# Epidemic spreading model

I'm studying a model in the field of complex systems regarding the epidemic spreading. The model is the susceptible-infected model, i.e., there is a population of N subjects and each of them can either be infected or susceptible so that: $$S(t)+I(t)=N$$ and $$N$$ is the total population. The model is discrete in the sense that infections happen in discrete time-step. The model defines a rate $$\beta$$ as $$\textit{the average number of people each infectious person spreads the disease to each day}$$ and writes the differential equation:

$$\frac{dI}{dt}=\beta \frac{S(t)}{N}I(t)$$

I don't understand this equation, as far as I am concerned, I would write an equation like this:

$$I(t+1)=I(t)+\beta I(t)$$ which means that the number of infected people at time $$t+1$$ is equal to the number of people who were infected at time $$t$$ plus the average number of susceptible people who are infected during a timestep (recover from the disease is not included in the model), but my formula would lead to $$\frac{dI}{dt}=\beta I(t)$$ so I don't understand why in this model they also include $$\frac{S(t)}{N}$$. As $$\beta$$ is defined, isn't the number of susceptible people, in a certain way, already included in that rate? Why do I have to multiply by $$S(t)$$? Being the definition of $$\beta$$ I think that $$\beta I(t)$$ already gives the average number of people getting infected in a timestep, where am I wrong? How should a person approach to this kind of problems if he wants to write a differential equation?

## 2 Answers

You almost nailed it. Notice that $$I(t+1)-I(t) = \frac {dI}{dt}$$

Now, why they use $$\beta \alpha I(t)$$ instead of simply $$\beta I$$,- the reason is simple. Limiting reason is that infection each day can't spread to more people than there are uninfected/susceptible ones. Besides as pool of infected persons grows,- chances grows that typical infected person will make contact with already infected person, so $$\beta$$ must go down with time, due to these pair of reasons. This is accumulated in a $$\alpha = S/N$$ coefficient which decreases available pool of persons $$\alpha I$$ for infection in each step.

EDIT

To get impression how $$I(t)$$ chart looks like, you can try to integrate such expression :

$$\int \frac {dI}{I} = \int \beta \frac SN dt \tag 1$$

Knowing that $$S=N-I(t)$$, and assuming a-priory that $$I(t)$$ "should go" like $$e^{\beta t}$$ if boundaries would be ignored, we can re-write (1) as :

$$\int \frac {dI}{I} = \int \beta \left( 1- \frac {e^{\beta t}}{N} \right) dt \tag 2$$

Integrating both sides and solving for $$I(t)$$ we get :

$$I\left(t\right) = I_0 \exp\left(\beta t-\frac{\exp\left(\beta t\right)}{N}\right) \tag 3$$

which looks like

It may not be really like that, but I hope you get the reasons of model behind.

• Thank you for your answer, so the problem is the DE I propose has an exponential growth solution which is incompatible with a finite number of people; to take into account this last fact we can multiply by the coefficient $\alpha$. But still I can't figure out why we multiply exactly for $S/N$. Commented May 24 at 13:19
• Exactly. For the choosing $\alpha$ to be specifically S/N ratio, I think because it cancels exponential growth of infected population nicely on small S numbers. Commented May 24 at 16:08

Let me first note that SI model is a simplified version of a more general SIR model (in fact, there is a whole zoo of related models.)

The number of persons that get infected at time step $$t$$ is proportional to the number of susceptible persons $$S(t)$$ and the number of infected persons $$I(t)$$. Indeed, the probability that an infected person meets a susceptible person and transmits infection is the probability that one person is infected, $$I(t)/N$$ times the probability that the other person is susceptible $$S(t)/N$$. So the change is $$I(t+1) - I(t) \propto \frac{I(t)}{N}\frac{S(t)}{N}$$

$$\beta$$ is just a proportionality coefficient that assures correct rate, so that the results agree with the experimental data (e.g., in a densely populated city people run into each other more frequently than in countryside, which results in higher $$\beta$$. Or it may account for seasonal conditions, e.g., in winter people spend more time indoors and transmission of infection is more likely. It also accounts for the infectivity of the disease.)