Question

A SIS disease spreads through a population of size K = 30, 000 individuals. The average time of recovery is 10 days and the infectious contact rate is 0.2 × 10^(−4) individuals^(−1) day^(−1) . (a) The disease has reached steady-state. How many individuals are infected with the disease? (b) What is the minimum percentage reduction in the infectious contact rate that is required to eliminate the disease? (c) By implementing a raft of measures it is proposed to reduce the value of the infectious contact rate to one percent of its initial value. Will this be sufficient to eliminate the disease within twenty-eight days? (d) Is it feasible to eliminate the disease within twenty-eight days solely by reducing the value of the pairwise contact rate? (e) How may days will the ‘raft of measures’ have to be maintained if we are to eliminate the disease?

Answer 1

According to the SIS model, the modelling equations are

where, is the susceptible population, is the infected population,

In steady state,

Therefore,

Therefore, either or , Given, Therefore, . Therefore, at steady state 1/4th of the population is susceptible. So, not everyone is infected.

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According to the SIS model, the modelling equations are

where, is the susceptible population, is the infected population,

Given, ,

(a) In steady state,

Therefore,

This means either the infected population goes to zero or,

Clearly, since , we approach a solution where the infected population reaches 0.

Therefore, there are no infected individuals.

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(b) Clearly, the steady state solution has I = 0. So no change is required.

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(c) The new contact rate is

We require that

Here is the MATLAB Code to simulate this system, in order to find if the proposal is feasible

SIS.m --

function dSIdt = SIS(~, SI)
    beta = 0.2e-6;
    gamma = 0.1;
    K = 30000;
    dSIdt = zeros(2, 1);
    dSIdt(1) = -(beta/K)*SI(2)*SI(1) + gamma*SI(2);
    dSIdt(2) = (beta/K)*SI(2)*SI(1) - gamma*SI(2);
end

solution.m --

init = [0 30000];
t = [0 28];
[T, SI] = ode45(@SIS, t, init);

% Plot Solution
plot(T, SI(:,1), 'b-'); % Plot S
hold on;
plot(T, SI(:,2), 'r-'); % Plot I
legend;

disp( SI(end, 2) );

We find that at the end of 28 days the population infected is

1824.3 Therefore, the measures are not suitable.

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(d) The effect of the contact rate is dominant only when there is a large enough susceptible population. Since 28 days is not enough of a recovery time for the susceptible population to have effect on the rate of change. It is not feasible to solely rely on reducing the contact rate.

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(e) Using numerical simulation for different values of time using the above MATLAB Code, we have

for t = [0 96], that SI(end, 2) = 0.92181.

Therefore, we require 96 days.