Zoonotic Disease, Avian Influenza, and Nonautonomous Models

Abstract

This chapter treats avain influenza (AI) as an example of zoonotic diseases. Simple AI model, involving poultry and humans, is introduced and fitted to H5N1 human case data. Tools from Chap. 6 are employed to evaluate multiple AI control strategies. The chapter argues that more detailed H5N1 data suggest that there is seasonality in the transmission of AI. Non-autonomous models as a modeling tool to seasonality are introduced. Tools for the analysis of non-autonomous models, such as the Poincare map, are discussed and their use is illustrated on a simple non-autonomous SI model. Simulations suggest that the simple non-autonomous model with disease-induced mortality is capable of exhibiting chaotic behavior.

Appendix

In this appendix, we include Matlab code that fits model (11.1)–(11.2) to the data in Table 11.1.

1 function Ch11fitting_model1

2

3 clear all

4 close all

5 clc

6

7 load AFluDatCumHalf14.txt %Imports the data file

8

9 tdata = AFluDatCumHalf14(:,1);

10 qdata = AFluDatCumHalf14(:,2);

11

12 tforward = (0:0.01:9)’;

13 tmeasure = [1:50:901]’;

14

15 format long

16

17 function dy = model_1(t,y,k)

18

19 Lb = 1200;

20 L = 1000;

21 mb = 1/2;

22 nb = 36.5;

23 mu = 1/65;

24 nu = 36.5;

25

26 dy = zeros(5,1);

27

28 dy(1) = Lb −k(1) * y(1) * y(2)−mb * y(1);

29 dy(2) = k(1) * y(1) * y(2)−(nb+mb) * y(2);

30 dy(3) = L − k(2) * y(3) * y(2)−mu * y(3);

31 dy(4) = k(2) * y(3) * y(2)−(mu+nu) * y(4);

32 dy(5) = k(2) * y(3) * y(2);

33

34 end

35

36 function q = model1(k,tdata)

37

38 [T,Y] = ode23s(@(t,y)(model_1(t,y,k)),tforward,

39 [2400 k(3) 65000 0.0007 .0007]);

40

41 q = Y(tmeasure(:),5);

42

43 end

44

45 k = [0.0158 0.00000001063 0.9]; % Initial values for parameters

46

47 lb = [0.0 0.0 0.0];

48

49

50

51 for i = 1:5

52

53 [k,resnorm] = lsqcurvefit(@model1,k,tdata,qdata,lb,[],...

54 optimset(’Disp’,’iter’,’TolX’,10̂(−20),’TolFun’,10̂(−20)))

55 end

56

57

58 [T,Y] = ode15s(@(t,y)(model_1(t,y,k)),tforward,[2400 ...

59 k(3) 65000 .0007 .0007]);

60

61

62

63 figure(1)

64

65 plot(tdata,qdata,’r.’);

66 hold on

67 plot(tforward,Y(:,5),’b−’);

68

69

70 end

71

72 end

Problems

11.1. For model (11.1)–(11.2), obtain a list of the countries that have human cases from the main source of data [166]. Use [58] to obtain the number of poultry units for these countries. Use [167] to determine the human population of the affected countries. Parameterize model (11.1)–(11.2) with these data.

11.2. Consider the AI model with pandemic strain [80]

$$\displaystyle{ \mathbf{DomesticBirds:}\left \{\begin{array}{l} S_{d}'(t) =\varLambda _{d} -\beta _{d}S_{d}I_{d} -\mu _{d}S_{d}, \\ I_{d}'(t) =\beta _{b}S_{d}I_{d} - (\mu _{d} +\nu _{d})I_{d}.\end{array} \right. }$$

(11.20)

The spillover model for humans with pandemic strain takes the form

$$\displaystyle{ \mathbf{Humans:}\left \{\begin{array}{l} S'(t) =\varLambda -\beta SI_{d} -\beta _{Z}SZ -\mu S, \\ I'(t) =\beta SI_{d} - (\mu +\nu +\rho )I, \\ Z'(t) =\rho I +\beta _{Z}SZ - (\mu +\nu _{Z})Z.\end{array} \right. }$$

(11.21)

where Z is the number of individuals infected by the pandemic strain.

• (a) Compute the reproduction numbers of the avian and the pandemic strains.

• (b) Fit the model to the data in Table 11.1. Take ν _z  = 36. 5. Estimate the reproduction numbers from the fit. The reproduction number of the pandemic strain should be between 1. 5 and 3.

• (c) Compute the invasion number of the pandemic strain.

• (d) Compute the elasticity of the pandemic invasion number with respect to I _d ^∗. Culling facilitates invasion, but how pronounced is that effect?

11.3. Consider the reproduction number

$$\displaystyle{\mathcal{R}_{0} = \frac{<\beta _{d} >\varLambda _{d}} {\mu _{d}^{2}}.}$$

Using the periodicity properties of $\sin$, simplify it as much as possible.

11.4. Show that model (11.14) has as a reproduction number

$$\displaystyle{\mathcal{R}_{0} = \frac{<\beta _{d} >\varLambda _{d}} {\mu _{d}^{2}}.}$$

11.5. Prove equality (11.16).

11.6. Let β(t), μ(t), and γ(t) be periodic with period T. Consider the model

$$\displaystyle{I'(t) =\beta (t)(1 - I(t))I(t) - (\mu (t) +\gamma (t))I(t).}$$

• (a) Compute the reproduction number of this model.

• (b) Show that $\mathcal{R}_{0}$ computed in (a) gives a threshold, that is, the DFE is locally asymptotically stable if $\mathcal{R}_{0} < 1$ and unstable if $\mathcal{R}_{0} > 1$.

• (c) Sow that the DFE is globally asymptotically stable if $\mathcal{R}_{0} < 1$.

11.7. Let β(t), μ(t), and γ(t) be periodic with period T. Consider the model (see Problem 11.6)

$$\displaystyle{I'(t) =\beta (t)(1 - I(t))I(t) - (\mu (t) +\gamma (t))I(t).}$$

• (a) Show that the model has a unique periodic solution if $\mathcal{R}_{0} > 1$.

• (b) Show that the periodic solution is globally asymptotically stable.

11.8. Let β(t), μ(t), and γ(t) be periodic with period T. Consider the two-strain model

$$\displaystyle{ \begin{array}{l} I_{1}'(t) =\beta _{1}(t)(1 - I_{1}(t) - I_{2}(t))I_{1}(t) - (\mu _{1}(t) +\gamma _{1}(t))I_{1}(t), \\ I_{2}'(t) =\beta _{2}(t)(1 - I_{1}(t) - I_{2}(t))I_{2}(t) - (\mu _{2}(t) +\gamma _{2}(t))I_{2}(t). \end{array} }$$

(11.22)

• (a) Define the reproduction number of each strain.

• (b) Let ξ ₁(t) and ξ ₂(t) be the periodic solutions of strain one alone and strain two alone. Define the invasion numbers of strain one and strain two.

Hint: You have to look at the stability of the solution (ξ ₁(t), 0). Define the Floquet exponent for strain two and read off the invasion number.