Question

Some relatively mild illnesses, like the common cold, return to infect you again and again. For a while, right after you recover from a cold, you are immune. But that doesn’t last; after some weeks or months, depending on the illness, you become susceptible again. This means there is now a flow from the recovered population to the susceptible. These exercises ask you to modify the basic S-I-R model to describe an illness where immunity is temporary.

c. We saw in the text that the model for an illness that confers permanent immunity has a threshold value for S in the sense that when S is above the threshold, I increases, but when it is below, I decreases. Does this model have the same feature? If so, what is the threshold value?

Answer 1

The SIR model is considered as an ideological model which calcuates the hypothetical count consisting of the number of people who are infected communicable disease in an exclusive population over a certain period of time. If people bounce back in terms of health with permanent immunity, the basic model is SIR model. If people bounce back in terms of health with temporary immunity so that over the course of time they become vulnerable again, then the basic model is SIRS model. If people do not recover at all, then the basic model is an SI model.

There are time delays present in the establishment of the SIR model with a non linear proportion function. The dynamical nature of the model is taken into consideration and the basic reproductive number R is proposed. For a certain value of time lag T, oscillations takes place which deestablish the system, and periodic solutions take place whcih emerges due to hopf bifurcation. The conditions have been studied in terms of threshold parameter R, that insured the asymptotic determination of infection free and native symmetry.

It has been acclaimed that when R<1, the infection fades out and when R>1, the infection becomes endemic and remains constant within the population.

In SIR models, the people in the bounce back recovery state achieve complete immunity to the pathogens whereas in SIRS model, the immunity is achieved over the period of time and the people are susceptible to getting reinfected.