I want to model the epidemic of 2019-nCov for my IB Maths IA(project) using the SEIRS Model. However, I encountered a difficulty in explaining why S*I=contact rate. I did some research and many papers just say it is so and do not explain the derivation or why it is so. Any help is appreciated!!!!!
MANY THANKS!!!
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We assume people make contact (in ways that could transmit virus) with other people randomly and at a constant rate. Each time there is such a contact, say between individuals $i$ and $j$, the probability that $i$ is susceptible and $j$ is infective is $S I$, where $S$ and $I$ are the proportions of susceptibles and infectives in the population. Similarly the probability that $i$ is infective and $j$ is susceptible. So the rate of transmission of the virus is some constant times $SI$.
Of course this is a rather crude model. A better (but more complicated) model might take into account the fact that there is not just one homogeneous population but a large numbers of sub-populations, where the rate of contact between individuals depends on which sub-populations they belong to.
The underlying assumptions are (1) the total population is constant and (2) the infected and susceptible populations are distributed evenly across space (homogenous mixing).
In this case, the correct form of the contact rate is $S\frac{I}{S+I}$, which represents the probability that a susceptible person comes in contact with an infected individual in a well-mixed population.
If the total population is constant, then $S+I = N$ (a constant). So the constant term is usually absorbed into the probability of infection term, for example, $\beta\frac{SI}{S+I} = \frac{\beta}{N}SI =:\hat\beta SI$. Just be careful with the unit.
A side note, if you include more types in your model (like $R$ or $E$), it still holds. For example, $N = S+I+R+E$. Then your probability of a susceptible person meeting an infected person is $\frac{SI}{S+I+R+E}$, so this gives the same thing: $\beta\frac{SI}{S+I+R+E} = \frac{\beta}{N}SI =:\hat\beta SI$