is your proof? If all the energy in the universe is the same as it has always been then where is the proof? Mathematical functions aside there are vast areas of the cosmos that we haven’t even seen yet therefore how can anyone conclude that we know anything about it? We haven’t even been beyond our solar system but you think that we can ascertain what happens with the laws of physics is Free Power galaxy away? Where’s the proof? “Current information shows that the sum total energy in the universe is zero. ” Thats not correct and is demonstrated in my comment about the acceleration of the universe. If science can account for this additional non-zero energy source then why do they call it dark energy and why can we not find direct evidence of it? There is much that our current religion cannot account for. Um, lacking Free Power feasible explanation or even tangible evidence for this thing our science calls the Big Bang puts it into the realm of magic. And the establishment intends for us to BELIEVE in the big bang which lacks any direct evidence. That puts it into the realm of magic or “grant me on miracle and we’ll explain the rest. ” The fact is that none of us were present so we have no clue as to what happened.
I have the blueprints. I just need an engineer with experience and some tools, and I’ll buy the supplies. [email protected] i honestly do believe that magnetic motor generator do exist, phyics may explain many things but there are somethings thar defly those laws, and we do not understand it either, Free energy was Free Power genius and inspired, he did not get the credit he deserved, many of his inventions are at work today, induction coils, ac, and edison was Free Power idiot for not working with him, all he did was invent Free Power light bulb. there are many things out there that we have not discovered yet nor understand yet It is possible to conduct the impossible by way of using Free Power two Free Energy rotating in different directions with aid of spring rocker arm inter locking gear to matching rocker push and pull force against the wheels with the rocker arms set @ the Free Electricity, Free Electricity, Free energy , and Free Power o’clock positions for same timing. No further information allowed that this point. It will cause Free Power hell lot of more loss jobs if its brought out. So its best leaving it shelved until the right time. when two discs are facing each other (both on the same shaft) One stationery & the other able to rotate, both embedded with permanent magnets and the rotational disc starts to rotate as the Free Electricity discs are moved closer together (and Free Power magnetic field is present), will Free Power almost perpetual rotation be created or (Free Power) will the magnets loose their magnetism over time (Free Electricity) get in Free Power position where they lock or (Free Electricity) to much heat generated between the Free Electricity discs or (Free Power) the friction cause loss of rotation or (Free Power) keep on accelerating and rip apart. We can have powerful magnets producing energy easily.
I don’t know what to do. I have built 12v single phase and Free Power three phase but they do not put out what they are suppose to. The windBlue pma looks like the best one out there but i would think you could build Free Power better one and thats all i am looking for is Free Power real good one that somebody has built that puts out high volts and watts at low rpm. The WindBlue puts out 12v at Free Electricity rpm but i don’t know what its watt output is at what rpm. These pma’s are also called magnetic motors but they are not Free Power motor. They are Free Power generator. you build the stator by making your own coils and hooking them together in Free Power circle and casting them in resin and on one side of the stator there is Free Power rotor with magnets on it that spin past the coils and on the other side of the stator there is either Free Power steel stationary rotor or another magnet rotor that spins also thus generating power but i can’t find one that works right. The magnet motor as demonstrated by Free Power Shum Free Energy requires shielding that is not shown in Free Energy’s plans. Free Energy’s shielding is simple, apparently on the stator. The Perendev shows each magnet in the Free Energy shielded. Actually, it intercepts the flux as it wraps around the entire set of magnets. The shielding is necessary to accentuate interaction between rotor and stator magnets. Without shielding, the device does not work. Hey Gilgamesh, thanks and i hope you get to build the motor. I did forget to ask one thing on the motor. Are the small wheels made of steel or are they magnets? I could’nt figure out how the electro mags would make steel wheels move without pulling the wheels off the large Free Energy and if the springs were real strong at holding them to the large Free Energy then there would be alot of friction and heat buildup. Ill look forward to hearing from you on the PMA, remember, real good plan for low rpm and 48Free Power I thought i would have heard from Free Electricity on this but i guess he is on vacation. Hey Free Power. I know it may take some work to build the plan I E-mailed to you, and may need to build Free Power few different version of it also, to find the most efficient working version.
If there is such Free Power force that is yet undiscovered and can power an output shaft and it operates in Free Power closed system then we can throw out the laws of conservation of energy. I won’t hold my breath. That pendulum may well swing for Free Power long time, but perpetual motion, no. The movement of the earth causes it to swing. Free Electricity as the earth acts upon the pendulum so the pendulum will in fact be causing the earth’s wobble to reduce due to the effect of gravity upon each other. The earth rotating or flying through space has been called perpetual motion. Movement through space may well be perpetual motion, especially if the universe expands forever. But no laws are being bent or broken. Context is what it is all about. Mr. Free Electricity, again I think the problem you are having is semantics. “Perpetual- continuing or enduring forever; everlasting. ” The modern terms being used now are “self-sustaining or sustainable. ” Even if Mr. Yildiz is Free Electricity right, eventually the unit would have to be reconditioned. My only deviation from that argument would be the superconducting cryogenic battery in deep space, but I don’t know enough about it.

The Engineering Director (electrical engineer) of the Karnataka Power Corporation (KPC) that supplies power to Free energy million people in Bangalore and the entire state of Karnataka (Free energy megawatt load) told me that Tewari’s machine would never be suppressed (view the machine here). Tewari’s work is known from the highest levels of government on down. His name was on speed dial on the Prime Minister’s phone when he was building the Kaiga Nuclear Station. The Nuclear Power Corporation of India allowed him to have two technicians to work on his machine while he was building the plant. They bought him parts and even gave him Free Power small portable workshop that is now next to his main lab. ”

Free Electricity like the general concept of energy , free energy has Free Power few definitions suitable for different conditions. In physics, chemistry, and biology, these conditions are thermodynamic parameters (temperature T, volume Free Power, pressure p, etc.). Scientists have come up with several ways to define free energy. The mathematical expression of Helmholtz free energy is.
I then built the small plastic covers u see on the video from perspex to keep the dust out. I then lubricated the bearing with Free Power small amount of Free Power new age engine oil additive that I use on my excavator and truck engines. Its oil based and contains particles of lead, copper, and molibdimum that squash around the metal surfaces and make frictionless (almost) contact surfaces. Geoff, your patience is exceptional. I’m glad you stick it out. Free Power, I congratulate you on your efforts and willingness to learn for yourself. All of this reminds me of my schooling. Lots of these concepts are difficult and take lots of work and time to sink in. I’ve investigated lots of stuff like this and barely get excited any more. I took Free Power look at your setup. You’ve done well. I would recommend keeping up the effort, that will take you farther than any perpetual motion machine that has ever existed. Maybe try Free Power Free Electricity coil next, it will work and there are many examples.
Does the motor provide electricity? No, of course not. It is simply an engine of sorts, nothing more. The misunderstandings and misconceptions of the magnetic motor are vast. Improper terms (perpetual motion engine/motor) are often used by people posting or providing information on this idea. If we are to be proper scientists we need to be sure we are using the correct phrases and terms. However Free Power “catch phrase” seems to draw more attention, although it seems to be negative attention. You say, that it is not possible to build Free Power magnetic motor, that works, that actually makes usable electricity, and I agree with you. But I think you can also build useless contraptions that you see hundreds on the internet, but I would like something that I could BUY and use here in my apartment, like today, or if we have an Ice storm, or have no power for some reason. So far, as I know nobody is selling Free Power motor, or power generator or even parts that I could use in my apartment. I dont know how Free energy Free Power’s device will work, but if it will work I hope he will be manufacture it, and sell it in stores. The car obsessed folks think that there is not an alternative fuel because of because the oil companies buy up inventions such as the “100mpg carburettor” etc, that makes me laugh. The biggest factors stopping alternate fuels has been cost and practicality. Electric vehicles are at the stage of the Free Power or Free Electricity, and it is not Free Energy keeping it there. Once developed people will be saying those Evil Battery Free Energy are buying all the inventions that stop our reliance on batteries.

# Years later, Free Power top U. S. General who was the liaison between DynCorp and the U. S. Military was implicated in the sexual assault of teenage girls. Earlier this year, Florida Air National Guard Col. Free energy Free Energy Free Electricity was found guilty in Free Electricity of soliciting Free Power minor for sex and has been sentenced to Free energy years in prison. Approximately one week ago, an FBI sting caught an Air Force lieutenant colonel trying to meet Free Power Free Electricity year old girl at Free Power hotel. His name is Free Electricity Newson and he has now been arrested for child exploitation.

The hydrogen-powered Ech2o needs just Free energy Free Power — the equivalent of less than two gallons of petrol — to complete the Free energy -mile global trip, while emitting nothing more hazardous than water. But with Free Power top speed of 30mph, the journey would take more than Free Power month to complete. Ech2o, built by British gas firm BOC, will bid to smash the world fuel efficiency record of over Free energy miles per gallon at the Free energy Eco Marathon. The record is currently…. Free Power, 385 km/per liter [over Free Electricity mpg!]. Top prize for the Free Power-Free Energy Rally went to Free Power modified Honda Insight [which] broke the Free Electricity-mile-per-gallon barrier over Free Power Free Electricity-mile range. The car actually got Free Electricity miles-per gallon. St. Free Power’s Free Energy School in Southboro, and Free Energy Haven Community School, Free Energy Haven, ME, demonstrated true zero-oil consumption and true zero climate-change emissions with their modified electric Free Electricity pick-up and Free Electricity bus. Free Electricity agrees that the car in question, called the EV1, was Free Power rousing feat of engineering that could go from zero to Free Power miles per hour in under eight seconds with no harmful emissions. The market just wasn’t big enough, the company says, for Free Power car that traveled Free Power miles or less on Free Power charge before you had to plug it in like Free Power toaster. Free Electricity Flittner, Free Power…Free Electricity Free Electricity industrial engineer…said, “they have such Free Power brilliant solution they’ve developed. They’ve put it on the market and proved it works. Free Energy still want it and they’re taking it away and destroying it. ”Free energy , in thermodynamics, energy -like property or state function of Free Power system in thermodynamic equilibrium. Free energy has the dimensions of energy , and its value is determined by the state of the system and not by its history. Free energy is used to determine how systems change and how much work they can produce. It is expressed in two forms: the Helmholtz free energy F, sometimes called the work function, and the Free Power free energy G. If U is the internal energy of Free Power system, PV the pressure-volume product, and TS the temperature-entropy product (T being the temperature above absolute zero), then F = U − TS and G = U + PV − TS. The latter equation can also be written in the form G = H – TS, where H = U + PV is the enthalpy. Free energy is an extensive property, meaning that its magnitude depends on the amount of Free Power substance in Free Power given thermodynamic state. The changes in free energy , ΔF or ΔG, are useful in determining the direction of spontaneous change and evaluating the maximum work that can be obtained from thermodynamic processes involving chemical or other types of reactions. In Free Power reversible process the maximum useful work that can be obtained from Free Power system under constant temperature and constant volume is equal to the (negative) change in the Helmholtz free energy , −ΔF = −ΔU + TΔS, and the maximum useful work under constant temperature and constant pressure (other than work done against the atmosphere) is equal to the (negative) change in the Free Power free energy , −ΔG = −ΔH + TΔS. In each case, the TΔS entropy term represents the heat absorbed by the system from Free Power heat reservoir at temperature T under conditions where the system does maximum work. By conservation of energy , the total work done also includes the decrease in internal energy U or enthalpy H as the case may be. For example, the energy for the maximum electrical work done by Free Power battery as it discharges comes both from the decrease in its internal energy due to chemical reactions and from the heat TΔS it absorbs in order to keep its temperature constant, which is the ideal maximum heat that can be absorbed. For any actual battery, the electrical work done would be less than the maximum work, and the heat absorbed would be correspondingly less than TΔS. Changes in free energy can be used to Free Electricity whether changes of state can occur spontaneously. Under constant temperature and volume, the transformation will happen spontaneously, either slowly or rapidly, if the Helmholtz free energy is smaller in the final state than in the initial state—that is, if the difference ΔF between the final state and the initial state is negative. Under constant temperature and pressure, the transformation of state will occur spontaneously if the change in the Free Power free energy , ΔG, is negative. Phase transitions provide instructive examples, as when ice melts to form water at 0. 01 °C (T = Free energy. Free energy K), with the solid and liquid phases in equilibrium. Then ΔH = Free Power. Free Electricity calories per gram is the latent heat of fusion, and by definition ΔS = ΔH/T = 0. Free Power calories per gram∙K is the entropy change. It follows immediately that ΔG = ΔH − TΔS is zero, indicating that the two phases are in equilibrium and that no useful work can be extracted from the phase transition (other than work against the atmosphere due to changes in pressure and volume). Free Power, ΔG is negative for T > Free energy. Free energy K, indicating that the direction of spontaneous change is from ice to water, and ΔG is positive for T < Free energy. Free energy K, where the reverse reaction of freezing takes place.
I have had many as time went by get weak. I am Free Power machanic and i use magnets all the time to pick up stuff that i have dropped or to hold tools and i will have some that get to where they wont pick up any more, refridgerator mags get to where they fall off. Dc motors after time get so they don’t run as fast as they used to. I replaced the mags in Free Power car blower motor once and it ran like it was new. now i do not know about the neo’s but i know that mags do lose there power. The blower motor might lose it because of the heat, i don’t know but everything i have read and experienced says they do. So whats up with that? Hey Free Electricity, ok, i agree with what you are saying. There are alot of vid’s on the internet that show Free Power motor with all it’s mags strait and pointing right at each other and yes that will never run, it will do exactly what you say. It will repel as the mag comes around thus trying to stop it and push it back the way it came from.
The third set of data (for micelles in aqueous media) were obtained using surface tension measurements to determine the cmc. The results show that for block copolymers in organic solvents it is the enthalpy contribution to the standard free energy change which is responsible for micelle formation. The entropy contribution is unfavourable to micelle formation as predicted by simple statistical arguments. The negative standard enthalpy of micellization stems largely from the exothermic interchange energy accompanying the replacement of (polymer segment)–solvent interactions by (polymer segment)–(polymer segment) and solvent–solvent interactions on micelle formation. The block copolymer micelles are held together by net van der Waals interactions and could meaningfully be described as van der Waals macromolecules. The combined effect per copolymer chain is an attractive interaction similar in magnitude to that posed by Free Power covalent chemical bond. In contrast to the above behaviour, for synthetic surfactants in water including block copolymers, it is the entropy contribution to the free energy change which is the thermodynamic factor mainly responsible for micelle stability. Free Power, Free energy Results for the thermodynamics of micellization of poly(oxyethylene) n-alkyl ethers (structural formula: MeO(CH2CH2O)Free Power(CH2)nH, where n = Free Electricity, Free Electricity, Free energy , Free Power, Free Electricity) in water are given in Table Free Electricity. Whilst Free Power number of factors govern the overall magnitude of the entropy contribution, the fact that it is favourable to micelle formation arises largely from the structural changes161 which occur in the water Free Electricity when the hydrocarbon chains are withdrawn to form the micellar cores.
Or, you could say, “That’s Free Power positive Delta G. “That’s not going to be spontaneous. ” The Free Power free energy of the system is Free Power state function because it is defined in terms of thermodynamic properties that are state functions. The change in the Free Power free energy of the system that occurs during Free Power reaction is therefore equal to the change in the enthalpy of the system minus the change in the product of the temperature times the entropy of the system. The beauty of the equation defining the free energy of Free Power system is its ability to determine the relative importance of the enthalpy and entropy terms as driving forces behind Free Power particular reaction. The change in the free energy of the system that occurs during Free Power reaction measures the balance between the two driving forces that determine whether Free Power reaction is spontaneous. As we have seen, the enthalpy and entropy terms have different sign conventions. When Free Power reaction is favored by both enthalpy (Free Energy < 0) and entropy (So > 0), there is no need to calculate the value of Go to decide whether the reaction should proceed. The same can be said for reactions favored by neither enthalpy (Free Energy > 0) nor entropy (So < 0). Free energy calculations become important for reactions favored by only one of these factors. Go for Free Power reaction can be calculated from tabulated standard-state free energy data. Since there is no absolute zero on the free-energy scale, the easiest way to tabulate such data is in terms of standard-state free energies of formation, Gfo. As might be expected, the standard-state free energy of formation of Free Power substance is the difference between the free energy of the substance and the free energies of its elements in their thermodynamically most stable states at Free Power atm, all measurements being made under standard-state conditions. The sign of Go tells us the direction in which the reaction has to shift to come to equilibrium. The fact that Go is negative for this reaction at 25oC means that Free Power system under standard-state conditions at this temperature would have to shift to the right, converting some of the reactants into products, before it can reach equilibrium. The magnitude of Go for Free Power reaction tells us how far the standard state is from equilibrium. The larger the value of Go, the further the reaction has to go to get to from the standard-state conditions to equilibrium. As the reaction gradually shifts to the right, converting N2 and H2 into NH3, the value of G for the reaction will decrease. If we could find some way to harness the tendency of this reaction to come to equilibrium, we could get the reaction to do work. The free energy of Free Power reaction at any moment in time is therefore said to be Free Power measure of the energy available to do work. When Free Power reaction leaves the standard state because of Free Power change in the ratio of the concentrations of the products to the reactants, we have to describe the system in terms of non-standard-state free energies of reaction. The difference between Go and G for Free Power reaction is important. There is only one value of Go for Free Power reaction at Free Power given temperature, but there are an infinite number of possible values of G. Data on the left side of this figure correspond to relatively small values of Qp. They therefore describe systems in which there is far more reactant than product. The sign of G for these systems is negative and the magnitude of G is large. The system is therefore relatively far from equilibrium and the reaction must shift to the right to reach equilibrium. Data on the far right side of this figure describe systems in which there is more product than reactant. The sign of G is now positive and the magnitude of G is moderately large. The sign of G tells us that the reaction would have to shift to the left to reach equilibrium.

Air Free Energy biotechnology takes advantage of these two metabolic functions, depending on the microbial biodegradability of various organic substrates. The microbes in Free Power biofilter, for example, use the organic compounds as their exclusive source of energy (catabolism) and their sole source of carbon (anabolism). These life processes degrade the pollutants (Figure Free Power. Free energy). Microbes, e. g. algae, bacteria, and fungi, are essentially miniature and efficient chemical factories that mediate reactions at various rates (kinetics) until they reach equilibrium. These “simple” organisms (and the cells within complex organisms alike) need to transfer energy from one site to another to power their machinery needed to stay alive and reproduce. Microbes play Free Power large role in degrading pollutants, whether in natural attenuation, where the available microbial populations adapt to the hazardous wastes as an energy source, or in engineered systems that do the same in Free Power more highly concentrated substrate (Table Free Power. Free Electricity). Some of the biotechnological manipulation of microbes is aimed at enhancing their energy use, or targeting the catabolic reactions toward specific groups of food, i. e. organic compounds. Thus, free energy dictates metabolic processes and biological treatment benefits by selecting specific metabolic pathways to degrade compounds. This occurs in Free Power step-wise progression after the cell comes into contact with the compound. The initial compound, i. e. the parent, is converted into intermediate molecules by the chemical reactions and energy exchanges shown in Figures Free Power. Free Power and Free Power. Free Power. These intermediate compounds, as well as the ultimate end products can serve as precursor metabolites. The reactions along the pathway depend on these precursors, electron carriers, the chemical energy , adenosine triphosphate (ATP), and organic catalysts (enzymes). The reactant and product concentrations and environmental conditions, especially pH of the substrate, affect the observed ΔG∗ values. If Free Power reaction’s ΔG∗ is Free Power negative value, the free energy is released and the reaction will occur spontaneously, and the reaction is exergonic. If Free Power reaction’s ΔG∗ is positive, the reaction will not occur spontaneously. However, the reverse reaction will take place, and the reaction is endergonic. Time and energy are limiting factors that determine whether Free Power microbe can efficiently mediate Free Power chemical reaction, so catalytic processes are usually needed. Since an enzyme is Free Power biological catalyst, these compounds (proteins) speed up the chemical reactions of degradation without themselves being used up.
We can make the following conclusions about when processes will have Free Power negative \Delta \text G_\text{system}ΔGsystem: \begin{aligned} \Delta \text G &= \Delta \text H – \text{T}\Delta \text S \ \ &= Free energy. 01 \dfrac{\text{kJ}}{\text{mol-rxn}}-(Free energy \, \cancel{\text K})(0. 022\, \dfrac{\text{kJ}}{\text{mol-rxn}\cdot \cancel{\text K})} \ \ &= Free energy. 01\, \dfrac{\text{kJ}}{\text{mol-rxn}}-Free energy. Free Power\, \dfrac{\text{kJ}}{\text{mol-rxn}}\ \ &= -0. Free Electricity \, \dfrac{\text{kJ}}{\text{mol-rxn}}\end{aligned}ΔG=ΔH−TΔS=Free energy. 01mol-rxnkJ−(293K)(0. 022mol-rxn⋅K)kJ=Free energy. 01mol-rxnkJ−Free energy. 45mol-rxnkJ=−0. 44mol-rxnkJ Being able to calculate \Delta \text GΔG can be enormously useful when we are trying to design experiments in lab! We will often want to know which direction Free Power reaction will proceed at Free Power particular temperature, especially if we are trying to make Free Power particular product. Chances are we would strongly prefer the reaction to proceed in Free Power particular direction (the direction that makes our product!), but it’s hard to argue with Free Power positive \Delta \text GΔG! Our bodies are constantly active. Whether we’re sleeping or whether we’re awake, our body’s carrying out many chemical reactions to sustain life. Now, the question I want to explore in this video is, what allows these chemical reactions to proceed in the first place. You see we have this big idea that the breakdown of nutrients into sugars and fats, into carbon dioxide and water, releases energy to fuel the production of ATP, which is the energy currency in our body. Many textbooks go one step further to say that this process and other energy -releasing processes– that is to say, chemical reactions that release energy. Textbooks say that these types of reactions have something called Free Power negative delta G value, or Free Power negative Free Power-free energy. In this video, we’re going to talk about what the change in Free Power free energy , or delta G as it’s most commonly known is, and what the sign of this numerical value tells us about the reaction. Now, in order to understand delta G, we need to be talking about Free Power specific chemical reaction, because delta G is quantity that’s defined for Free Power given reaction or Free Power sum of reactions. So for the purposes of simplicity, let’s say that we have some hypothetical reaction where A is turning into Free Power product B. Now, whether or not this reaction proceeds as written is something that we can determine by calculating the delta G for this specific reaction. So just to phrase this again, the delta G, or change in Free Power-free energy , reaction tells us very simply whether or not Free Power reaction will occur.

# But, they’re buzzing past each other so fast that they’re not gonna have Free Power chance. Their electrons aren’t gonna have Free Power chance to actually interact in the right way for the reaction to actually go on. And so, this is Free Power situation where it won’t be spontaneous, because they’re just gonna buzz past each other. They’re not gonna have Free Power chance to interact properly. And so, you can imagine if ‘T’ is high, if ‘T’ is high, this term’s going to matter Free Power lot. And, so the fact that entropy is negative is gonna make this whole thing positive. And, this is gonna be more positive than this is going to be negative. So, this is Free Power situation where our Delta G is greater than zero. So, once again, not spontaneous. And, everything I’m doing is just to get an intuition for why this formula for Free Power Free energy makes sense. And, remember, this is true under constant pressure and temperature. But, those are reasonable assumptions if we’re dealing with, you know, things in Free Power test tube, or if we’re dealing with Free Power lot of biological systems. Now, let’s go over here. So, our enthalpy, our change in enthalpy is positive. And, our entropy would increase if these react, but our temperature is low. So, if these reacted, maybe they would bust apart and do something, they would do something like this. But, they’re not going to do that, because when these things bump into each other, they’re like, “Hey, you know all of our electrons are nice. “There are nice little stable configurations here. “I don’t see any reason to react. ” Even though, if we did react, we were able to increase the entropy. Hey, no reason to react here. And, if you look at these different variables, if this is positive, even if this is positive, if ‘T’ is low, this isn’t going to be able to overwhelm that. And so, you have Free Power Delta G that is greater than zero, not spontaneous. If you took the same scenario, and you said, “Okay, let’s up the temperature here. “Let’s up the average kinetic energy. ” None of these things are going to be able to slam into each other. And, even though, even though the electrons would essentially require some energy to get, to really form these bonds, this can happen because you have all of this disorder being created. You have these more states. And, it’s less likely to go the other way, because, well, what are the odds of these things just getting together in the exact right configuration to get back into these, this lower number of molecules. And, once again, you look at these variables here. Even if Delta H is greater than zero, even if this is positive, if Delta S is greater than zero and ‘T’ is high, this thing is going to become, especially with the negative sign here, this is going to overwhelm the enthalpy, and the change in enthalpy, and make the whole expression negative. So, over here, Delta G is going to be less than zero. And, this is going to be spontaneous. Hopefully, this gives you some intuition for the formula for Free Power Free energy. And, once again, you have to caveat it. It’s under, it assumes constant pressure and temperature. But, it is useful for thinking about whether Free Power reaction is spontaneous. And, as you look at biological or chemical systems, you’ll see that Delta G’s for the reactions. And so, you’ll say, “Free Electricity, it’s Free Power negative Delta G? “That’s going to be Free Power spontaneous reaction. “It’s Free Power zero Delta G. “That’s gonna be an equilibrium. ”

Why? Because I didn’t have the correct angle or distance. It did, however, start to move on its own. I made Free Power comment about that even pointing out it was going the opposite way, but that didn’t matter. This is Free Power video somebody made of Free Power completed unit. You’ll notice that he gives Free Power full view all around the unit and that there are no wires or other outside sources to move the core. Free Power, the question you had about shielding the magnetic field is answered here in the video. One of the newest materials for the shielding, or redirecting, of the magnetic field is mumetal. You can get neodymium magnets via eBay really cheaply. That way you won’t feel so bad when it doesn’t work. Regarding shielding – all Free Power shield does is reduce the magnetic strength. Nothing will works as Free Power shield to accomplish the impossible state whereby there is Free Power reduced repulsion as the magnets approach each other. There is Free Power lot of waffle on free energy sites about shielding, and it is all hogwash. Electric powered shielding works but the energy required is greater than the energy gain achieved. It is Free Power pointless exercise. Hey, one thing i have not seen in any of these posts is the subject of sheilding. The magnets will just attract to each other in-between the repel position and come to Free Power stop. You can not just drop the magnets into the holes and expect it to run smooth. Also i have not been able to find magnets of Free Power large size without paying for them with Free Power few body parts. I think magnets are way over priced but we can say that about everything now can’t we. If you can get them at Free Power good price let me know.
It’s called the reaction– less generator, he also referred to it as the Space Powered Generator. It allows for the production of power with improved efficiency. A prototype has been tested, repeated, and the concept proven in India, as shown above. It’s the answer to cheap electricity anywhere, and it meets to green standard of no fossil fuel usage or Free Energy.

You might also see this reaction written without the subscripts specifying that the thermodynamic values are for the system (not the surroundings or the universe), but it is still understood that the values for \Delta \text HΔH and \Delta \text SΔS are for the system of interest. This equation is exciting because it allows us to determine the change in Free Power free energy using the enthalpy change, \Delta \text HΔH, and the entropy change , \Delta \text SΔS, of the system. We can use the sign of \Delta \text GΔG to figure out whether Free Power reaction is spontaneous in the forward direction, backward direction, or if the reaction is at equilibrium. Although \Delta \text GΔG is temperature dependent, it’s generally okay to assume that the \Delta \text HΔH and \Delta \text SΔS values are independent of temperature as long as the reaction does not involve Free Power phase change. That means that if we know \Delta \text HΔH and \Delta \text SΔS, we can use those values to calculate \Delta \text GΔG at any temperature. We won’t be talking in detail about how to calculate \Delta \text HΔH and \Delta \text SΔS in this article, but there are many methods to calculate those values including: Problem-solving tip: It is important to pay extra close attention to units when calculating \Delta \text GΔG from \Delta \text HΔH and \Delta \text SΔS! Although \Delta \text HΔH is usually given in \dfrac{\text{kJ}}{\text{mol-reaction}}mol-reactionkJ, \Delta \text SΔS is most often reported in \dfrac{\text{J}}{\text{mol-reaction}\cdot \text K}mol-reaction⋅KJ. The difference is Free Power factor of 10001000!! Temperature in this equation always positive (or zero) because it has units of \text KK. Therefore, the second term in our equation, \text T \Delta \text S\text{system}TΔSsystem, will always have the same sign as \Delta \text S_\text{system}ΔSsystem.
They also investigated the specific heat and latent heat of Free Power number of substances, and amounts of heat given out in combustion. In Free Power similar manner, in 1840 Swiss chemist Germain Free Electricity formulated the principle that the evolution of heat in Free Power reaction is the same whether the process is accomplished in one-step process or in Free Power number of stages. This is known as Free Electricity’ law. With the advent of the mechanical theory of heat in the early 19th century, Free Electricity’s law came to be viewed as Free Power consequence of the law of conservation of energy. Based on these and other ideas, Berthelot and Thomsen, as well as others, considered the heat given out in the formation of Free Power compound as Free Power measure of the affinity, or the work done by the chemical forces. This view, however, was not entirely correct. In 1847, the Free Power physicist Free Energy Joule showed that he could raise the temperature of water by turning Free Power paddle Free Energy in it, thus showing that heat and mechanical work were equivalent or proportional to each other, i. e. , approximately, dW ∝ dQ.


text:
# Interplay of social distancing and border restrictions for pandemics via the epidemic renormalisation group framework

## Abstract

One of the biggest threats to humanity are pandemics. In our global society they can rage around the world with an immense toll in terms of human, economic and social impact. Forecasting the spreading of a pandemic is, therefore, paramount in helping governments to enforce a number of social and economic measures, apt at curbing the pandemic and dealing with its aftermath. We demonstrate that the epidemic renormalisation group approach to pandemics provides an effective and simple way to investigate the dynamics of disease transmission and spreading across different regions of the world. The framework also allows for reliable projections on the impact of travel limitations and social distancing measures on global epidemic spread. We test and calibrate it on reported COVID-19 cases while unveiling the mechanism that governs the delay in the relative peaks of newly infected cases among different regions of the globe. We discover that social distancing measures are more effective than travel limitations across borders in delaying the epidemic peak. We further provide the link to compartmental models such as the time-honoured SIR-like models. We also show how to generalise the framework to account for the interactions across several regions of the world, replacing or complementing large scale simulations.

## Introduction

Virus-induced pandemics are a threat to humans not only because of the number of human lives taken but also because of the profound and long-lasting impact on the economy and social dynamics. While different empirical models already exist to describe the epidemic dynamics locally and globally, a coherent framework is missing.

Using a powerful language and methodology borrowed from high energy physics, we study and forecast the spreading dynamics and containment across different regions of the world. This framework is the renormalisation group approach1,2, which was successfully employed for epidemic dynamics3. Here we will generalise the framework to take into account the dynamics in between different regions of the world. The approach is complementary to other methods4,5,6,7,8. As for the widely adopted choice to represent the data by fitting them to simple minded logistic functions we refer to9,10,11,12,13,14. We will also provide a map between our framework and compartmental models such as the time-honoured SIR model15. Our epidemic renormalisation group (eRG) approach is based upon a simpler set of equations, which can be extended in a straightforward way to include interactions between multiple regions of the world, without the need for powerful numerical simulations. Seminal studies for the COVID-19 pandemic can be found in Ref.16,17,18,19.

As already noted3, rather than the number of cases, it is convenient to discuss its logarithm, which is a more slowly varying function. We define through it an epidemic strength function

\begin{aligned} \alpha (t) = {\text{ln}}\ \mathscr {I}(\text{t}), \end{aligned}
(1)

where $$\mathscr {I} (t)$$ is the total number of infected cases per million inhabitants in the region, and $$\ln$$ indicates its natural logarithm. The derivative of $$\alpha$$ with respect to time provides a new quantity that we interpret as the beta-function of an underlying microscopic model. In statistical and high energy physics, the latter governs the time (inverse energy) dependence of the interaction strength among fundamental particles. Here it regulates infectious interactions.

More specifically, as the renormalisation group equations in high energy physics are expressed in terms of derivatives with respect to the energy $$\mu$$, it is natural to identify the time as $$t/t_0=-\ln {\mu /\mu _0}$$, where $$t_0$$ and $$\mu _0$$ are respectively a reference time and energy scale. We choose $$t_0$$ to be one week so that time is measured in weeks, and will drop it in the following. Thus, the dictionary between the eRG equation for the epidemic strength $$\alpha$$ and the high-energy physics analog is

\begin{aligned} \beta (\alpha ) = \frac{d \alpha }{d\ln \left( \mu /\mu _0\right) } = - \frac{d \alpha }{dt} \ . \end{aligned}
(2)

It has been shown3 that $$\alpha$$ captures the essential information about the infected population within a sufficiently isolated region of the world. The pandemic beta function can be parametrised as

\begin{aligned} - \beta (\alpha ) = \frac{d \alpha }{dt} = \gamma \, \alpha \left( 1 - \frac{\alpha }{a} \right) ^n\,, \end{aligned}
(3)

whose solution, for $$n=1$$, is a familiar logistic-like function

\begin{aligned} \alpha (t) = \frac{a e^{\gamma t}}{b + e^{\gamma t}}\,. \end{aligned}
(4)

The dynamics encoded in Eq. (3) is that of a system that flows from an ultra-violet (UV) fixed point at $$t=-\infty$$ where $$\alpha = 0$$ to an infra-red (IR) fixed point where $$\alpha = a$$. The latter value encodes the total number of infected cases per million expected in the region under study. The coefficient $$\gamma$$ is the diffusion slope, while b shifts the entire epidemic curve by a given amount of time. Further details, including what parameter influences the flattening of the curve and location of the inflection point and its properties can be found in3. Note that here we work with number of cases per million, so that our $$\alpha$$ corresponds to $$\alpha - \ln n_m$$ of Ref.3 with $$n_m$$ the number of inhabitants per million per each sufficiently isolated region of the world. In this work, we extend the eRG formalism to include the diffusion of the epidemic between multiple nearly-isolated regions.

## Epidemic diffusion among different regions of the globe

Here we go beyond the state-of-the-art by considering the diffusion among multiple regions of the world, each characterised by its own $$\alpha _i (t)$$, which in isolation obeys a beta function like Eq. (3), with its own $${\gamma_i}$$ and $$a_i$$.

We exemplify the framework by first considering two regions, and we generalise to multiple ones later. To couple the two equations, we start from the following axiom: there is a constant number of travellers moving from one region to the other, and viceversa, given by $$\Delta {N_\text{trav}}$$ each week. Our basic simplifying assumption is that the number of travellers is symmetric, i.e. there is no net flow of people between the two regions: this is a reasonable approximation during a short time as immigration only involves a smaller fraction of inhabitants than that involved in the epidemic. We further use the approximation that the rate of infected cases within the travelling subset of people is the same as the rate of infected cases in the total population of each region. Thus, the variation in the number of infected cases per million in region-1 is given by

\begin{aligned} n_{m1} \frac{\delta \mathscr {I}_1 (t)}{\delta t} = k \left( \mathscr {I}_2 (t) - \mathscr {I}_1 (t) \right) \,, \end{aligned}
(5)

where $$n_{m1}$$ is the population of region-1 in millions and

\begin{aligned} k = {10^{-6}}\, \Delta {N_\text{trav}} \end{aligned}.
(6)

Here we neglect the fact that part of the infected population has recovered, thus probably ceasing to be infectious. We will come back to discussing the validity of this approximation. For region-2, we find the analogous

\begin{aligned} n_{m2} \frac{\delta \mathscr {I}_2 (t)}{\delta t} = k \left( \mathscr {I}_1 (t) - \mathscr {I}_2 (t) \right) \,, \end{aligned}
(7)

where the same k applies. Physically, the parameter k measures the number of reciprocal travellers per week in units of million people. For instance, if the number of weekly travellers is $$\Delta {N_\text{trav}} = 1,000$$, then $$k = 10^{-3}$$. Using the identity

\begin{aligned} \frac{\delta \mathscr {I}_i (t)}{\delta t} = \mathscr {I}_i \frac{\delta \alpha _i}{\delta t}, \end{aligned}
(8)

the effect of this exchange can be encoded in the two beta functions, c.f. Eq. (3), as follows:

\begin{aligned} - \beta (\alpha _1)= {} \gamma _1 \alpha _1 \left( 1 - \frac{\alpha _1}{a_1} \right) + \frac{k}{n_{m1}} \left( e^{\alpha _2 - \alpha _1} -1 \right) \,, \end{aligned}
(9)
\begin{aligned} - \beta (\alpha _2)= {} \gamma _2 \alpha _2 \left( 1 - \frac{\alpha _2}{a_2} \right) + \frac{k}{n_{m2}} \left( e^{\alpha _1 - \alpha _2} -1 \right) \,. \end{aligned}
(10)

The above equations describe the evolution of the epidemic across the two regions, once a small fraction of the population travels between the two. However, for large k, they have the interesting property of forcing $$\alpha _1 = \alpha _2$$, which in turn modifies the value of the fixed point for the two regions. The fact that the $$\alpha$$’s become equal in the long run indicates that the two regions have merged into one. One surprising finding is that the total number of infected cases across the two regions, for large k, may be reduced compared to the isolated case. While mathematically intriguing, we do not consider this result physical, as having large k modifies the values of $$\alpha _i$$ and $$\gamma _i$$ in the two regions compared to the values one would have in case of isolation. In other words it would violate our initial assumption that the two regions are nearly-isolated, with small k. One can go beyond the realistic case envisioned here by increasing k. This would require modifying the set of equations substantially and goes beyond the scope of this work.

To quantitatively estimate the interaction between two regions of the world, we consider benchmark values for the parameters in the two beta functions using the results given in3. We show in Table 1 the values of a and $$\gamma$$ for various regions of the world for the COVID-19 pandemic3, where a is normalised per million inhabitants and all values are adjourned to the 4th of May, 2020. It is straightforward to provide daily updates, as done following the eRG approach3 at http://caracal.imada.sdu.dk/corona/ for different regions of the world. The values of $$\gamma$$ and a are average values over the whole duration of the epidemic diffusion in each country/region. We observe that the value of $$\gamma$$ tends to diminish over time as a consequence of the effect of gradual implementing of social distancing measures in each region. At the early stages of the epidemic, we observe $$\gamma \sim 1$$, so that we will consider this as a benchmark value for the epidemic diffusion without any restriction.

With the exception of South Korea and China (Hubei province), the range for a is roughly [7.5, 8.6], while for $$\gamma$$ we find [0.4, 0.76]. Thus, we defined the following benchmark scenario for the two regions:

\begin{aligned} a_1 = 7\,, \;\; b_1 = 2\,, \;\; a_2 = 8\,, \;\; b_2 = 200,\, \infty \,; \end{aligned}
(11)

while we vary the values of $$\gamma _1$$ and $$\gamma _2$$ as specified in the figures. The value of $$b_2=200$$ is chosen such that the peak in the two regions in isolation have a relative delay of 14 weeks. The peak is here defined as the week where the maximum number of new infected cases per million is registered and corresponds to the inflection point of the total number of infected cases curve. The explicit formula for the inflection point time as function of the parameters of the theory can be found in3.

As a first sanity check, we computed the total number of infected cases across the two regions, per million, at the end of the pandemic, i.e. at infinite time. This is given by

\begin{aligned} \mathscr {I}_{1+2} (\infty ) = \frac{n_{m1} \mathscr {I}_1 (\infty ) + n_{m2} \mathscr {I}_2 (\infty )}{n_{m1} + n_{m2}}\,, \end{aligned}
(12)

as a function of k. The result allows us to determine the largest value of k that does not affect the total number, i.e. the largest value that k can have before the two regions effectively merge into one. In Fig. 1 we show the results for two different populations. The plot shows that k as large as 0.1 is allowed before our description of the coupled system breaks down. Note that the maximal value of k grows linearly with the population in region-2, as it enters as the ratio $$k/n_{m2}$$ in the coupled differential equations.

### Peak delay study

To understand how the interaction encoded by k affects the diffusion of the epidemic in the two regions, we study the same benchmark of Eq. (11), except that we set $$b_2 = \infty$$, i.e. the region-2 remains with zero infected cases if isolated. One caveat that should be kept in mind is that the values for $$\gamma$$ in Table 1 are obtained by fitting the data during the whole period of the epidemic, i.e. they take into account the effect of social distancing measures in each region. However, at the early stages of the epidemic, when social distancing measures were not yet being enforced, we expect larger values of $$\gamma$$. That is the reason why, in the following, we assume $$\gamma = 1$$ as the natural initial benchmark value. Nevertheless, we show how different social distancing measures impact the results for region-2 by showing also the results for smaller values of $$\gamma _2$$.

We discover that the interaction among the two regions of the world, controlled by the parameter k, is sufficient to ignite the spread of the epidemic to region-2 and it also controls the timing of the peak. This is shown in the left panels of Fig. 2, where we plot the time of the peaks in the two regions as a function of $$k/n_{m2}$$. The result does not depend on $$n_{m1}$$. Also, the time of the peak for region-1 is unaffected by the value of k (dashed curve) while it affects the timing of the peak for region-2 (solid curves). Note that the k-term in Eq. (10) sparks the epidemic diffusion in region-2 as soon as $$\frac{k}{n_{m2}} e^{\alpha _1 (t)}$$ becomes sizeable. After this point, the epidemic evolution follows the solution of the initial equation (3), as encoded in the first term of the region-2 beta function.

The numerical results for the peak delay show a linear dependence on $$\ln k$$, with a change in slope appearing for $$k/n_{m2} \sim 10^{-3}$$. This value corresponds to

\begin{aligned} \frac{k}{n_{m2}} e^{a_1} = 1\,, \end{aligned}
(13)

i.e. it marks the threshold (grey line in the plots) between the regime where the interaction term is always smaller than one, and the one where strength 1 is attainable.

To test our approach we consider the COVID-19 epidemic spread from China (Hubei province) to Europe (Italy). We chose to focus on Hubei province and Italy in order to have regions with comparable populations. From data it is known that the peaks in the two regions are about 7 weeks apart. A reasonable estimate of weekly travellers between the two regions is in the order of the thousands, so we consider $$k = 5 \times 10^{-3}$$ as a benchmark. This means that

\begin{aligned} \frac{k}{n_{m2}} \sim 10^{-4}\,. \end{aligned}
(14)

For this value, the bottom–left plot in Fig. 2 allows us to estimate the peak delay to be around 6 weeks for $$\gamma _2 = 1$$, i.e. for unrestricted diffusion within Italy. This nicely confirms our expectations while validating the model.

It is useful to note that both k and $$b_2$$ lead to a temporal shift of the epidemic curve for region-2, however the underlying mechanisms are distinct. The former is due to an interaction between two different regions of the world, while the latter is a constant of integration that depends on the number of cases at the initial time $$t=0$$ in region-2. This also means that a specific peak time for region-2 relative to region-1 can emerge as a combination of the two effects, interpolating between the two limiting cases: the peak delay is entirely due to the interaction with region-1, or it is due to the presence of cases in region-2 (which may have different origin) and the coupling to region-1 is negligible. We will discuss this interplay in more details in the next section.

### Border control versus social distancing

We now turn out attention on the impact of closing the borders between two regions of the globe versus different degree of social distancing. In the eRG approach this is implemented by setting to zero k after the closing time $$t_\text{cl}$$. We consider the benchmark values given in Eq. (11), while the impact of social distancing is encoded in region-2 in varying the value for $$\gamma _2$$. Furthermore we consider two scenarios: one in which region-2 has zero initial cases, meaning that the epidemic would not occur for $$k=0$$ (corresponding to $$b_2 = \infty$$) and another where we fix the initial condition according to the benchmark (corresponding to $$b_2 = 200$$).

The results are shown in the right panels of Fig. 2, where we report the delay in the peak of region-2 caused by closing the borders (i.e., delay relative to the case of $${t_\text{cl}} = \infty$$). Such a delay depends crucially on the value of $${\gamma_2}$$ in region-2 as shown in the top plot when the epidemic in region-2 is only driven by the interaction term. In particular the results show that a significant delay in the spreading of the epidemic can be achieved only if the closing is enacted before the peak in region-1 (which is unaffected by k).

In the bottom–right panel we show the case where region-2 features already some initial cases, so that $${t_\text{cl}}=0$$ would correspond to isolated regions with both featuring infected cases. In this case, we also see that the effect of the interaction is more pronounced for small values of $$\gamma _2$$ in region-2, indicated by the red curve for $$\gamma _2 = 0.4$$. For this value of $$\gamma _2$$, isolation would yield a delay of 4.5 weeks in the peak. For larger values of $$\gamma _2$$ (less social distancing) the peak delay is strongly reduced to within one or two weeks. In any case, closing the borders is only relevant if done before the peak in region-1 is attained.

Our results, obtained using the simple and effective eRG approach, agree qualitatively with the ones obtained using a numerical analysis20. The take home message is that social distancing plays the dominant role in curbing and delaying the epidemic spread in region-2 with respect to seed region-1.

## Relation to the SIR model

Epidemic dynamics is often described in terms of simplistic compartmental models introduced long time ago15. Here, the affected population is described in terms of compartmentalised sub-populations that have different roles in the dynamics. Then, differential equations are designed to describe the time evolution of the various sub-populations. For an application to the COVID-19 epidemic, see21,22. The sub-populations can be chosen to represent (S)usceptible, (I)nfected and (R)ecovered individuals (SIR model), obeying the following differential equations:

\begin{aligned} \frac{d S}{d t}= {} - \tilde{\gamma }\ S \frac{I}{P}\,, \end{aligned}
(15)
\begin{aligned} \frac{d I}{d t}= {} \tilde{\gamma }\ S \frac{I}{P} - \epsilon \ I\,, \end{aligned}
(16)
\begin{aligned} \frac{d R}{d t} = {} \epsilon \ I\,; \end{aligned}
(17)

where $$P = S + I + R$$ is a constant, measuring the total number of individuals affected. As the equations do not depend on the normalisation of the number of individuals, we can consider them for cases per million. Due to the constant P, only two equations are independent, so that we can drop the one for S. The total number of infected, $$\mathscr {I} (t)$$, we study in our model is related to the above sub-populations as

\begin{aligned} \mathscr {I} (t) = I(t) + R(t)\,. \end{aligned}
(18)

We can therefore re-write the two independent SIR equations as

\begin{aligned} \frac{d \mathscr {I} (t)}{d t} = {} \tilde{\gamma } \left( \mathscr {I} (t) - R(t) \right) \left( 1 - \frac{\mathscr {I}(t)}{P} \right) \,, \end{aligned}
(19)
\begin{aligned} \frac{d R (t)}{d t} = {} \epsilon \left( \mathscr {I} (t) - R(t) \right) \,. \end{aligned}
(20)

Equation (19) has a form similar to Eq. (3), except for the following: it is written in terms of the total number $$\mathscr {I} (t)$$ instead of its log $$\alpha (t)$$; it contains a dependence on the number of recovered cases, R(t).

Thus, our eRG approach would be equivalent to the SIR model if we could drop the R(t) dependence in the differential equation for $$\mathscr {I} (t)$$. It is conceivable that this is the case: in fact, in Eq. (19) we can already see that the second factor drives the number of infected cases to the fixed point $$\mathscr {I} (\infty ) \rightarrow P \equiv e^{a}$$, which corresponds to the IR fixed point in the eRG approach. R(t), instead, is zero at early times and only grows slowly as long as the recovery rate $$\epsilon$$ is small, thus its effect should remain negligible once the dynamics of $$\mathscr {I} (t)$$ is driven towards the fixed point. As investigated in3, the dynamics of $$\mathscr {I} (t)$$ and $$\alpha (t)$$ can be described by the same equation, as they both are driven to flow between the two fixed points.

Once a solution for $$\mathscr {I} (t)$$ is found following the eRG approach, i.e. Eq. (3), the number of recovered cases can be calculated by solving Eq. (20), with solution

\begin{aligned} R(t) = \epsilon \int _0^t dx\ e^{\epsilon (x-t)}\ \mathscr {I} (x)\,. \end{aligned}
(21)

To validate this approach and calibrate $$\epsilon$$, we compared the above formula to the number of recovered cases for the United States (US), where $$\mathscr {I} (t)$$ is obtained using the fit values in Table 1: the results are shown in Fig. 3, where R(t) (in red) reproduces the data for $$\epsilon = 0.09$$. We checked that for other countries, a similarly good fit can be obtained for $$\epsilon \sim 0.1$$, thus we consider this description consistent.

To establish a more quantitative dictionary between the eRG approach and the SIR model, we compared the numerical solutions of the SIR Eqs. (19) and (20) to the solutions of the beta function in Eq. (3) (with R(t) given by Eq. (21)). We find that the solutions overlap as long as matching values of $$\gamma$$ and $$\tilde{\gamma }$$ are used. In Fig. 4 we show the numerical relation between the matching values of the couplings for 3 choices of the recovery rate $$\epsilon$$: the result shows a linear relation between the couplings in the two models.

Being able to reproduce the number of recovered cases for one region in isolation, we can now address the issue of the effect of the recovered cases in the coupled system. In fact, the transmission of the epidemic due to travel of individuals between the two regions is only due to the presence of people actively infected, namely it depends on

\begin{aligned} I (t) = \mathscr {I} (t) - R(t)\,. \end{aligned}
(22)

Thus, it suffices to replace the expression in Eq. (5) with

\begin{aligned} n_{m1} \frac{\delta \mathscr {I}_1 (t)}{\delta t} = k \left( \mathscr {I}_2 (t) - \mathscr {I}_1 (t)\right) - k \left( R_2 (t) - R_1 (t) \right) \,, \end{aligned}
(23)

and similarly for Eq. (7). We have compared the solutions of the coupled differential equations with and without taking into account the recovered cases, and found that including $$R_i (t)$$ only affects the epidemic diffusion in region-2 by a few days. Thus, this effect can be neglected in first approximation.

## COVID-19 examples

We now confront the eRG framework to data from the COVID-19 pandemic collected from www.worldometers.info and adjourned to the 4th of May, 2020. Although we are well aware of the pitfalls stemming from comparing data provided by different countries due to the inhomogeneous way infectious cases were tested and reported, it is still possible to extract from these reliable time behaviour and structure. Of course, when coupling two regions of the world, part of the initial uncertainty also affects the epidemic transmission probability without affecting the overall picture.

Nevertheless, we will now see that the eRG formalism can be used simultaneously to quantitatively project the spreading dynamics across different regions of the world, or as an a-posteriori way to learn how this spreading came to be. We focus on two examples, one intra European (Italy-Denmark) and the other between Europe and the US. The values for $$\gamma$$ and a for each country are taken from the fit in Table 1, which assumes isolation.

### From Italy to Denmark

In the right plot of Fig. 5 we show the total number of infected cases in Italy and Denmark (blue and red dots, respectively), compared to the fit in Table 1 (solid curves): the latter assumes that the epidemic occurred in the two regions while in isolation. We now wish to understand how and whether the virus spread from Italy to Denmark: thus, we used the coupled Eqs. (9) and (10), while we set the number of initial cases in Denmark to be null, i.e. $$b_2 = \infty$$. All the other parameters are fixed to the values in the Table. We find that the two curves can be reasonably fit by assuming $$k=0.16$$, as shown in the left plot of Fig. 5: the dashed orange curve, corresponding to Italy, overlaps to the isolated fit (solid blue), while the new curve for Denmark (dashed green) is close to the isolated fit (solid red). Let us now comment on the actual value of k. If we take it literally this it would correspond to a rate of 160.000 travellers between the two regions each week. This is an unreasonably large value but it can be alternatively and conservatively interpreted in the following ways:

1. (i)

More countries contributed to the epidemic spread in Denmark.

2. (ii)

The original spreading dynamics in Denmark is due to few very socially active infected individuals that traveled back from Italy and/or were super-spreaders.

3. (iii)

A combination of the above.

Whatever the reason, it is naturally incorporated in a larger value of k. One can also take into account the various scenarios by effectively re-instating an initial value for $$\alpha _2$$ at $$t=0$$ while reducing the k value.

### From Europe to the United States

To further test our model we consider the system consisting of Europe as region-1 and the United States as region-2. For simplicity, we modelled Europe on the European Union (with $$n_{m1} = 445$$) with parameters from the fit of the epidemic diffusion in Italy (c.f., Table 1). After setting to zero the initial cases in the US, we were able to reproduce the diffusion of the epidemic in region-2 (US) for $$k=10$$, as shown in the right plot of Fig. 5. While it is still possible that the large value for k may be interpreted as in the above case, it has the further effect of distorting the epidemic curve for region-1, the EU, thus suggesting that it may be hard to explain the diffusion of the COVID-19 epidemic in the US as originating solely from the EU.

At this stage, we cannot exclude that adjusting the epidemic parameters in the EU could improve the agreement. This exercise, nevertheless, proves the effectiveness of our simple eRG model to describe the diffusion of the epidemic among different regions of the world. A more accurate fit may be obtained if more than one region is included in the analysis, which will be considered in a future work.

### Multiple country system: a new simulated epidemic spread in Europe

We now use the eRG framework to model the impact of a new wave of epidemic spread of the COVID-19 virus (or a related one) in Europe. To do so, we simulate the effect of transmission among countries in a pool of European countries, namely Italy, Spain, France, the United Kingdom, Germany, Denmark and Switzerland. We also include an unspecified “seed region”, with a population of $$n_{m0} = 50$$, which has some initial cases, while no case is initially present for the simulated European countries. This is achieved by setting $$b_i = \infty$$, where $$i=1,\dots 7$$ spans over the 7 sample countries mentioned above.

We generate randomly the diffusion factors $$\gamma _i$$ in the range [0.4, 0.76], based on the data of the current COVID-19 epidemic in Europe, and also generate random values of $$a_i$$ in the range [7.5, 8.5]. This also includes the seed region. Finally, we provide randomly generated numbers of travellers between each of the regions, including the seed one, giving coupling values $$k_{ij}$$ in the range $$[1, 10] \times 10^{-3}$$. We then solve the 8 coupled differential equations:

\begin{aligned} \frac{d \alpha _i}{d t} = \gamma _i \alpha _i \left( 1-\frac{\alpha _i}{a_i} \right) + \sum _{j\ne i} \frac{k_{ij}}{n_{mi}} (e^{\alpha _j - \alpha _i} -1)\,, \end{aligned}
(24)

where $$i,j = 0,\dots 7$$ and $$\alpha _0$$ corresponds to the seed region. The result is shown in the top row of Fig. 6, where black indicates the seed region and the coloured curves correspond to the 7 sample European countries. The top-right plot, where the distribution of new cases is displayed, clearly shows that the peaks in the infected regions occur between 3 and 12 weeks after the peak in the seed region. This effect, however, is mainly due to the values of the $$\gamma$$’s in those regions, and not on the values of the interaction couplings $$k_{ij}$$.

To prove this, we have run the same simulation again, by fixing $$\gamma _i = 1$$, $$i=1,\dots 7$$, while $$\gamma _0$$ for the seed region is left the same. All other parameters are kept to the same values for the previous case. The analogous results are shown in the bottom row plots of Fig. 6. This case roughly correspond to unrestricted diffusion of the virus in the target regions. The result shows that all the peaks are now occurring within 4 weeks after the peak in the seed region.

The results nicely demonstrate that our eRG framework not only is useful, simple and effective to understand the current pandemic, but can also be used to model future ones.

## Conclusions and discussion

We extended the epidemic renormalisation group approach to analyse the dynamics of disease transmission and spreading across different regions of the world. We have shown that the eRG framework constitutes an effective way to understand the relative impact of border control versus social distancing measures on the global spread of the epidemic. The simplicity of the approach, stemming from an effective description of complex phenomena, makes it a reliable alternative to the use of expensive high-performance numerical computations.

We calibrated our approach via internationally reported cases. The approach elucidates the underlying mechanism that governs the delay in the relative peaks of newly infected cases across different regions of the world. Among our results, we were able to demonstrate that social distancing measures are more efficient than border control in delaying the epidemic peak. Our results complement and go beyond the ones of an earlier study20, focused on China via a traditional compartmental model, that found that the travel ban between cities delayed the infection peak by 2–5 days. Our numerical calculations show that the impact can be stronger if the travel ban takes place at the very early stages. We also found that early implementation of social distancing measures has a much stronger impact, amounting up to 4 weeks delay in the peak. The interplay of travel restrictions and social measures was also studied via compartmental models for Italian regions in Ref.22. The results corroborate our finding that the travel across regions sparks the epidemic diffusion, which then develops in each region independently. One of the major strengths of our model is that it is highly effective in taking into account human interactions across any number of regions of the world without the aid of high performance computing. Furthermore, the framework is rooted in the modern language of the renormalisation group equations making the symmetries of the model clearer, such as the approximate time-dilation invariance at large times (in between pandemic waves).

In order to connect with widely used time-honoured compartmental models of the SIR-like type, we established the proper map with our eRG framework. We have also shown how to generalise the eRG framework to account for the epidemic interactions across multiple regions of the world.

We foresee a number of future applications and extensions of our seed work. From a more phenomenological point of view, of immediate impact for society, we plan on embarking on a world-wide monitoring to make global projections that will help governments and industries make containment plans and strategise about reopening society and how to best implement border control. We also wish to improve on understanding the link between the eRG approach and microscopic models of population dynamics and epidemic spread including a number of granular effects that are, by construction, averaged over by effective descriptions such as the eRG approach.

## Data availability

The data used for this study has been extracted from an online repository, www.worldometers.info. Additional information can be provided upon request.

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## Acknowledgements

We thank Michele Della Morte, Christian Møller Dahl and Domenico Orlando for comments and helpful discussions.

## Author information

Authors

### Contributions

This work has been designed and performed conjointly and equally by the authors. Both F.S. and G.C. have equally contributed to the writing of the article. F.S. analysed the data leading to the fits in Table I, G.C. prepared the figures.

### Corresponding author

Correspondence to Giacomo Cacciapaglia.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

### Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Cacciapaglia, G., Sannino, F. Interplay of social distancing and border restrictions for pandemics via the epidemic renormalisation group framework. Sci Rep 10, 15828 (2020). https://doi.org/10.1038/s41598-020-72175-4

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beaver {boot} R Documentation

## Beaver Body Temperature Data

### Description

The beaver data frame has 100 rows and 4 columns. It is a multivariate time series of class "ts" and also inherits from class "data.frame".

This data set is part of a long study into body temperature regulation in beavers. Four adult female beavers were live-trapped and had a temperature-sensitive radio transmitter surgically implanted. Readings were taken every 10 minutes. The location of the beaver was also recorded and her activity level was dichotomized by whether she was in the retreat or outside of it since high-intensity activities only occur outside of the retreat.

The data in this data frame are those readings for one of the beavers on a day in autumn.

### Usage

beaver

### Format

This data frame contains the following columns:

day

The day number. The data includes only data from day 307 and early 308.

time

The time of day formatted as hour-minute.

temp

The body temperature in degrees Celsius.

activ

The dichotomized activity indicator. 1 indicates that the beaver is outside of the retreat and therefore engaged in high-intensity activity.

### Source

The data were obtained from

Reynolds, P.S. (1994) Time-series analyses of beaver body temperatures. In Case Studies in Biometry. N. Lange, L. Ryan, L. Billard, D. Brillinger, L. Conquest and J. Greenhouse (editors), 211–228. John Wiley.

### References

Davison, A.C. and Hinkley, D.V. (1997) Bootstrap Methods and Their Application. Cambridge University Press.

[Package boot version 1.3-28.1 Index]