Epidemic spreading model

Question

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?

Answer 1

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.

Answer 2

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.)