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One of the basic assumptions of the SIR model is that individuals who recover from the disease never get it again. However, diseases continually evolve, and new strains can emerge that can infect those who have recovered from the previous strain. In this exercise, we modify the SIR model so that recovered become susceptible again in a linear rate. We obtain the system of equations
(a) Show that the sum S(t) + I(t) + R(t) is constant as a function of t for this model.
(b) Derive a system in the two dependent variables S and I using the fact that R = 1 − (S + I) What are the equilibrium points for this model of the two variables S and I?
(Hint: Both S and I are nonnegative, and S(t) + I(t)
(d) Fix
(e) How does the system change if we fix
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Differential Equations
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