Transmission Dynamics of Tuberculosis with Age-specific Disease Progression

Abstract

Demographic structure and latent phenomenon are two essential factors determining the rate of tuberculosis transmission. However, only a few mathematical models considered age structure coupling with disease stages of infectious individuals. This paper develops a system of delay partial differential equations to model tuberculosis transmission in a heterogeneous population. The system considers demographic structure coupling with the continuous development of disease stage, which is crucial for studying how aging affects tuberculosis dynamics and disease progression. Here, we determine the basic reproduction number, and several numerical simulations are used to investigate the influence of various progression rates on tuberculosis dynamics. Our results support that the aging effect on the disease progression rate contributes to tuberculosis permanence.

Appendix

A. Numerical Scheme

We first define a function Z(t, a) as

$$\begin{aligned} Z(t,a)&= I(t,a,0)\\&= \int ^{\infty }_0\int ^{s_{\text {max}}}_0\int ^{a_{\text {max}}}_0 S(t-\tau ,a-\tau )I(t-\tau ,a',s)\\&\quad \times T(a-\tau ,a',s) \mathrm {d}a'\mathrm {d}s\alpha \mathrm {e}^{-\alpha \tau }\mathrm {e}^{-\int ^a_{a-\tau } \delta _S(\mu )\mathrm {d}\mu }\mathrm {d}\tau . \end{aligned}$$

We rewrite system (1) into the following system

$$\begin{aligned}&\frac{\partial S(t,a)}{\partial t}+\frac{\partial S(t,a)}{\partial a}\nonumber \\&\quad =-\int ^{s_{\text {max}}}_0\int ^{a_{\text {max}}}_0 S(t,a)I(t,a',s)T(a,a',s)\mathrm {d}a'\mathrm {d}s+\beta (t,a)\nonumber \\&\qquad -\delta _S(a)S(t,a) +\int ^{s_{\text {max}}}_0\gamma (a,s)I(t,a,s)\mathrm {d}s,\nonumber \\&\frac{\partial I(t,a,s)}{\partial t}+\frac{\partial I(t,a,s)}{\partial a} +v(a)\frac{\partial I(t,a,s)}{\partial s}\nonumber \\&\quad =-\delta _I(a,s)I(t,a,s)-\gamma (a,s)I(t,a,s),\nonumber \\&\frac{\partial Z(t,a)}{\partial t}+\frac{\partial Z(t,a)}{\partial a}\nonumber \\&\quad = \alpha \int ^{s_{\text {max}}}_0\int ^{a_{\text {max}}}_0 S(t,a)I(t,a',s)T(a,a',s)\mathrm {d}a'\mathrm {d}s-(\alpha +\delta _S(a))Z(t,a).\nonumber \\ \end{aligned}$$

(A.10)

The initial and boundary conditions are

$$\begin{aligned} S(0,a)&= S_0(a),\ \ I(0,a,s) = I_0(a,s),\ \ Z(0,a) = Z_0(a),\\ S(t,0)&= \beta (t),\ \ I(t,0,s) = 0,\ \ I(t,a,0) = Z(t,a),\ \ Z(t,0) = 0. \end{aligned}$$

We first discretize the space of time, age and stage into mesh as followed

$$\begin{aligned} t_i&= i\varDelta t,\ \ \ i = 0,1,2,\dots , N,\\ a_j&= j\varDelta a,\ \ \ j = 0,1,2,\dots , L_1,\\ s_k&= k\varDelta s,\ \ \ k = 0,1,2,\dots , L_2, \end{aligned}$$

where $\varDelta t, \varDelta a, \varDelta s$ are the step sizes of the time, age and stage, respectively. We suppose that all the age densities distribute in the interval $[0,a_{\max }]$ and all the stage densities distribute in the interval $[0,s_{\max }]$ and $t\in [0,T]$, where T is the time span of the prediction period for the disease transmission. We have the following equalities for $N, L_1, L_2$

$$\begin{aligned} N = \frac{T}{\varDelta t},\ \ \ L_1 = \frac{a_{\max }}{\varDelta a}, \ \ \ L_2 = \frac{s_{\max }}{\varDelta s}, \end{aligned}$$

that is, $N, L_1, L_2$ are the number of steps from 0 to T, $a_{\max }$ and $s_{\max }$ separately.

Here, we simulate the dynamics of system in the population with stationary structure and hence we suppose that the immigration rate and the birth rate are only related with age in the following simulations, that is, $\beta (t,a) = \beta (a)$. Using i, j, k represent $t_i, a_j,s_k$ for simplification, we have the following identities for the boundary conditions

$$\begin{aligned} S^i_0&= S(t_i,0) = \beta _0 = S(t,0),\\ I^i_{0,k}&= I(t_i,0,s_k) = 0 = I(t,0,s),\\ I^i_{j,0}&= I(t_i,a_j,0) = Z(t_i,a_j) = I(t,a,0),\\ Z^i_0&=Z(t_i,0) = 0 = Z(t,0),\ i = 0,1,2\dots ,N, \ j = 0,1,2,\dots ,L_1,\\ k&= 0,1,2\dots ,L_2, \end{aligned}$$

where $\beta _0$ is a positive constant representing the birth rate of the population. On the basis of the finite difference method for derivative and the trapezoidal rule for approximating integral, we discretize system (A.10) as follows.

$$\begin{aligned} \frac{S^{i+1}_j-S^i_j}{\varDelta t}+\frac{S^i_j-S^i_{j-1}}{\varDelta a}= & {} -S^i_jG^i_j+\beta _j-{\delta _S}_jS^i_j+H^i_j,\nonumber \\ \end{aligned}$$

(A.11)

$$\begin{aligned} \frac{I^{i+1}_{j,k}-I^i_{j,k}}{\varDelta t}+\frac{I^i_{j,k}-I^i_{j-1,k}}{\varDelta a}+v_j\frac{I^i_{j,k}-I^i_{j,k-1}}{\varDelta s}= & {} -{\delta _I}_{j,k}I^i_{j,k}-\gamma _{j,k}I^i_{j,k}, \end{aligned}$$

(A.12)

$$\begin{aligned} \frac{Z^{i+1}_j-Z^i_j}{\varDelta t}+\frac{Z^i_j-Z^i_{j-1}}{\varDelta a}= & {} \alpha S^i_jG^i_j-(\alpha +{\delta _S}_j)Z^i_j. \end{aligned}$$

(A.13)

Here,

$$\begin{aligned} G^i_j&= \frac{1}{4}\varDelta a\varDelta s\bigg [I^i_{0,0}T_{0,0,j}+ I^i_{0,L_2}T_{0,L_2,j}+2\sum \limits _{k=1}^{L_2-1}I^i_{0,k}T_{0,k,j}\bigg ]\\&\quad +\frac{1}{4}\varDelta a\varDelta s\bigg [I^i_{L_1,0}T_{L_1,0,j}+ I^i_{L_1,L_2}T_{L_1,L_2,j}+2\sum \limits _{k=1}^{L_2-1}I^i_{L_1,k}T_{L_1,k,j}\bigg ]\\&\quad +\frac{1}{2}\varDelta a\varDelta s\bigg [\sum \limits _{j_1 = 1}^{L_1-1}I^i_{j_1,0}T_{j_1,0,j} +\sum \limits _{j_1 = 1}^{L_1-1}I^i_{j_1,L_2}T_{j_1,L_2,j} +2\sum \limits _{k = 1}^{L_2-1}\sum \limits _{j_1 = 1}^{L_1-1}I^i_{j_1,k} T_{j_1,k,j}\bigg ],\\ H^i_j&= \frac{1}{2}\varDelta s\bigg [\gamma _{j,0}I^i_{j,0}+\gamma _{j,L_2} I^i_{j,L_2}+2\sum \limits _{k = 1}^{L_2-1}\gamma _{j,k}I^i_{j,k}\bigg ]. \end{aligned}$$

For $i = 0$, we utilize the initial epidemic data

$$\begin{aligned} S^0_j&= S(0,a_j) = S(0,a),\\ I^0_{j,k}&= I(0,a_j,s_k) = I(0,a,s),\\ Z^0_j&= Z(0,a_j) = Z(0,a),\ j=0,1,2,\dots ,L_1,\ k=0,1,2,\dots ,L_2. \end{aligned}$$

For $i = 1$, we do the following

• 1. Calculating $I^1_{j,0}$ for all j with

  $$\begin{aligned} I^1_{j,0} = S^0_jG^0_j, \end{aligned}$$

  and

  $$\begin{aligned} Z^1_j = I^1_{j,0}. \end{aligned}$$

• 2. Calculating $S^1_j$ for all j with formula (A.11) and $I^1_{j,k}$ for all k with formula (A.12).

For $2\le i\le N$, we do the following

• 1. Calculating $S^i_j$, $I^i_{j,k}$ with the formula in (A.11) and (A.12) separately.

• 2. Calculating $Z^i_j$ with formula (A.13) and then replacing $I^i_{j,0}$ with $Z^i_j$, that is, $I^i_{j,0} = Z^i_j$ for the boundary condition of infectious individuals.

In our numerical simulations, we set $\varDelta t = 0.1$ and $\varDelta a =\varDelta s = 0.01$.