Class-Age Structured Epidemic Models

Abstract

This chapter introduces and studies class-age structured models. These include models structured by age-since-infection, age-since-recovery, etc. The chapter first derives and studies an SIR model with age-since-infection and mass action incidence. A reproduction number is derived, and its threshold properties are explained. A time-since-recovery model is introduced. Destabilization of this model is studied, and oscillatory solutions are presented. Numerical methods for class-age structured models are introduced. Matlab code for simulations is given.

Appendices

Appendix

In this appendix we include the Matlab code that executes the numerical method in the section.

1  function [S, I] = sir3(M,N,dt)

2 

3 

4 

5 T = dt * N;

6 G = dt * M;

7 

8 Lambda = 1000/365;

9 mu = 1/(70*365);

10 alpha = 0.520548;

11 beta = 0.0000482876;

12 

13 S = zeros(N,1);

14 I = zeros(N,1);

15 rold = zeros(N,1);

16 rnew = zeros(N,1);

17 

18 S(1) = 7000;

19 I(1) = 30000;

20 

21 for i = 1:M

22     rold(i) = 10;

23 end

24 

25 t = 0:dt:T;

26 ttau = 0:dt:G;

27 

28 for n = 1:N

29 

30     S(n+1) = (Lambda * dt+S(n))/(1+beta * I(n) * dt + mu * dt);

31 

32     Int = 0.0;

33 

34     for i = 1:M

35 

36         tau = i * dt;

37 

38         if tau < 30

39 

40             q = 0.0;

41 

42         elseif   tau ≥ 30

43 

44             q = 1;

45 

46         end

47 

48      rnew(i+1) = rold(i)/(1 + q * beta * I(n) * dt + mu * dt);

49 

50      Int = Int + q * rnew(i+1) * dt;

51 

52     end

53 

54     I(n+1) = (I(n) + beta * S(n+1) * I(n) * dt + I(n) * dt * beta * Int)/(1 + (mu + alpha) * dt);

55 

56     rnew(1) = alpha * I(n+1);

57 

58 

59     for i = 1:M

60 

61         rold(i) = rnew(i);

62 

63     end

64 

65 end

66 

67 plot(t, I, ’−r’)

68 xlim([90000,91000])

69 

70 

71 end

Problems

13.1. Consider the SIR model with age-of-infection (13.4). Assume that the transmission rate and the recovery rate are given by the following functions:

$$ \displaystyle{ \gamma (\tau ) = \left \{\begin{array}{ll} 0&\qquad \tau \leq A\\ \gamma &\qquad \tau > A \end{array} \right. } $$

(13.49)

and β(τ) = β τ e ^{k τ}.

• (a) Compute the probability of survival in the infectious class: $\pi (\tau ) = e^{-\mu \tau }e^{-\int _{0}^{\tau }\gamma (s)ds }$.

• (b) Compute the reproduction number in terms of k and A.

• (c) Compute the endemic equilibrium in terms of k and A.

13.2. Backward Bifurcation in the Time-Since-Recovery Model

Consider Eq. (13.27) with γ(τ) = γ, a constant.

• (a) Show that if γ ≤ β and $\mathcal{R}_{0} > 1$, the equation (13.27) has a unique nonzero solution. Furthermore, show that if γ ≤ β and $\mathcal{R}_{0} < 1$, the equation (13.27) has no solutions.

• (b) Show that if γ > β and $\mathcal{R}_{0} < 1$, the equation (13.27) may have two solutions.

• (c) For α = 0. 05, μ = 1∕(365 ∗ 70), β = 0. 021, and γ = 0. 025, use a computer algebra system to draw the backward bifurcation diagram of I ^∗ with respect to $\mathcal{R}_{0}$.

13.3. Consider the model with time-since-recovery (13.23). Show that if $\mathcal{R}_{0} < 1$, the disease-free equilibrium is locally asymptotically stable. Furthermore, show that if $\mathcal{R}_{0} > 1$, the disease-free equilibrium is unstable.

13.4. HIV/AIDS Model

Consider the following model of HIV:

$$ \displaystyle{ \left \{\begin{array}{l} S'(t) =\varLambda -\frac{S(t)} {N(t)}\int _{0}^{\infty }\beta (\tau )i(\tau,t)\,d\tau -\mu S(t), \\ i_{\tau }(\tau,t) + i_{t}(\tau,t) = -\gamma (\tau )i(\tau,t) -\mu i(\tau,t), \\ i(0,t) = \frac{S(t)} {N(t)}\int _{0}^{\infty }\beta (\tau )i(\tau,t)\,d\tau, \end{array} \right. } $$

(13.50)

where S(t) are the susceptible individuals, i(τ, t) is the density of the infected individuals, N(t) is the total population size. We have to use standard incidence in HIV models. (Why?)

• (a) Compute $\mathcal{R}_{0}$ and the disease-free equilibrium. Show that if $\mathcal{R}_{0} < 1$, the disease-free equilibrium is locally stable and that otherwise, it is unstable.

• (b) Compute the endemic equilibrium.

• (c) Derive the characteristic equation of the endemic equilibrium.

• (d) Take β(τ) = τ e ^{−c τ}. Is the endemic equilibrium stable or unstable in this case?

13.5. HIV/AIDS Model

Consider the following model of HIV:

$$ \displaystyle{ \left \{\begin{array}{l} S'(t) =\varLambda -\frac{S(t)} {N(t)}\int _{0}^{\infty }\beta (\tau )i(\tau,t)\,d\tau -\mu S(t), \\ i_{\tau }(\tau,t) + i_{t}(\tau,t) = -\gamma (\tau )i(\tau,t) -\mu i(\tau,t), \\ i(0,t) = \frac{S(t)} {N(t)}\int _{0}^{\infty }\beta (\tau )i(\tau,t)\,d\tau, \end{array} \right. } $$

(13.51)

where S(t) are the susceptible individuals, i(τ, t) is the density of the infected individuals, N(t) is the total population size. Assume

$$ \displaystyle{\beta (\tau ) =\tau e^{-c\tau }.} $$

• (a) Compute $\mathcal{R}_{0}$ and the disease-free equilibrium.

• (b) Compute the endemic equilibrium.

• (c) Derive the characteristic equation of the endemic equilibrium.

• (d) Is the endemic equilibrium stable or can it become unstable in this case?

13.6. HIV/AIDS Model

Consider the following model of HIV:

$$ \displaystyle{ \left \{\begin{array}{l} S'(t) =\varLambda -\frac{S(t)} {N(t)}\int _{0}^{\infty }\beta (\tau )i(\tau,t)\,d\tau -\mu S(t), \\ i_{\tau }(\tau,t) + i_{t}(\tau,t) = -\gamma (\tau )i(\tau,t) -\mu i(\tau,t), \\ i(0,t) = \frac{S(t)} {N(t)}\int _{0}^{\infty }\beta (\tau )i(\tau,t)\,d\tau, \end{array} \right. } $$

(13.52)

where S(t) are the susceptible individuals, i(τ, t) is the density of the infected individuals, N(t) is the total population size.

• (a) Derive a numerical method for model (13.52).

• (b) Write a Matlab code to simulate the method. How do you know whether your code computes correctly? Compare the equilibrium computed by the code with the one that you computed in Problem 13.5.

13.7. Time-Since-Vaccination Model

Many vaccines wane, and the waning depends on the time elapsed since the individual was vaccinated. Consider the following model with time-since-vaccination τ:

$$ \displaystyle{ \left \{\begin{array}{l} S'(t) =\varLambda -\beta S(t)I(t) - (\mu +\psi )S(t) +\int _{ 0}^{\infty }\omega (\tau )v(\tau,t)\,d\tau, \\ I'(t) =\beta S(t)I(t) - (\mu +\alpha )I, \\ v_{\tau } + v_{t} = -\omega (\tau )v(\tau,t) -\mu v(\tau,t), \\ v(0,t) =\alpha I +\psi S,\end{array} \right. } $$

(13.53)

where v(τ, t) is the density of vaccinated individuals structured by the time-since-vaccination τ, and ψ is the vaccination rate.

• (a) Interpret all terms in the model. What is the assumed efficacy of the vaccine in this model?

• (b) Compute the reproduction number $\mathcal{R}_{0}(\psi )$.

• (c) Compute the disease-free equilibrium. Show that if $\mathcal{R}_{0}(\psi ) < 1$, the disease-free equilibrium is locally asymptotically stable; otherwise, the disease-free equilibrium is unstable.

• (d) Compute the endemic equilibrium.

13.8. Time-Since-Vaccination Model

Many vaccines wane, and the waning depends on the time elapsed since the individual was vaccinated. Consider the following model with time-since-vaccination τ:

$$ \displaystyle{ \left \{\begin{array}{l} S'(t) =\varLambda -\beta S(t)I(t) - (\mu +\psi )S(t) +\int _{ 0}^{\infty }\omega (\tau )v(\tau,t)\,d\tau, \\ I'(t) =\beta S(t)I(t) - (\mu +\alpha )I, \\ v_{\tau } + v_{t} = -\omega (\tau )v(\tau,t) -\mu v(\tau,t), \\ v(0,t) =\alpha I +\psi S,\end{array} \right. } $$

(13.54)

where v(τ, t) is the density of vaccinated individuals structured by the time-since-vaccination τ, and ψ is the vaccination rate.

• (a) Write a numerical method for the model above. Show that your numerical method preserves the positivity of solutions.

• (b) Write a Matlab code to simulate the model. How do you know whether your code computes correctly? Compare the equilibrium computed by the code with the one that you computed in Problem 13.7.

13.9. Time-Since-Infection Model of Vector-Borne Disease

Consider the following model of a vector-borne disease, structured by time-since-infection τ:

$$ \displaystyle{ \left \{\begin{array}{l} S_{v}' =\varLambda _{v} - S_{v}\int _{0}^{\infty }\beta _{ H}(\tau )i(\tau,t)d\tau -\mu _{v}S_{v}, \\ I_{v}' = S_{v}\int _{0}^{\infty }\beta _{ H}(\tau )i(\tau,t)d\tau -\mu _{v}I_{v}, \\ S_{H}' =\varLambda _{H} -\beta _{v}S_{H}I_{v} -\mu _{H}S_{H}, \\ i_{\tau } + i_{t} = -(\alpha _{H}(\tau ) +\mu _{H})i(\tau,t), \\ i(0,t) =\beta _{v}S_{H}I_{v}, \\ R_{H}' =\int _{ 0}^{\infty }\alpha _{ H}(\tau )i(\tau,t)d\tau -\mu _{H}R_{H},\end{array} \right. } $$

(13.55)

where S _v , I _v are the susceptible and infected vectors, S _H , i(τ, t), and R _H are the susceptible, infected, and recovered humans.

• (a) Compute the reproduction number $\mathcal{R}_{0}$.

• (b) Compute the disease-free equilibrium. Show that if $\mathcal{R}_{0} < 1$, the disease-free equilibrium is locally asymptotically stable; otherwise, it is unstable.

• (c) Compute the endemic equilibrium.

• (d) Derive the characteristic equation of the endemic equilibrium. Can you show local stability of the endemic equilibrium?