Explain the full step solutions
Consider the SIR epidemic model with vaccination, given by the following set of assumptions:
1. The population is fixed at size N = S + I + R at all times.
2. Each time period, each infected individual causes S/N new infections of previously susceptible individuals.
3. Each time period, a fraction of infected individuals recover and become resistant.
4. Each time period, p fraction of susceptible individuals are vaccinated and become resistant.
5. Each time period, b fraction of all individuals die and are "reborn" as susceptible.
Furthermore, we assume that 0 < p < 1 and 0 < b < 1. Additionally, for this model, the basic reproduction number satisfies:
Ro = (b * S) / (p + b)
1. Write out a system of difference equations capturing the model given above:
S(t+1) = (1 - p) * S(t) - I(t) * S(t) / N + b * R(t) + I(t)
I(t+1) = N * R(t+1) = (1 - b) * R(t) * (1 - p) * S(t)
R(t+1) = N * I(t) + (1 - b) * R(t)
2. Show that if S0, I0, R0 > 0 and S0 + I0 + R0 = N, then solutions are positive for all time St, It, Rt > 0.
Use the fact that 0 < p < 1 to show that if St > 0, then St+1 > 0. Use the fact that 0 < b < 1 to show that if It > 0, then It+1 > 0. Use the fact that b < 1 to show that if Rt > 0, then Rt+1 > 0.
3. Express the three-dimensional SIR model as a two-dimensional SI model by making the substitution R(t) = N * (I(t) - S(t)):
S(t+1) = (1 - p) * S(t) - I(t) * S(t) / N + (1 - b) * I(t)
I(t+1) = N * I(t) + (1 - b) * R(t) * (1 - p) * S(t)
4. Show that this system has two equilibria (S, I), where either I = 0 or I > 0. Show that the positive equilibrium exists if and only if Ro > 1. Solve for the equilibria.
5. Show that the disease-free equilibrium I = 0 is locally asymptotically stable if Ro < 1. Use the eigenvalues of the Jacobian.