# 两地区人口流动肺结核动力学模型研究Dynamic Model of Tuberculosis in Population Migration between Two Regions

DOI: 10.12677/AAM.2019.83061, PDF, 下载: 402  浏览: 605  国家自然科学基金支持

Abstract: To study the effects of population migration and vaccination on the spread of tuberculosis, a dy-namic SEIR epidemic model of the two regions with different population densities is established. The effective reproduction number is calculated, and the vaccination strategy for each region is considered in model. We numerically simulate the sensitivity of the effective reproduction number with respect to migration rate and vaccination rate. The results show that migration from areas with high population density to areas with low population density can reduce the effective re-production number and disease scale. The migration from low-density areas to high-density areas has the opposite result, but they cannot achieve the purpose of eliminating tuberculosis through migration. Appropriate application of vaccines to people in the high-density area can eliminate tuberculosis.

1. 引言

2. 模型建立及再生数的计算

$\begin{array}{l}\frac{\text{d}{S}_{11}}{\text{d}t}={A}_{11}-{\sigma }_{12}{S}_{11}+{\rho }_{12}{S}_{12}-{S}_{11}\underset{j=1}{\overset{2}{\sum }}{k}_{1}{\beta }_{11j}\frac{{I}_{j1}}{{N}_{1}^{t}}-{\chi }_{11}{S}_{11}-d{S}_{11}\\ \frac{\text{d}{S}_{12}}{\text{d}t}={A}_{12}+{\sigma }_{12}{S}_{11}-{\rho }_{12}{S}_{12}-{S}_{12}\underset{j=1}{\overset{2}{\sum }}{k}_{2}{\beta }_{12j}\frac{{I}_{j2}}{{N}_{2}^{t}}-{\chi }_{12}{S}_{12}-d{S}_{12}\\ \frac{\text{d}{S}_{22}}{\text{d}t}={A}_{22}-{\sigma }_{21}{S}_{22}+{\rho }_{21}{S}_{21}-{S}_{22}\underset{j=1}{\overset{2}{\sum }}{k}_{1}{\beta }_{22j}\frac{{I}_{j2}}{{N}_{2}^{t}}-{\chi }_{22}{S}_{22}-d{S}_{22}\\ \frac{\text{d}{S}_{21}}{\text{d}t}={A}_{21}+{\sigma }_{21}{S}_{22}-{\rho }_{21}{S}_{21}-{S}_{21}\underset{j=1}{\overset{2}{\sum }}{k}_{1}{\beta }_{21j}\frac{{I}_{j1}}{{N}_{1}^{t}}-{\chi }_{21}{S}_{21}-d{S}_{21}\end{array}$

$\begin{array}{l}\frac{\text{d}{E}_{11}}{\text{d}t}={S}_{11}\underset{j=1}{\overset{2}{\sum }}{k}_{1}{\beta }_{11j}\frac{{I}_{j1}}{{N}_{1}^{t}}-{\sigma }_{12}{E}_{11}+{\rho }_{12}{E}_{12}-\left(\alpha +d\right){E}_{11}\\ \frac{\text{d}{E}_{12}}{\text{d}t}={S}_{12}\underset{j=1}{\overset{2}{\sum }}{k}_{2}{\beta }_{12j}\frac{{I}_{j2}}{{N}_{2}^{t}}+{\sigma }_{12}{E}_{11}-{\rho }_{12}{E}_{12}-\left(\alpha +d\right){E}_{12}\\ \frac{\text{d}{E}_{22}}{\text{d}t}={S}_{22}\underset{j=1}{\overset{2}{\sum }}{k}_{1}{\beta }_{22j}\frac{{I}_{j2}}{{N}_{2}^{t}}-{\sigma }_{21}{E}_{22}+{\rho }_{21}{E}_{21}-\left(\alpha +d\right){E}_{22}\\ \frac{\text{d}{E}_{21}}{\text{d}t}={S}_{21}\underset{j=1}{\overset{2}{\sum }}{k}_{1}{\beta }_{21j}\frac{{I}_{j1}}{{N}_{1}^{t}}+{\sigma }_{21}{E}_{22}-{\rho }_{21}{E}_{21}-\left(\alpha +d\right){E}_{21}\end{array}$

$\begin{array}{l}\frac{\text{d}{I}_{11}}{\text{d}t}=\alpha {E}_{11}-{\sigma }_{12}{I}_{11}+{\rho }_{12}{I}_{12}-\left(\gamma +d\right){I}_{11}\\ \frac{\text{d}{I}_{12}}{\text{d}t}=\alpha {E}_{12}+{\sigma }_{12}{I}_{11}-{\rho }_{12}{I}_{12}-\left(\gamma +d\right){I}_{12}\\ \frac{\text{d}{I}_{22}}{\text{d}t}=\alpha {E}_{22}-{\sigma }_{21}{I}_{22}+{\rho }_{21}{I}_{21}-\left(\gamma +d\right){I}_{22}\\ \frac{\text{d}{I}_{21}}{\text{d}t}=\alpha {E}_{21}+{\sigma }_{21}{I}_{22}-{\rho }_{21}{I}_{21}-\left(\gamma +d\right){I}_{21}\end{array}$

$\begin{array}{l}\frac{\text{d}{R}_{11}}{\text{d}t}=\gamma {I}_{11}-{\sigma }_{12}{R}_{11}+{\rho }_{12}{R}_{12}+{\chi }_{11}{S}_{11}-d{R}_{11}\\ \frac{\text{d}{R}_{12}}{\text{d}t}=\gamma {I}_{12}+{\sigma }_{12}{R}_{11}-{\rho }_{12}{R}_{12}+{\chi }_{12}{S}_{12}-d{R}_{12}\\ \frac{\text{d}{R}_{22}}{\text{d}t}=\gamma {I}_{22}-{\sigma }_{21}{R}_{22}+{\rho }_{21}{R}_{21}+{\chi }_{22}{S}_{22}-d{R}_{22}\\ \frac{\text{d}{R}_{21}}{\text{d}t}=\gamma {I}_{21}+{\sigma }_{21}{R}_{22}-{\rho }_{21}{R}_{21}+{\chi }_{21}{S}_{21}-d{R}_{21}\end{array}$ (1)

$\begin{array}{l}{N}_{1}^{t}={S}_{11}\left(t\right)+{S}_{21}\left(t\right)+{E}_{11}\left(t\right)+{E}_{21}\left(t\right)+{I}_{11}\left(t\right)+{I}_{21}\left(t\right)+{R}_{11}\left(t\right)+{R}_{21}\left(t\right)\\ {N}_{2}^{t}={S}_{12}\left(t\right)+{S}_{22}\left(t\right)+{E}_{12}\left(t\right)+{E}_{22}\left(t\right)+{I}_{12}\left(t\right)+{I}_{22}\left(t\right)+{R}_{12}\left(t\right)+{R}_{22}\left(t\right)\end{array}$

Table 1. The meaning of parameter in model

$\begin{array}{l}\frac{\text{d}{S}_{11}}{\text{d}t}={A}_{11}-{\sigma }_{12}{S}_{11}+{\rho }_{12}{S}_{12}-d{S}_{11}\\ \frac{\text{d}{S}_{12}}{\text{d}t}={A}_{12}+{\sigma }_{12}{S}_{11}-{\rho }_{12}{S}_{12}-d{S}_{12}\\ \frac{\text{d}{S}_{22}}{\text{d}t}={A}_{22}-{\sigma }_{21}{S}_{22}+{\rho }_{21}{S}_{21}-d{S}_{22}\\ \frac{\text{d}{S}_{21}}{\text{d}t}={A}_{21}+{\sigma }_{21}{S}_{22}-{\rho }_{21}{S}_{21}-d{S}_{21}\end{array}$ (2)

${S}_{011}^{*}=\frac{{\rho }_{12}\left({A}_{11}+{A}_{12}\right)+{A}_{11}d}{\left({\rho }_{12}+d\right)\left({\sigma }_{12}+d\right)-{\sigma }_{12}{\rho }_{12}},{S}_{012}^{*}=\frac{{A}_{12}d+{\sigma }_{12}\left({A}_{11}+{A}_{12}\right)}{\left({\rho }_{12}+d\right)\left({\sigma }_{12}+d\right)-{\sigma }_{12}{\rho }_{12}}$

${S}_{022}^{*}=\frac{{\rho }_{21}\left({A}_{21}+{A}_{22}\right)+{A}_{22}d}{\left({\rho }_{21}+d\right)\left({\sigma }_{21}+d\right)-{\sigma }_{21}{\rho }_{21}},{S}_{021}^{*}=\frac{{A}_{21}d+{\sigma }_{21}\left({A}_{21}+{A}_{22}\right)}{\left({\rho }_{21}+d\right)\left({\sigma }_{21}+d\right)-{\sigma }_{21}{\rho }_{21}}$

$\left(\begin{array}{cccc}-{\sigma }_{12}-d& {\rho }_{12}& 0& 0\\ {\sigma }_{12}& -{\rho }_{12}-d& 0& 0\\ 0& 0& -{\sigma }_{21}-d& {\rho }_{21}\\ 0& 0& {\sigma }_{21}& -{\rho }_{21}-d\end{array}\right)$

$\begin{array}{l}\frac{\text{d}{S}_{11}}{\text{d}t}={A}_{11}-{\sigma }_{12}{S}_{11}+{\rho }_{12}{S}_{12}-{\chi }_{11}{S}_{11}-d{S}_{11}\\ \frac{\text{d}{S}_{12}}{\text{d}t}={A}_{12}+{\sigma }_{12}{S}_{11}-{\rho }_{12}{S}_{12}-{\chi }_{12}{S}_{12}-d{S}_{12}\\ \frac{\text{d}{S}_{22}}{\text{d}t}={A}_{22}-{\sigma }_{21}{S}_{22}+{\rho }_{21}{S}_{21}-{\chi }_{22}{S}_{22}-d{S}_{22}\\ \frac{\text{d}{S}_{21}}{\text{d}t}={A}_{21}+{\sigma }_{21}{S}_{22}-{\rho }_{21}{S}_{21}-{\chi }_{21}{S}_{21}-d{S}_{21}\end{array}$

$\begin{array}{l}\frac{\text{d}{R}_{11}}{\text{d}t}={\sigma }_{12}{R}_{11}+{\rho }_{12}{R}_{12}+{\chi }_{11}{S}_{11}-d{R}_{11}\\ \frac{\text{d}{R}_{12}}{\text{d}t}={\sigma }_{12}{R}_{11}-{\rho }_{12}{R}_{12}+{\chi }_{12}{S}_{12}-d{R}_{12}\\ \frac{\text{d}{R}_{22}}{\text{d}t}={\sigma }_{21}{R}_{22}+{\rho }_{21}{R}_{21}+{\chi }_{22}{S}_{22}-d{R}_{22}\\ \frac{\text{d}{R}_{21}}{\text{d}t}={\sigma }_{21}{R}_{22}-{\rho }_{21}{R}_{21}+{\chi }_{21}{S}_{21}-d{R}_{21}\end{array}$ (3)

${S}_{111}^{*}=\frac{{A}_{11}\left({\chi }_{12}+d\right)+{\rho }_{12}\left({A}_{11}+{A}_{12}\right)}{\left({\rho }_{12}+{\chi }_{12}+d\right)\left({\sigma }_{12}+{\chi }_{11}+d\right)-{\sigma }_{12}{\rho }_{12}},{S}_{112}^{*}=\frac{{A}_{12}\left({\chi }_{11}+d\right)+{\sigma }_{12}\left({A}_{11}+{A}_{12}\right)}{\left({\rho }_{12}+{\chi }_{12}+d\right)\left({\sigma }_{12}+{\chi }_{11}+d\right)-{\sigma }_{12}{\rho }_{12}}$

${S}_{122}^{*}=\frac{{A}_{22}\left({\chi }_{21}+d\right)+{\rho }_{21}\left({A}_{22}+{A}_{21}\right)}{\left({\rho }_{21}+{\chi }_{21}+d\right)\left({\sigma }_{21}+{\chi }_{22}+d\right)-{\sigma }_{21}{\rho }_{21}},{S}_{121}^{*}=\frac{{A}_{21}\left({\chi }_{22}+d\right)+{\sigma }_{21}\left({A}_{22}+{A}_{21}\right)}{\left({\rho }_{21}+{\chi }_{21}+d\right)\left({\sigma }_{21}+{\chi }_{22}+d\right)-{\sigma }_{21}{\rho }_{21}}$

${R}_{111}^{*}=\frac{{\rho }_{12}{\chi }_{12}{S}_{112}^{*}+\left({\rho }_{12}+d\right){\chi }_{11}{S}_{111}^{*}}{d\left({\sigma }_{12}+{\rho }_{12}+d\right)},{R}_{112}^{*}=\frac{{\sigma }_{12}{\chi }_{11}{S}_{111}^{*}+\left({\sigma }_{12}+d\right){\chi }_{12}{S}_{112}^{*}}{d\left({\sigma }_{12}+{\rho }_{12}+d\right)}$

${R}_{122}^{*}=\frac{{\rho }_{21}{\chi }_{21}{S}_{121}^{*}+\left({\rho }_{21}+d\right){\chi }_{22}{S}_{22}^{*}}{d\left({\sigma }_{21}+{\rho }_{21}+d\right)},{R}_{121}^{*}=\frac{{\sigma }_{21}{\chi }_{22}{S}_{122}^{*}+\left({\sigma }_{21}+d\right){\chi }_{21}{S}_{121}^{*}}{d\left({\sigma }_{21}+{\rho }_{21}+d\right)}$

${N}_{1}^{t*}={S}_{111}^{*}+{S}_{121}^{*}\text{+}{R}_{111}^{*}+{R}_{121}^{*},{N}_{2}^{t*}={S}_{222}^{*}+{S}_{212}^{*}+{R}_{222}^{*}+{R}_{212}^{*}$

$\mathcal{F}=\left(\begin{array}{c}{S}_{11}\underset{j=1}{\overset{2}{\sum }}{k}_{1}{\beta }_{11j}\frac{{I}_{j1}}{{N}_{1}^{t}}\\ {S}_{12}\underset{j=1}{\overset{2}{\sum }}{k}_{2}{\beta }_{12j}\frac{{I}_{j2}}{{N}_{2}^{t}}\\ {S}_{22}\underset{j=1}{\overset{2}{\sum }}{k}_{1}{\beta }_{22j}\frac{{I}_{j2}}{{N}_{2}^{t}}\\ {S}_{21}\underset{j=1}{\overset{2}{\sum }}{k}_{1}{\beta }_{21j}\frac{{I}_{j1}}{{N}_{1}^{t}}\\ 0\\ 0\\ 0\\ 0\end{array}\right)$ , $\mathcal{V}=\left(\begin{array}{c}{\sigma }_{12}{E}_{11}-{\rho }_{12}{E}_{12}+\left(\alpha +d\right){E}_{11}\\ -{\sigma }_{12}{E}_{11}+{\rho }_{12}{E}_{12}+\left(\alpha +d\right){E}_{12}\\ {\sigma }_{21}{E}_{22}-{\rho }_{21}{E}_{21}+\left(\alpha +d\right){E}_{22}\\ -{\sigma }_{21}{E}_{22}+{\rho }_{21}{E}_{21}+\left(\alpha +d\right){E}_{21}\\ -\alpha {E}_{11}+{\sigma }_{12}{I}_{11}-{\rho }_{12}{I}_{12}+\left(\gamma +d\right){I}_{11}\\ -\alpha {E}_{12}-{\sigma }_{12}{I}_{11}+{\rho }_{12}{I}_{12}+\left(\gamma +d\right){I}_{12}\\ -\alpha {E}_{22}+{\sigma }_{21}{I}_{22}-{\rho }_{21}{I}_{21}+\left(\gamma +d\right){I}_{22}\\ -\alpha {E}_{21}-{\sigma }_{21}{I}_{22}+{\rho }_{21}{I}_{21}+\left(\gamma +d\right){I}_{21}\end{array}\right)$

$\begin{array}{l}{m}_{12}={\sigma }_{12}+\alpha +d,{n}_{12}={\rho }_{12}+\alpha +d,{m}_{21}={\sigma }_{21}+\alpha +d,{n}_{21}={\rho }_{21}+\alpha +d,\\ {p}_{12}={\sigma }_{12}+r+d,{q}_{12}={\rho }_{12}+r+d,{p}_{21}={\sigma }_{21}+r+d,{q}_{21}={\rho }_{21}+r+d\end{array}$

$\mathcal{F},\mathcal{V}$ 分别关于潜伏项 ${E}_{ij}$ 与染病项 ${I}_{ij}$ 求导得 $F,V$

$F=\left(\begin{array}{cc}0& {F}_{12}\\ 0& 0\end{array}\right)$ ，其中 ${F}_{12}=\left(\begin{array}{cccc}\frac{{S}_{11}^{*}{k}_{1}{\beta }_{111}}{{N}_{1}^{t*}}& 0& 0& \frac{{S}_{11}^{*}{k}_{1}{\beta }_{112}}{{N}_{1}^{t*}}\\ 0& \frac{{S}_{12}^{*}{k}_{2}{\beta }_{121}}{{N}_{2}^{t*}}& \frac{{S}_{12}^{*}{k}_{2}{\beta }_{122}}{{N}_{2}^{t*}}& 0\\ 0& \frac{{S}_{22}^{*}{k}_{2}{\beta }_{221}}{{N}_{2}^{t*}}& \frac{{S}_{22}^{*}{k}_{2}{\beta }_{222}}{{N}_{2}^{t*}}& 0\\ \frac{{S}_{21}^{*}{k}_{1}{\beta }_{211}}{{N}_{1}^{t*}}& 0& 0& \frac{{S}_{21}^{*}{k}_{1}{\beta }_{212}}{{N}_{1}^{t*}}\end{array}\right)$

$V=\left(\begin{array}{cc}{V}_{11}& 0\\ {V}_{21}& {V}_{22}\end{array}\right)$ ，其中 

${V}_{21}=\left(\begin{array}{cccc}-\alpha & 0& 0& 0\\ 0& -\alpha & 0& 0\\ 0& 0& -\alpha & 0\\ 0& 0& 0& -\alpha \end{array}\right)$ , ${V}_{22}=\left(\begin{array}{cccc}{p}_{12}& -{\rho }_{12}& 0& 0\\ -{\sigma }_{12}& {q}_{12}& 0& 0\\ 0& 0& {p}_{21}& -{\rho }_{21}\\ 0& 0& -{\sigma }_{21}& {q}_{21}\end{array}\right)$

${V}_{11}^{-1}=\left(\begin{array}{cccc}\frac{{n}_{12}}{{m}_{12}{n}_{12}-{\rho }_{12}{\sigma }_{12}}& \frac{{\rho }_{12}}{{m}_{12}{n}_{12}-{\rho }_{12}{\sigma }_{12}}& 0& 0\\ \frac{{\sigma }_{12}}{{m}_{12}{n}_{12}-{\rho }_{12}{\sigma }_{12}}& \frac{{m}_{12}}{{m}_{12}{n}_{12}-{\rho }_{12}{\sigma }_{12}}& 0& 0\\ 0& 0& \frac{{n}_{21}}{{m}_{21}{n}_{21}-{\rho }_{21}{\sigma }_{21}}& \frac{{\rho }_{21}}{{m}_{21}{n}_{21}-{\rho }_{21}{\sigma }_{21}}\\ 0& 0& \frac{{\sigma }_{21}}{{m}_{21}{n}_{21}-{\rho }_{21}{\sigma }_{21}}& \frac{{m}_{12}}{{m}_{21}{n}_{21}-{\rho }_{21}{\sigma }_{21}}\end{array}\right)$

${V}_{22}^{-1}=\left(\begin{array}{cccc}\frac{{q}_{12}}{{p}_{12}{q}_{12}-{\rho }_{12}{\sigma }_{12}}& \frac{{\rho }_{12}}{{p}_{12}{q}_{12}-{\rho }_{12}{\sigma }_{12}}& 0& 0\\ \frac{{\sigma }_{12}}{{p}_{12}{q}_{12}-{\rho }_{12}{\sigma }_{12}}& \frac{{p}_{12}}{{p}_{12}{q}_{12}-{\rho }_{12}{\sigma }_{12}}& 0& 0\\ 0& 0& \frac{{q}_{21}}{{p}_{21}{q}_{21}-{\rho }_{21}{\sigma }_{21}}& \frac{{\rho }_{21}}{{p}_{21}{n}_{21}-{\rho }_{21}{\sigma }_{21}}\\ 0& 0& \frac{{\sigma }_{21}}{{p}_{21}{q}_{21}-{\rho }_{21}{\sigma }_{21}}& \frac{{p}_{21}}{{p}_{21}{q}_{21}-{\rho }_{21}{\sigma }_{21}}\end{array}\right)$

${U}_{12}={m}_{12}{n}_{12}-{\rho }_{12}{\sigma }_{12},{U}_{21}={m}_{21}{n}_{21}-{\rho }_{21}{\sigma }_{21},{W}_{12}={p}_{12}{q}_{12}-{\rho }_{12}{\sigma }_{12},{W}_{21}={p}_{21}{q}_{21}-{\rho }_{21}{\sigma }_{21}$ ，有

$-{V}_{22}^{-1}{V}_{21}{V}_{11}^{-1}=\left(\begin{array}{cccc}\frac{\alpha \left({n}_{12}{q}_{12}+{\sigma }_{12}{\rho }_{12}\right)}{{W}_{12}{U}_{12}}& \frac{\alpha \left({\rho }_{12}{q}_{12}+{m}_{12}{\rho }_{12}\right)}{{W}_{12}{U}_{12}}& 0& 0\\ \frac{\alpha \left({\sigma }_{12}{\eta }_{12}+{\sigma }_{12}{p}_{12}\right)}{{W}_{12}{U}_{12}}& \frac{\alpha \left({\sigma }_{12}{\rho }_{12}+{p}_{12}{m}_{12}\right)}{{W}_{12}{U}_{12}}& 0& 0\\ 0& 0& \frac{\alpha \left({n}_{21}{q}_{21}+{\sigma }_{21}{\rho }_{21}\right)}{{W}_{21}{U}_{21}}& \frac{\alpha \left({\rho }_{21}{q}_{21}+{m}_{21}{\rho }_{21}\right)}{{W}_{21}{U}_{21}}\\ 0& 0& \frac{\alpha \left({\sigma }_{21}{\eta }_{21}+{\sigma }_{21}{p}_{21}\right)}{{W}_{21}{U}_{21}}& \frac{\alpha \left({\sigma }_{21}{\rho }_{21}+{p}_{21}{m}_{21}\right)}{{W}_{21}{U}_{12}}\end{array}\right)$

${K}_{11}=-{F}_{12}{V}_{22}^{-1}{V}_{21}{V}_{11}^{-1}$ ，则有

3. 数值模拟

Table 2. The value of parameter in simulation

Figure 1. Trend graph of total infected I with time at different migration rates ${\sigma }_{12}$

Figure 2. Trend graph of total infected I with time at different migration rates ${\sigma }_{21}$

Figure 3. Graph of relation between ${\sigma }_{12}$ and ${R}_{0}$

Figure 4. Graph of relation between ${\sigma }_{21}$ and ${R}_{0}$

Figure 5. Graph of of relation between ${\chi }_{11}$ and total infected I

Figure 6. Graph of relation between ${\chi }_{11}$ and ${R}_{v}$

Figure 7. Graph of relationship between vaccine rate in each group and ${R}_{v}$

4. 结论

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