[A series of posts explaining a paper on the mathematical modeling of the spread of antiviral resistance. Links to other posts in the series by clicking tags, "Math model series" or "Antiviral model series" under Categories, left sidebar. Preliminary post here. Table of contents at end of this post.]
We are almost ready to begin a detailed examination of the mathematical model presented in the paper, “Antiviral resistance and the control of pandemic influenza,” by Lipsitch et al., published in PLoS Medicine. The main model is presented in the first four paragraphs of the Methods.
Some preliminaries on homogeneous population models:
The first paragraph tells us that the presentation is of a slightly simpler model than used for the numerical simulations reported in Results. The authors have made the code for the model freely available in a form which can be pasted into one of the commonly used software packages for solving systems of ordinary differential equations (Berkeley Madonna version 8.2). This enables others to check, modify and build on their work. This is commendable practice and we hope to see it more often.
The description of the basic model begins in paragraph 2 of Methods. Its essential elements are displayed in Figure 1 (page 3), which we will examine in the next post. Before we dig into it in earnest, we need to talk a little more about this kind of model, known as a homogeneous population model. The word “homogeneous” means “the same throughout.” In this case it refers to the fact that the population being modeled is assumed composed of essentially identical individuals, each of which has an equal chance of coming in contact with any other person in the population (this is also called random mixing). This is an idealized situation and in the numerical simulations that are given in Results the authors refine it by putting people into age groups, allowing different contact rates in and between the groups.
It turns out adding an age structure makes relatively little difference in this case. The random mixing assumption is commonly used and allows certain kinds of mathematical analysis to be done (it is also the kind of assumption non-modelers make when they envisage scenarios like, “what happens if a school child brings flu to her classroom?”). There are other kinds of models that examine how disease spread is affected by non-random contact patterns. In this model it is not one of the things being investigated, although we will discuss it again in the second last post. It is best to think of modeling research in the same way you would a laboratory experiment where certain questions are isolated and examined while other elements are controlled. Modelers also try to isolate certain key questions they want to answer with the model and they devise systems that keep other things, like contact structure, constant. So think of the homogeneous (random mixing) assumption as similar to using a mouse to answer questions about a human. The fit isn’t perfect (a mouse isn’t a human), but you hope the essential features are sufficiently close that inferences from your mouse experiment give you usable information about human health and disease. At some point you have to ask about the consequences of your assumptions. But you don’t do everything at once. Just as biologists gradually build up a picture of what is happening by doing many different experiments, using different techniques and species, modelers do the same thing with different kinds of models. They are looking for patterns and inferences they can make on the basis of their different experimental systems, just like wet bench researchers. The hope is that with continued experience, the pieces of the jigsaw puzzle will start to fit together in a coherent picture.
The homogeneous population model is extremely important for understanding mathematical modeling of this sort because it gives modelers a means to describe how people move from being susceptible to infection, to being infected, to dying or recovering. The idea behind this kind of analysis came originally from chemistry, where it was called the Law of Mass Action. I’ll explain it in that context first because I think it is easier to visualize.
The Law of Mass Action in chemistry:
Chemists are interested in how chemicals react with each other in a solution or in a mixture of gases, so they reasoned this way. If two molecules are going to react they need to be in the same place (technically the same small volume) at the same time. This is the chemist’s version of being “in contact.” Chemicals don’t react with each other from across the room. Consider the probability that a molecule will be at some place at a particular time. It should be unaffected by the probability another molecule will also be there. As they randomly bounce around in solution or the gas they are “unaware” of the other molecules except if they bump into each other. This “unawareness” is expressed by saying the probabilities of being in a particular place at a particular time are independent of each other. The probability of a combined event for two independent events is the product of the probabilities of the events considered separately. This is the same as saying the probability of tossing two heads in a row (each toss having an independent probability of 1/2) is 1/4 (1/2 times 1/2). You can see the validity of this for yourself if you think of the four equally likely outcomes of the two coin tosses: HT, TH, TT and HH. The HH (Heads Heads) combo is one fourth of the total possible outcomes. Thus the probability that two reacting molecules will be in the same small volume at the same time is equal to the probability one is in a particular small volume times the probability the other is, too.
Now comes the crucial step. The probability that some molecule of chemical A will be in a particular small volume at a particular time depends on how many molecules of chemical A are in the gas or the solution. If there are twice as many, the probability is twice as much. This makes some intuitive sense. In your town, whatever the random chance of any person being at the corner of Lake and Spruce Streets at noon on July 12, if you double the number of people in town you will roughly double the chances that someone will be walking by that corner at that time. The same is true for chemical B. If you double the number of B molecules, you will double the chance one of them will be in the same place at a particular time as one of the A molecules. Chemists therefore reasoned that how fast a chemical reaction occurs (the number of such simultaneous encounters per hour, say) is proportional to the product of the number of molecules in the reacting volume of each. That’s because we are multiplying probabilities. This was an educated guess about how the chemical world would work. They called this broad generalization, The Law of Mass Action. And it seems to be the case, an empirical fact, not a true “Law.” Its main justification in chemistry is that it works. It describes the world. It turns out that it can also be made to follow mathematically from additional assumptions about random mixing and small encounter probabilities.
The Law of Mass Action at work in epidemiology:
It was but a short leap from this analogy to the idea of a Law of Mass Action for epidemiology. Instead of two different chemicals being in contact we have two people being in contact, one infected and one susceptible. The number of new cases of influenza in a day or a week will depend on how many contacts between an infected and a susceptible person there are in a day or a week. This is converted into a mathematical notion by expressing the [per capita] rate of new infections as being proportional to the number of susceptibles times the number of infectives. If you double either one, the [per capita] number of new infectives in a day or a week will also double. Again, while this makes some intuitive sense, it is an assumption. As in the chemistry example it can be made to follow from the earlier assumptions that all people in the population are exchangeable with one another (all molecules of chemical A are the same) and randomly mixed (susceptibles and infecteds come in contact unaware of each other’s status and with equal chance). There are ways around these assumptions but we will not deal with them here so as to keep the explanation as clear as we can make it. We will return to the assumption itself in the second last post.
We are now ready to examine the model in detail and explain the figure. That will begin in the next post.
[Addendum/clarification: Ethan in the comments pointed out an ambiguity in the second last paragraph, which I have now clarified with bracketed insertions of [per capita]. The chemistry example should be clear, as is, and the epidemiological analog parallels it. Hat tip to Ethan for pointing out the ambiguity.]
Table of contents for posts in the series: