I'm currently simulating an epidemic using an agent-based simulation. However, as a part of my computer science bachelor thesis, I am expected to be able to explain the SIR compartmental model. It seemed simple enough, but I have several questions I was not able to find answers to. Please, keep in mind that I have very limited knowledge of differential equations.
Firstly, there were two types of equations, and none of the authors of either type acknowledged the other type. Also, every paper claims that the basic reproduction number $R_0$, is $\frac{\beta}{\gamma}$, but this number will obviously be different for the two sets of equations.
The two types are:
$$\frac{dS}{dt}=-\frac{\beta IS}{N},$$
$$\frac{dI}{dt}=\frac{\beta IS}{N}-\gamma I,$$
$$\frac{dR}{dt}=\gamma I.$$
and the second one
$$\frac{dS}{dt}=-\beta IS,$$
$$\frac{dI}{dt}=\beta IS-\gamma I,$$
$$\frac{dR}{dt}=\gamma I.$$
The only difference is that in the first set, the flow from Susceptible to Infectious is divided by the total number of all people. Which set is the correct one? What is the "unit" of the $\beta$ and $\gamma$ constants? I suppose my question is basically aimed at the way these three equations are exactly derived. However, my questions are:
1.) Why do we multiply I and S together? If there were, lets say, 100000 Susceptible and 1000 Infected, this number suddenly becomes really large – 100000000. I understand that the more people there are in susceptible and infected compartments, the faster the disease spreads, but why not, for example, sum the two? Additionally, multiplying this number by a parameter $\beta$ does not really make sense to me as it would need to be extremely small in order for the equations to be useful. It seems arbitrary and I was unable to find the derivation of this. Furthermore, all the literature I found on the subject claims that $\beta$ is "exposure rate", most claim it's a probability but there were no real definitions of what exactly it means. I, therefore, ask – what exactly does $-\frac{\beta IS}{N}$ mean, and how is it derived beyond "more people mean more infections"
2.) In the first set of equations, why do we divide only the flow from susceptible to infected, but not the flow from infected to recovered? Also, why does the literature claim that $\gamma$ is just $\frac{1}{\text{Duration of recovery}}$? Again, it seems arbitrary and I don't know why it works.
I will appreciate any and all help, and also if there are any articles which can explain the function of these equations step by step, I will be glad to read them.
Thank you.
Best Answer
$S$ multiplies $I$ because of the law of mass action in chemical kinetics. These equations are written as if the individuals in population are similar to the molecules in a well mixed reactor (which is of course very far from reality, but produces a set of manageable differential equations).
From a slightly more mechanical point of view, consider the following simple model of disease transmission: it can be transmitted only by a direct contact, and there is fixed probability of transmission per contact, call it $p$. Then, in a population with not that many infectious we get that the rate of transmission (i.e., $-\dot S$) is given by $$ p\cdot c(N)\cdot\frac{S}{N}\cdot I, $$ where $N$ is the size of population, $c(N)$ is the average number of contacts per time unit, $S/N$ is the probability to meet a susceptible, and $I$ is the number (or density) of infectious.
Now to get your first model, let $c(N)=k$, fixed number that does not depend on $N$ (think about sexually transmitted disease), and hence $\beta=p\cdot k$. For your second model you have $c(N)=kN$ (the law of mass action: the number of contacts is proportional to the total population size), and hence $\beta=pk$, but this is a different $\beta$ (with different units!).
$R_0$ for the first model is indeed $\beta/\gamma$, whereas for the second one is $\beta N/\gamma$.
To see the meaning of $\gamma$, consider a simple ODE $\dot x=-\gamma x$, which has the solution $x(t)=x_0e^{-\gamma t}$, which can be rewritten as $$ \frac{x_0-x(t)}{x_0}=1-e^{-\gamma t}, $$ which gives you the fraction (probability) of population at time $t$ that experiences something (decay, disease, etc), and on the right you should recognize the cumulative distribution function for exponential distribution with parameter $\gamma$. It is well known fact that the mean of this distribution is $1/\gamma$, and hence this value gives you the average timespan of whatever you model.
Probably the best source that explains mathematical models of disease transmission is Mathematical Tools for Understanding Infectious Disease Dynamics (Princeton Series in Theoretical and Computational Biology, 7) by Odo Diekmann (Author), Hans Heesterbeek (Author), Tom Britton (Author), but this book does require some good mathematical background.