Question

Consider an epidemic that moves through an isolated population. We will make the following assumptions about the epidemic. - Individuals are infected at a rate proportional to the product of the number of infected and susceptible individuals. We assume that the constant of proportionality is α. - The length of the incubation period is negligible, and infectious individuals are immediately infectious. - On the average, an infected individual dies after 10 days. - Only a single individual is initially ill. - Infected individuals do not give birth, but susceptible individuals have a birth rate of 0.0003 per individual per year. Newborns are susceptible. If S(t) is the number of susceptible and I(t) is the number of infected people, then

ds/dt =−αSI+0.0003S
di/dt = αSI−0.1I ​
The constant α is a measure of the relative infectivity of the disease. Some diseases such as Ebola, a viral hemorrhagic fever, are extremely infectious with a mortaility rate of up to 90%. On the other hand, AIDS which is caused by the HIV virus, has a much lower transmission rate. The goal of this exercise is to examine the differences between the two. a. If α=0.05, draw the phase portrait. Be sure to label all nullclines and equilibrium solutions. Suppose that S(0)=1000 and I(0)=1. What happens to the solution curve as t→[infinity] ? b. If α=0.000001, draw the phase portrait. Be sure to label all nullclines and equilibrium solutions. Suppose that S(0)=30000 and I(0)=1. What happens to the solution curve as t→[infinity] ? c. What conclusions can you draw about the behavior of the two different epidemics?

Answer 1

The behavior of the two epidemics is different depending on the value of α. For α=0.05, the epidemic is relatively small with a maximum number of infected individuals of 1000000. On the other hand, for α=0.000001, the epidemic is much larger with a maximum number of infected individuals of 30000000.

First, let's analyze the equation that represents the rate of change of susceptible individuals, ds/dt = −αSI+0.0003S. The term -αSI represents the rate at which susceptible individuals become infected, and the term 0.0003S represents the rate at which susceptible individuals are born. Thus, this equation takes into account both the infection rate and the birth rate of susceptible individuals.

Next, let's analyze the equation that represents the rate of change of infected individuals, di/dt = αSI−0.1I. The term αSI represents the rate at which susceptible individuals become infected, and the term -0.1I represents the rate at which infected individuals die. Thus, this equation takes into account both the infection rate and the mortality rate of infected individuals.

Now, let's consider the behavior of the epidemics for different values of α.

a. For α=0.05, the phase portrait can be plotted by finding the nullclines and equilibrium solutions. The nullcline for ds/dt is given by ds/dt=0, which is S=0 or I=1000000. The nullcline for di/dt is given by di/dt=0, which is S=0 or I=0. The equilibrium solutions can be found by setting both ds/dt and di/dt to 0, which gives S=0 and I=1000000. Thus, the phase portrait consists of two equilibrium points, one at (0, 1000000) and the other at (0, 0).

When S(0)=1000 and I(0)=1, the solution curve will initially move away from the equilibrium point at (0, 0) and towards the equilibrium point at (0, 1000000). As t→∞, the solution curve will approach the equilibrium point at (0, 1000000). This means that the number of susceptible individuals will decrease to 0, while the number of infected individuals will increase to 1000000, resulting in an epidemic.

b. For α=0.000001, the phase portrait can be plotted in a similar manner as in part a. The nullcline for ds/dt is given by ds/dt=0, which is S=0 or I=30000000. The nullcline for di/dt is given by di/dt=0, which is S=0 or I=0. The equilibrium solutions can be found by setting both ds/dt and di/dt to 0, which gives S=0 and I=30000000. Thus, the phase portrait consists of two equilibrium points, one at (0, 30000000) and the other at (0, 0).

When S(0)=30000 and I(0)=1, the solution curve will initially move away from the equilibrium point at (0, 0) and towards the equilibrium point at (0, 30000000). As t→∞, the solution curve will approach the equilibrium point at (0, 30000000). This means that the number of susceptible individuals will decrease to 0, while the number of infected individuals will increase to 30000000, resulting in a much larger epidemic compared to the previous case.

c. From the analysis of the phase portraits, we can conclude that the behavior of the two epidemics is different depending on the value of α. For α=0.05, the epidemic is relatively small with a maximum number of infected individuals of 1000000. On the other hand, for α=0.000001, the epidemic is much larger with a maximum number of infected individuals of 30000000. This shows that the value of α, which represents the relative infectivity of the disease, has a significant impact on the behavior and severity of the epidemic.