Question

.Consider the following continuous-time model of the dynamics of an epidemic system made of a growing susceptible population (x) and an infected population(y). Populations are nonnegative (i.e., x≥0, y≥0),and all the parameters are positive (i.e.,a> 0, b> 0, c> 0). dx/dt = ax ( 1 - ( x + y ) ) - b x y + c y dy/dt = bxy - cy

It is assumed that(i)the susceptible population grows until the total population (x+ y) reaches1(= the carrying capacity of the environment), (ii)the infection of the disease changes susceptible individuals into infected ones, and (iii)the infected individuals recover and become susceptible again at a certain rate.

Answer the following.

1.Explain how each of the above three assumptions were represented in the equations. (5x3 = 15points)

2.Find the equilibrium points. There are three such points. (5x3 = 15points)

3.Calculate the Jacobian matrix of the model. Keep a, b and c as symbols. (10 points)

4.Conduct linear stability analysis for each equilibrium point and discuss the conditions under which each equilibrium point is stable/unstable. (5x3 = 15points)

5.Identify the critical condition of parameter values at which a bifurcation occurs.Note that all the parameters are positive.(5points)

6.Draw the phase spaces of this model using Python for several different parameter values to confirm the prediction of bifurcation derived in the previous question.(10points)

Answer 1