Dynamics of a multi-strain malaria model with diffusion in a periodic environment

Figures & data

Table 1. The potential dynamical outcomes of system (6(6) $\{\begin{array}{rl}& \frac{\mathrm{\partial }{u}_1(t,x)}{\mathrm{\partial }t}=D_h\mathrm{\Delta }{u}_1(t,x)+\frac{c_1\beta (t,x)l[N(x)-u_1(t,x)-u_2(t,x)]u_4(t,x)}{p[u_1(t,x)+u_2(t,x)]+l[N(x)-u_1(t,x)-u_2(t,x)]}\\ & -({\mu }_h(x)+{\varrho }_1(x))u_1(t,x),x\in \mathrm{\Omega },\\ & \frac{\mathrm{\partial }{u}_2(t,x)}{\mathrm{\partial }t}=D_h\mathrm{\Delta }{u}_2(t,x)+\frac{c_2\beta (t,x)l[N(x)-u_1(t,x)-u_2(t,x)]u_5(t,x)}{p[u_1(t,x)+u_2(t,x)]+l[N(x)-u_1(t,x)-u_2(t,x)]}\\ & -({\mu }_h(x)+{\varrho }_2(x))u_2(t,x),x\in \mathrm{\Omega },\\ & \frac{\mathrm{\partial }{u}_3(t,x)}{\mathrm{\partial }t}=D_v\mathrm{\Delta }{u}_3(t,x)+\mathrm{\Lambda }(t,x)\\ & -\frac{{\alpha }_1\beta (t,x)p{u}_1(t,x)u_3(t,x)}{p[u_1(t,x)+u_2(t,x)]+l[N(x)-u_1(t,x)-u_2(t,x)]}\\ & -\frac{{\alpha }_2\beta (t,x)p{u}_2(t,x)u_3(t,x)}{p[u_1(t,x)+u_2(t,x)]+l[N(x)-u_1(t,x)-u_2(t,x)]}-{\mu }_v(t,x)u_3(t,x),x\in \mathrm{\Omega },\\ & \frac{\mathrm{\partial }{u}_4(t,x)}{\mathrm{\partial }t}=D_v\mathrm{\Delta }{u}_4(t,x)-{\mu }_v(t,x)u_4(t,x)+(1-{\tau }_1^{\mathrm{\prime }}(t)){\int }_{\mathrm{\Omega }}\mathrm{\Gamma }(t,t-{\tau }_1(t),x,y)\\ & \cdot \frac{{\alpha }_1\beta (t-{\tau }_1(t),y)p{u}_1(t-{\tau }_1(t),y)u_3(t-{\tau }_1(t),y)}{\begin{array}{c}p[u_1(t-{\tau }_1(t),y)+u_2(t-{\tau }_1(t),y)]\\ +l[N(y)-u_1(t-{\tau }_1(t),y)-u_2(t-{\tau }_1(t),y)]\end{array}}\mathrm{d}y,x\in \mathrm{\Omega },\\ & \frac{\mathrm{\partial }{u}_5(t,x)}{\mathrm{\partial }t}=D_v\mathrm{\Delta }{u}_5-{\mu }_v(t,x)u_5(t,x)+(1-{\tau }_2^{\mathrm{\prime }}(t)){\int }_{\mathrm{\Omega }}\mathrm{\Gamma }(t,t-{\tau }_2(t),x,y)\\ & \cdot \frac{{\alpha }_2\beta (t-{\tau }_2(t),y)p{u}_2(t-{\tau }_2(t),y)u_3(t-{\tau }_2(t),y)}{\begin{array}{c}p[u_1(t-{\tau }_2(t),y)+u_2(t-{\tau }_2(t),y)]\\ +l[N(y)-u_1(t-{\tau }_2(t),y)-u_2(t-{\tau }_2(t),y)]\end{array}}\mathrm{d}y,x\in \mathrm{\Omega },\\ & \frac{\mathrm{\partial }{u}_1(t,x)}{\mathrm{\partial }\nu }=\frac{\mathrm{\partial }{u}_2(t,x)}{\mathrm{\partial }\nu }=\frac{\mathrm{\partial }{u}_3(t,x)}{\mathrm{\partial }\nu }=\frac{\mathrm{\partial }{u}_4(t,x)}{\mathrm{\partial }\nu }=\frac{\mathrm{\partial }{u}_5(t,x)}{\mathrm{\partial }\nu }=0,x\in \mathrm{\partial }\mathrm{\Omega },\end{array}$(6) ).

Case	Strain 1	Strain 2	Expected outcome	Comment
1	$R_1<1$	$R_2<1$	$u_1\to 0$, $u_2\to 0$, $u_4\to 0$, $u_5\to 0$	Theorem 5.1, Figure 1
2	$R_1<1$	$R_2>1$	$u_1\to 0$, $u_2$ persists, $u_4\to 0$, $u_5$ persists	Theorem 5.4, Figure 2
3	$R_1>1$	$R_2<1$	$u_1$ persists, $u_2\to 0$, $u_4$ persists, $u_5\to 0$	Theorem 5.5
			(i) $u_1\to 0$, $u_2$ persists, $u_4\to 0$, $u_5$ persists	Theorem 5.6 (ii), Figure 3
4	$R_1>1$	$R_2>1$	(ii) $u_1$ persists, $u_2\to 0$, $u_4$ persists, $u_5\to 0$	Theorem 5.6 (i), Figure 4
			(iii) $u_1$ persists, $u_2$ persists, $u_4$ persists, $u_5$ persists	Theorem 5.7, Figure 5

Figure 1. The evolution of infection compartments of humans and mosquitoes when $R_1<1$ and $R_2<1$. (a) The evolution of $u_1$. (b) Then evolution of $u_4$. (c) Then evolution of $u_2$. (d) Then evolution of $u_5$.

Figure 2. The evolution of infection compartments of humans and mosquitoes when $R_1<1$ and $R_2>1$. (a) The evolution of $u_1$. (b) The evolution of $u_4$. (c) The evolution of $u_2$. (d) The evolution of $u_5$.

Figure 3. The evolution of infection compartments of humans and mosquitoes when $R_1>1$ and $R_2>1$. (a) The evolution of $u_1$. (b) Then evolution of $u_4$. (c) Then evolution of $u_2$. (d) Then evolution of $u_5$.

Figure 4. The evolution of infection compartments of humans and mosquitoes when $R_1>1$ and $R_2>1$. (a) The evolution of $u_1$. (b) The evolution of $u_4$. (c) The evolution of $u_2$. (d) The evolution of $u_5$.

Figure 5. The evolution of infection compartments of humans and mosquitoes when $R_1>1$ and $R_2>1$. (a) The evolution of $u_1$. (b) The evolution of $u_4$. (c) The evolution of $u_2$. (d) The evolution of $u_5$.

Figure 6. The evolution of infection compartments of humans and mosquitoes when $D_h=0$ and $D_v=0$. (a) The evolution of $u_1$. (b) The evolution of $u_4$. (c) The evolution of $u_2$. (d) The evolution of $u_5$.

Figure 7. The evolution of infection compartments of humans and mosquitoes with $D_h=0$ and $D_v=0.0125$. (a) The evolution of $u_1$. (b) The evolution of $u_4$. (c) The evolution of $u_2$. (d) The evolution of $u_5$.

Figure 8. The evolution of infection compartments of humans and mosquitoes with $D_h=0.1$ and $D_v=0$. (a) The evolution of $u_1$. (b) The evolution of $u_4$. (c) The evolution of $u_2$. (d) The evolution of $u_5$.

Figure 9. Increasing medical resources in the environment of spatial heterogeneity and its influence on the $R_1$.

Figure 10. For fixed medical resources, the spatial distribution of ${\varrho }_1$ and the corresponding value of $R_1$.

Figure 11. The evolution of infection compartments of humans and mosquitoes with different vector-bias parameter. Among them, green indicates that there is a vector-bias effect ($l/p=3/4$), and red indicates that there is no vector-bias effect ($l/p=1$). (a) With vector-bias effect. (b) Without vector-bias effect. (c) With vector-bias effect. (d) Without vector-bias effect.

Figure 12. The evolution infection compartments of humans and mosquitoes with seasonal temperature changes and without seasonal temperature changes. Among them, red indicates that the parameters depend on temperature, and green indicates that the parameters do not depend on temperature.

Figure A1. A small interval of development level of infection.

Figure A2. The parameters change with time. (a) Change of contact rate $\beta (t)$ with time. (b) Change of EIP ${\tau }_1(t)$ with time. (c) Change of the death rate of mosquitoes ${\mu }_v(t)$ with time.

Table F1. Definition and value of parameters.

Parameter	Definition	Value (range)	Dimension	References
N	Total human	1205709	Dimensionless	[35]
${\mu }_h$	Human natural death rate	0.00157	${\mathrm{Month}}^{-1}$	[35]
${\varrho }_i$	Recovery rate	(0.04258,0.568)	${\mathrm{Month}}^{-1}$	[40]
l/p	Vector bias parameter	$(0,1)$	Dimensionless	[5]
$c_i$	Transmission probability from infectious mosquitoes to humans	(0.01,0.27)	Dimensionless	[6]
${\alpha }_i$	Transmission probability from infectious humans to mosquitoes	(0.072,0.64)	Dimensionless	[6]
$D_h$	Human diffusion rate	0.1	${\mathrm{km}}^2{\mathrm{Month}}^{-1}$	[40]
$D_v$	Mosquito diffusion rate	0.0125	${\mathrm{km}}^2{\mathrm{Month}}^{-1}$	[40]
β	Mosquito biting rate	(A15(A15) $\begin{array}{rl}\beta (t)& =6.983-1.993\cos (\pi t/6)-0.4247\cos (\pi t/3)-0.128\cos (\pi t/2)-0.04095\cos (2\pi t/3)\\ & +0.0005486\cos (5\pi t/6)-1.459\sin (\pi t/6)-0.007642\sin (\pi t/3)-0.0709\sin (\pi t/2)\\ & +0.05452\sin (2\pi t/3)-0.06235\sin (5\pi t/6){\mathrm{Month}}^{-1},\end{array}$(A15) )	${\mathrm{Month}}^{-1}$	[35]
${\mu }_v$	Mosquito death rate	(A16(A16) $\begin{array}{rl}{\mu }_v(t)& =3.086+0.04788\cos (\pi t/6)+0.01942\cos (\pi t/3)+0.007133\cos (\pi t/2)\\ & +0.0007665\cos (2\pi t/3)-0.001459\cos (5\pi t/6)+0.02655\sin (\pi t/6)\\ & +0.01819\sin (\pi t/3)+0.01135\sin (\pi t/2)\\ & +0.005687\sin (2\pi t/3)+0.003198\sin (5\pi t/6){\mathrm{Month}}^{-1},\end{array}$(A16) )	${\mathrm{Month}}^{-1}$	[35]
${\tau }_1$	EIP of mosquitoes infected with strain 1	(A17(A17) $\begin{array}{rl}{\tau }_1(t)& =1/30.4[17.25+8.369\cos (\pi t/6)+4.806\sin (\pi t/6)+3.27\cos (\pi t/3)+2.857\sin (\pi t/3)\\ & +1.197\cos (\pi t/2)+1.963\sin (\pi t/2)+0.03578\cos (2\pi t/3)+1.035\sin (2\pi t/3)\\ & -0.3505\cos (5\pi t/6)+0.6354\sin (5\pi t/6)-0.3257\cos (\pi t)+0\sin (\pi t)]\mathrm{Month},\end{array}$(A17) )	$\mathrm{Month}$	[35]
${\tau }_2$	EIP of mosquitoes infected with strain 2	(A18(A18) $\begin{array}{rl}{\tau }_2(t)& =0.8/30.4[17.25+8.369\cos (\pi t/6)+4.806\sin (\pi t/6)+3.27\cos (\pi t/3)+2.857\sin (\pi t/3)\\ & +1.197\cos (\pi t/2)+1.963\sin (\pi t/2)+0.03578\cos (2\pi t/3)+1.035\sin (2\pi t/3)\\ & -0.3505\cos (5\pi t/6)+0.6354\sin (5\pi t/6)-0.3257\cos (\pi t)+0\sin (\pi t)]\mathrm{Month},\end{array}$(A18) )	$\mathrm{Month}$	[35]
Λ	Recruitment rate of adult female mosquitoes	(A19(A19) $\mathrm{\Lambda }(t)=\hat{k}\times \beta (t)({\mathrm{km}}^2\mathrm{Month}{)}^{-1},\mathrm{where}\hat{k}=5\times 1205709.$(A19) )	${\mathrm{Month}}^{-1}$	[35]

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