Linearisation of an epidemic model. Consider the following model for the spread of a disease, with susceptibles constantly renewed by births at a constant overall birth rate of μ: S^'=-βS I μ, I^'=βS I-γI (a) Show that the only equilibrium point is (γ/ β, μ/ γ). (b) Linearise the system and determine the behaviour of the trajectories near the equilibrium point. (c) Discuss, briefly, the implications for the time-dependent behaviour of both S(t) and I(t) . | Numerade
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(Epidemic model revisited) In Exercise 3.7.6, you analyzed the Kermack-McKendrick model of an epidemic by reducing it to a certain first-order system. In this problem you'll see how much easier the analysis becomes in the phase plane. As before, let x(t) ≥0 denote the size of the healthy population and y(t) ≥0 denote the size of the sick population. Then the model is ẋ=-k x y, ẏ=k x y-ℓy where k, ℓ>0 . (The cquation for z(t), the number of deaths, plays no role in the x, y dynamics so we omit
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4 one model for the spread of an epidemic that the rate of spread is jointly proportional to the number of infected people and the number of uninfected people in an isolated town of 5000 inh 43771
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