### Logistic growth: More examples of logistic growth

### Epidemiology

We look at two mathematical models for the spread of an epidemic: the *SI* model and *SIS* model with constant coefficients. In both models, we consider the outbreak of an infectious disease in a population of individuals. We denote the number of infected individuals at time \(t\) as \(I(t)\) and the number of susceptible individuals as \(S(t)\). We assume in both models the following:

- The total number of individuals in the population is constant, say \(N\). In formulas, we may always use: \[S(t)+I(t)=N\]
- All susceptible individuals are equally susceptible to the infectious disease and all infected individuals may equally easily infect other susceptible individuals.
- The disease is spread through meetings between susceptible and infected individuals.
- The disease is transmitted instantaneously upon contact between individuals; there is no delay in time.

SI model for constant population In the *SI* model we assume that there is no birth, death or migration, and that every individual is either susceptible or infected. Assume that each individual per time interval of length \(\triangle t\) meets on average \(c\cdot \triangle t\) other individuals. Then during the time interval \((t,t+\triangle t)\) \(c\cdot I(t)\cdot \triangle t\) meetings take place between two individuals of which at least one individual is infected. The probability that the other person is not infected, is equal to the quotient of the number of susceptible individuals and the total population size, so \(S(t)/N\). Because \(S(t)=N-I(t)\) we can write this probability as \(\bigl(N-I(t)\bigr)/N\). The number of meetings between susceptible and infected individuals during the time interval \((t,t+\triangle t)\) is approximately equal to \[\frac{c}{N}\cdot I(t)\cdot \bigl(N-I(t)\bigr)\cdot \triangle t\] We now assume that the number of new infections, \(\triangle I(t)\), during the time interval \((t,t+\triangle t)\) is proportional to the number of meetings: \[\triangle I(t) =\alpha\cdot \frac{c}{N}\cdot I(t)\cdot \bigl(N-I(t)\bigr)\cdot \triangle t\] for some constant \(\alpha\). We rewrite this as \[\frac{\triangle I(t)}{\triangle t} =r\cdot I(t)\cdot \left(1-\frac{I(t)}{N}\right)\] for some constant \(r\). If we take the limit \(\triangle t\rightarrow 0\), we get the differential equation \[\frac{\dd I}{\dd t}=r\cdot I\cdot \left(1-\frac{I}{N}\right)\] So we have a logistic equation for the number of infected individuals with carrying capacity \(N\) .

*SIS* model for constant population In the *SI* model, every individual will eventually be infected in the long run, and will remain so. The *SIS* model takes into account that an infected individual recovers from the illness and will again be susceptible. We assume that the rate of recovery for infected individuals (that are then susceptible again) is proportional to the number of infected individuals. The reciprocal of this proportionality factor \(\gamma\) is equal to the average length of time an individual remains infected after infection, i.e., \(\gamma^{-1}=\text{the average duration of infection}\). The number of infected individuals \(I(t)\) then satisfies the differential equation \[\begin{aligned}\frac{\dd I}{\dd t} &=r\cdot I\cdot \left(1-\frac{I}{N}\right)-\gamma\cdot I\\ \\ &= (r-\gamma)\cdot I\cdot \left(1-\frac{I}{N\cdot\bigl(1-\frac{\gamma}{r}\bigr)}\right)\end{aligned}\] So we have again a logistic equation with carrying capacity \(N\cdot\bigl(1-\frac{\gamma}{r}\bigr)\) and intrinsic rate coefficient \(r-\gamma\) for the number of infected individuals. Dependent on the size of the parameter \(\gamma\), the number of infected individuals eventually reaches a maximum value, or this number decays to zero.