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The following provides an introduction in the modeling of infectious diseases.
When a disease breaks out in a group of humans, it would be of great interest if we could predict the amount of infected people throughout the outbreak. The more information we can predict, the better the government can react to limit the disease’s impact. Predicting the flow of an infectious disease is one of the tasks which concerns the research.
Suppose first we have a healthy population of 100 humans to which we add 1 infected person, named Bob. The healthy persons can be infected by Bob. We denote the class of healthy, but susceptible, people by $latex S $. In this case $S = 100$. In the same way, we denote the class of infected people by $I$. Similar, this gives $I=1$. Suppose now that Bob infects 5 new people every day. We can model this by the set of equations
\[I_{t+1} = I_t +5t\]
\[S_{t+1} = S_{t} – 5t \]
$S_t$ now represents the amount of healthy people left after $t$ days, similar for $I_t$.
Social contacts and infection spread
This isn’t really realistic. It is wrong to assume that each and every day 5 new infections takes place. To create a new infected human there has to be contact between two persons: one from the $S$ class and another from the $I$ class. In our case Bob can make contact to any of the 100 people in the $S$ class. So there are 100 possible contacts which may or may not cause the infection to be transmitted. Consider now the case in which there is a human Alice infected by Bob. We then have $S=99$ and $I=2$. Now Bob can make contact to the 99 remaining people in the $S$-class, but so can Alice. This gives 198 possible contacts. More generally the amount of contacts is given by the product $SI$. Not every contact makes a new infection, but we may assume each day there will be new infections proportional to the amount of contacts $SI$. We model this by a constant $b$. The model becomes
$$ S_{t+1} = S_{t} – b S_{t} I_{t} $$
$$ I_{t+1} = I_t +b S_{t} I_{t} $$
In many cases, people can recover from there infection and become immune for any further ones in the future. This makes a new class denoted by $R$. Each day, some infected people get cured. The amount of which this happens is proportional to $I$. By introducing the constant $g$, we can state that $R$ increases each day with $gI$. However a gain in $R$ gives an equal decrease in the $I$ class. Put all together, we constructed the SIR model.
$$ S_{t+1} = S_{t} – b S_{t} I_{t} $$
$$ I_{t+1} = I_t +b S_{t} I_{t}-gI_{t} $$
$$R_{t+1} = R_t+gI_{t}$$
To get familiar with the numerical simulation of such models, we present a short simulation of the SIR model with fictitious parameters.
Let’s further discuss the SIR model. First of all the model is deterministic, this means that given the initial conditions it always gives the same results.
This can be improved for the aim of our purposes. It depends on a lot of factors for a contact to transmit the disease. We have to include some randomness in this process. Instead of saying that after each day $bSI$ new infections takes place, we take a number from a probability distribution with mean $bSI$ and let that number be the amount of newly created infections, analog for the number of people recovering. We did this for the Poisson and the binomial distribution. We present two simulations which models the SIR-equations with the above considerations.
Each simulation starts with a part modeling the SIR model and finishes with a part modeling the SIRS model. The SIRS model is an extension of the SIR model in which people can lose their immunity, resulting in a migration from the $R$-class to the $S$-class.