<![CDATA[The Mathematics of Biodiversity (Part 3)]]>

<![CDATA[The Mathematics of Biodiversity (Part 3)]]>We tend to think of biodiversity as a good thing, but sometimes it’s deadly. Yesterday Andrei Korobeinikov gave a talk on ‘Viral evolution within a host’, which was mainly about AIDS.

The virus that causes this disease, HIV, can reproduce very fast. In an untreated patient near death there are between 10¹⁰ and 10¹² new virions per day! Remember, a virion is an individual virus particle. The virus also has a high mutation rate: about 3 × 10^-5 mutations per generation for each base—that is, each molecule of A,T,C, or G in the RNA of the virus. That may not seem like a lot, but if you multiply it by 10¹² you’ll see that a huge number of new variations of each base arise within the body of a single patient.

So, evolution is at work within you as you die.

And in fact, many scientists believe that the diversity of the virus eventually overwhelms your immune system! Although it’s apparently not quite certain, it seems that while the body generates B cells and T cells to attack different variants of HIV as they arise, they eventually can’t keep up with the sheer number of variants.

Of course, the fact that the HIV virus attacks the immune system makes the disearse even worse. Here in blue you see the number of T cells per cubic millimeter of blood, and in red you see the number of virions per cubic centimeter of blood for a typical untreated patient:

Mathematicians and physicists have looked at some very simple models to get a qualitative understanding of these issues. One famous paper that started this off is:

• Lev S. Tsimring, Herbert Levine and David A. Kessler, RNA virus evolution via a fitness-space model, Phys. Rev. Lett. 76 (1996), 4440–4443.

The idea here is to say that at any time $t$ the viruses have a probability density $p(r,t)$ of having fitness $r$ . In fact the different genotypes of the virus form a cloud in a higher-dimensional space, but these authors are treating that space is 1-dimensional, with fitness as its one coordinate, just to keep things simple. They then write down an equation for how the population density changes with time:

$\displaystyle{\frac{\partial }{\partial t}p(r,t) = (r - \langle r \rangle)\, p(r,t) + D \frac{\partial^2 }{\partial r}p(r,t) - \frac{\partial}{\partial r}(v_{\mathrm{drift}}\, p(r,t)) }$

This is a replication-mutation-drift equation. If we just had

$\displaystyle{\frac{\partial }{\partial t}p(r,t) = (r - \langle r \rangle)\, p(r,t) }$

this would be a version of the replicator equation, which I explained recently in Information Geometry (Part 9). Here

$\displaystyle{ \langle r \rangle = \int_0^\infty r p(r,t) dr }$

is the mean fitness, and the replicator equations says that the fraction of organisms of a given type grows at a rate proportional to how much their fitness exceeds the mean fitness: that’s where the $(r - \langle r \rangle)$ comes from.

If we just had

$\displaystyle{\frac{\partial }{\partial t}p(r,t) = D \frac{\partial^2 }{\partial r^2}p(r,t) }$

this would be the heat equation, which describes diffusion occurring at a rate $D$ . This models the mutation of the virus, though not in a very realistic way.

If we just had

$\displaystyle{\frac{\partial}{\partial t} p(r,t) = - \frac{\partial}{\partial r}(v_{\mathrm{drift}} \, p(r,t)) }$

the fitness of the virus would increase at rate equal to the drift velocity $v_{\mathrm{drift}}$ .

If we include both the diffusion and drift terms:

$\displaystyle{\frac{\partial }{\partial t} p(r,t) = D \frac{\partial^2 }{\partial r^2}p(r,t) - \frac{\partial}{\partial r}(v_{\mathrm{drift}} \, p(r,t)) }$

we get the Fokker–Planck equation. This is a famous model of something that’s spreading while also drifting along at a constant velocity: for example, a drop of ink in moving water. Its solutions look like this:

Here we start with stuff concentrated at one point, and it spreads out into a Gaussian while drifting along.

By the way, watch out: what biologists call ‘genetic drift’ is actually a form of diffusion, not what physicists call ‘drift’.

More recently, people have looked at another very simple model. You can read about it here:

• Martin A. Nowak, and R. M. May, Virus Dynamics, Oxford University Press, Oxford, 2000.

In this model the variables are:

• the number of healthy human cells of some type, $\mathrm{H}(t)$

• the number of infected human cells of that type, $\mathrm{I}(t)$

• the number of virions, $\mathrm{V}(t)$

These are my names for variables, not theirs. It’s just a sick joke that these letters spell out ‘HIV’.

Chemists like to describe how molecules react and turn into other molecules using ‘chemical reaction networks’. You’ve seen these if you’ve taken chemistry, but I’ve been explaining more about the math of these starting in Network Theory (Part 17). We can also use them here! Though May and Nowak probably didn’t put it this way, we can consider a chemical reaction network with the following 6 reactions:

• the production of a healthy cell:

$\longrightarrow \mathrm{H}$

• the infection of a healthy cell by a virion:

$\mathrm{H} + \mathrm{V} \longrightarrow \mathrm{I}$

• the production of a virion by an infected cell:

$\mathrm{I} \longrightarrow \mathrm{I} + \mathrm{V}$

• the death of a healthy cell:

$\mathrm{H} \longrightarrow$

• the death of a infected cell:

$\mathrm{I} \longrightarrow$

• the death of a virion:

$\mathrm{V} \longrightarrow$

Using a standard recipe which I explained, we can get from this chemical reaction network to some ‘rate equations’ saying how the number of healthy cells, infected cells and virions changes with time:

$\displaystyle{ \frac{d\mathrm{H}}{dt} = \alpha - \beta \mathrm{H}\mathrm{V} - \gamma \mathrm{H} }$

$\displaystyle{ \frac{d\mathrm{I}}{dt} = \beta \mathrm{H}\mathrm{V} - \delta \mathrm{I} }$

$\displaystyle{ \frac{d\mathrm{V}}{dt} = - \beta \mathrm{H}\mathrm{V} + \epsilon \mathrm{I} - \zeta \mathrm{V} }$

The Greek letters are constants called ‘rate constants’, and there’s one for each of the 6 reactions. The equations we get this way are exactly those described by Nowak and May!

What Andrei Korobeinikov is to unify the ideas behind the two models I’ve described here. Alas, I don’t have the energy to explain how. Indeed, I don’t even have the energy to explain what the models I’ve described actually predict. Sad, but true.

I don’t see anything online about Korobeinikov’s new work, but you can read some of his earlier work here:

• Andrei Korobeinikov, Global properties of basic virus dynamics models.

• Suzanne M. O’Regan, Thomas C. Kelly, Andrei Korobeinikov, Michael J. A. O’Callaghan and Alexei V. Pokrovskii, Lyapunov functions for SIR and SIRS epidemic models, Appl. Math. Lett. 23 (2010), 446-448.

The SIR and SIRS models are models of disease that also arise from chemical reaction networks. I explained them back in Network Theory (Part 3). That was before I introduced the terminology of chemical reaction networks… back then I was talking about ‘stochastic Petri nets’, which are an entirely equivalent formalism. Here’s the stochastic Petri net for the SIRS model:

Puzzle: Draw the stochastic Petri net for the HIV model discussed above. It should have 3 yellow circles and 6 aqua squares.

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