# Hydrobates

A mathematician thinks aloud

## Mathematical models for tuberculosis

In previous posts I made some remarks on mathematical models for diseases and/or the immune system. I also had a post about tuberculosis. Now I came across the web page of Denise Kirschner where there are a lot of links to her publications on modelling TB, the immune system, HIV and related topics. You can also see a related video of a talk of hers (from June 19, 2007) on the web site http://videocast.nih.gov.

In a paper of Wigginton and Kirschner (Journal of Immunology 66, 1951) the authors introduce a mathematical model to describe the interactions of the immune system and the TB bacterium within the lung. This is a system of twelve ODE. The unknowns are two populations of bacteria (inside or outside macrophages), three populations of macrophages, three populations of T cells (Th0, Th1 and Th2) and the concentrations of four cytokines (interferon $\gamma$, IL-4, IL-10 and IL-12.) A lot of detail has been included concerning models for the interaction between the different players and in extracting values from the literature for the many parameters which occur. The goal is to understand the different outcomes of disease: acute infection, latent infection and reactivation.

The ODE are solved on the computer. As far as I could see there has been no general mathematical analysis of the properties of solutions of this system done. It may just be too big and complicated but I would be interested to see if something could be done in that direction. The numerical results apparently show convergence to a stationary state and convergence to a limit cycle in different situations. This model has been further extended by Kirschner and collaborators in other papers. In one paper a model with two compartments is introduced (lung and lymph node) where dendritic cells are also included. Another paper includes CD8+ T cells and TNF$\alpha$. What I like about this work is that it seems to be making real contact between mathematical modelling and the details of immunology, going beyond the simplest model systems.

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This entry was posted on December 30, 2008 at 9:58 am and is filed under diseases, dynamical systems, immunology, mathematical biology. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

### 2 Responses to “Mathematical models for tuberculosis”

1. lvs Says:
December 30, 2008 at 6:56 pm | Reply

wow I never thought one could use it for TB! Did these guys make very strict assumptions or is it a fairly realistic model for TB.

2. hydrobates Says:
December 31, 2008 at 8:19 am | Reply

My impression is that this is a fairly realistic model although of course, as in any model of such a complex phenomenon, many things have to be left out.
The fact that there was an invited lecture about this modelling effort at the National Institutes of Health is a sign that medical researchers take it seriously.

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