Question

Consider a disease with an exposed class that is also infectious. Individuals move from the susceptible to the exposed class at rate βSI + αSE, where β is the transmission rate of the I class and α is the transmission rate of the E class. Individuals from the E class move into the I class at rate δE. Individuals in the I class recover at rate γI. We will assume that removed individuals are immune and that the total population size is constant.

(a) Draw the compartmental diagram corresponding to the disease described above.

(b) Write down the system of ODEs corresponding to the compartmental diagram.

(c) Nondimensionalise the model by converting the S, E, I, and R subpopulations into proportions of the total population, N. Use the recovery rate to scale time.

(d) Argue that the equation for r can be ignored in the analysis of the solutions, reducing the problem to a third-order system.

(e) Write down the Jacobian for the third-order system at any point (s, e, i).

(f) Show that (1,0,0) is a disease free steady state.

(g) Determine the stability of the disease free steady state. What is the basic reproductive number? What is the meaning of this number?

(h) How would you determine the final size and peak size of the epidemic? Give a detailed explanation.

Answer 1

Answer to above question

a) Compartmental diagram of a disease     

SUSCEPTIBLE    ------------ EXPOSED---------- INFECTIOUS-----     RECOVERED--------------SUSCEPTIBLE                   INFECTIOUS (I=0) --------------------------- INFECTIOUS (I=1) EXPOSED (E=0) ----------------------------------------   EXPOSED(E=1)

b) system of ODEs for above diagram

dS/dt= /N

dE/dt= +/N

dI/dt= /N

dR/dt=/N

c) nondimensionalising above model

dR/dT= dE /dN - dS/dN   - dI/dN

d) if dR/dt=0 then above equation changes to

dE/dN -dS/dN-dI/dN=0

dE/dN = dS/N +dI/dN

It is a third order reaction.