Question

Question 4: The SIR disease model tries to predict the sizes of three subgroups--Susceptible, Infected, and Recovered--of a population of constant size N subjected to a disease pathogen. The model is:

dS/dT=-pbST

dl/dt=pbSI-kI

dR/dT=kI

b,k>0]

0 1

Where S, I, and R are the sizes of the three subgroup populations at time t,b is a parameter that quantifies the rate at which encounters among population members occur, p is the probability of transmission in an encounter between an infected and a suceptible individual, and k is the recovery rate parameter.

(D) Does the model clearly limit how large R can be? What do you expect will happen to I after a long time?

(F) Suppose that scientists discover that the immunity to a disease is lost at a rate proportional to the recovered population with rate q. Rewrite the SIR model equations to include this additional mechanistic process.

(G) Suppose that the original SIR model is changed by including vaccination of susceptible at rate vS, where v is the vaccination rate constant. Rewrite the model equations for this revised conceptual model.

t stands for time.

Please help with bolded--questions D, F, and G. Thank you so much!

Answer 1

(D) The model clearly limits the value of R to be equal to size of population = N and the value of I after long run will become zero. Its mean when R=N then I=0

(F) Immunity lose will affect the rate of change of susceptible and thus the rate of change of S with time will have added factor qR

dS/dT = -pbST + qR

dl/dt = pbSI-kI

dR/dT = kI

b,k > 0

(G) The equations after taking vaccination in to account will be as follows:

dS/dT = -pbST - vS

dl/dt = pbSI-kI

dR/dT = kI

b,k > 0