Answer to Question #163082 in Differential Equations for Ajay

Question

The differential equations

dS/dt=−βSI + λS

dI/dt= βSI + γI

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

of infectives, β is the contact rate, γ is the removal rate and λ 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) First the first equation $\gamma I$ gives rise to the susceptibles. Since this is an SIS model, instead of it to leads to removal, it return back to susceptibles. Also in the second equation.

(II) This is an is an SIS model in which infectives

return to the susceptible class on recovery because the disease confers no immu-

nity against reinfection. There is a continuing flow of new susceptibles, namely recovered infectives. This differs from the SIR model only in that the recovered members return to the

class S at a rate γI instead of passing to the class R.

The population N=S+I is always constant.