Question

22. Consider the general S-I-R model for a measles-like illness:

S′ = −aSI,
I′= aSI−bI,

R′ = bI.

a) The threshold level for S—below which the number of infected will only decline—can be expressed in terms of the transmission coefficient a and the recovery coefficient b. What is that expression?

b) Consider two illnesses with the same transmission coefficient a; assume they differ only in the length of time someone stays ill. Which one has the lower threshold level for S? Explain your reasoning.

Can someone please thoroughly explain how to reach b? This question was answered before but I could not understand how the answer was reached as the person didn't use any words as to how he reached the conclusions that he did. Thank you!

Answer 1

We have the general S-I-R model where S represents the susceptability,I represents infected and R represents the recovered persons

acc to the model,Rate at which the infected population grows is given by I'=aSI-bI

so I will decrease if I' decrease and I will increase if I' will increase

there are three steps for threshold level S

Ist step:Given the current value of S,I, and R we get the current S' I' and R' by using the rate equations

S'=-aSI, I'=aSI-bI, R'=bI

2nd step:now we find the changes S I R over the course of the day by using equations

S persons=S'(persons/day)one day

I persons=I'(persons/day)one day

Rpersons=R'(persons/day)one day

3rd step:given the currrent values of delta S,delta I,delta R we find the new S,I,R

new S= current S+S

new I=current I +I

new R=current R+R