Answer to Question #209284 in Differential Equations for Amit

Question

The differential equations dS/dt = -bSI+aS

dI/dt = bSI-cI

model a disease spread by contact, where S is the number of susceptibles, I is the number

of infectives, b is the contact rate, c is the removal rate and a is the birth rate of

susceptibles.

(i) Identify which term in the RHS of each differential equation arises from the birth of

susceptibles.

(ii) Discuss the model given by the above two differential equations.

Answer 1

i) For the first differential equation, the term aS is the one that has to do with the birth of susceptibles. While the second differential equation does not have such a term. In other words it is zero in the second differential equation.

ii) $\dfrac{dS}{dt} = -bSI +aS$

This differential equation explains that the rate of change of the susceptible population with respect to time is dependent on the increase in the birth of susceptibles (aS) and the decreases in the rate of susceptibles becoming infected (-bSI)

$\dfrac{dI}{dt} = bSI -cI$

This differential equation explains that the rate of change of the infected with respect to time is dependent on the increase in the number of susceptibles becoming infected (bSI) and decrease in the infected that are being cured or removed from the process (-cI)

And we also have that

$\dfrac{d(S+I)}{dt} = aS-cI$

Equilibrium will be acheived if the above is equal to zero i.e., aS=cI, this implies that the population of those that are susceptible by birth is equal to the population of those that the infection is removed from them.