A mathematical model of malaria transmission in a periodic environment

Figures & data

Figure 1. Transfer diagram: the solid arrows represent the transition from one class to another and the blue dashed arrow represents the eggs laying of female mosquitoes.

Table 1. Values for constant parameters of the model.

Symbols	Biological descriptions	Values	Sources	Dimensions
Λ	constant recruitment rate for humans	5000	estimated	/month
$d_h$	human death rate	$0.00167$	estimated	/month
${\alpha }_e$	transfer rate of humans from $I_e$ to $R_a$	0.03	[10]	/month
${\alpha }_a$	transfer rate of humans from $E_a$ to $R_a$	0.3	[10]	/month
${\gamma }_e$	disease-induced death rate for non-immune	0.00054	estimated	/month
${\gamma }_a$	disease-induced death rate for semi-immune	0.00027	estimated	/month
${\nu }_e$	transfer rate of humans from $E_e$ to $I_e$	3	[7]	/month
${\nu }_a$	transfer rate of humans from $E_a$ to $I_a$	2.7	[7]	/month
${\beta }_a$	per capita rate of loss of immunity for humans	0.081	[10]	/month
$c_{ev}$	probability of transmission of infection from $I_e$ to $S_v$	0.45	[10]	dimensionless
$c_{av}$	probability of transmission of infection from $I_a$ to $S_v$	0.35	[10]	dimensionless
${\overline{c}}_{av}$	probability of transmission of infection from $R_h$ to $S_v$	0.002	[10]	dimensionless
p	probability of recruitment for humans	0.4	estimated	/month
$r_v$	transfer rate from L to adult	–	estimated	/month
$d_v$	death rate for adult vectors	–	estimated	/month
$c_{ve}$	probability of transmission of infection from $I_v$ to $S_e$	0.07	[10]	dimensionless
$c_{va}$	probability of transmission of infection from $I_v$ to $S_a$	0.022	[10]	dimensionless
${\nu }_v$	transfer rate of mosquitoes from $E_v$ to $I_v$	2.49	[10]	/month
$\beta \left(t\right)$	biting rate of mosquitoes	–	estimated	/month
K	breeder site occupied by immature mosquitoes	150,000	estimated	dimensionless
b	eggs laying rate	–	estimated	/month
$d_l$	death rate of immature mosquitoes	–	estimated	/month

Figure 2. Transfer diagram: the black dashed arrows indicate the direction of the infection, the solid arrows represent the transition from one class to another and the blue dashed arrow represents the eggs laying of female mosquitoes.

Figure 3. Distribution of infected non-immune with $d_l=4,r_v=15,d_v=3.5,b=180,{\gamma }_e=0.00054,{\gamma }_a=0.00027$ and $a_1=5,b_1=3$. The initial conditions are given by $S_e\left(0\right)=500,E_e\left(0\right)=250,I_e\left(0\right)=150,S_a\left(0\right)=1000,E_a\left(0\right)=500,I_a\left(0\right)=1000,R_a\left(0\right)=2000,S_v\left(0\right)=10,000,E_v\left(0\right)=4000,I_v\left(0\right)=2000\mathrm{and}L\left(0\right)=15,000$. We obtain $\kappa =40.6015$ and ${\mathcal{R}}_0=1.7127>1$. (a) Exposed non-immune and (b) infectious non-immune.

Figure 4. Distribution of infected semi-immune with $d_l=4,r_v=15,d_v=3.5,b=180,{\gamma }_e=0.00054,{\gamma }_a=0.00027$ and $a_1=5,b_1=3$. The initial conditions are given by $S_e\left(0\right)=500,E_e\left(0\right)=250,I_e\left(0\right)=150,S_a\left(0\right)=1000,E_a\left(0\right)=500,I_a\left(0\right)=1000,R_a\left(0\right)=2000,S_v\left(0\right)=10,000,E_v\left(0\right)=4000,I_v\left(0\right)=2000\mathrm{and}L\left(0\right)=15,000$. We obtain $\kappa =40.6015$ and ${\mathcal{R}}_0=1.7127>1$. (a) Exposed semi-immune and (b) infectious semi-immune.

Figure 5. Distribution of infected mosquitoes with $d_l=4,r_v=15,d_v=3.5,b=180,{\gamma }_e=0.00054,{\gamma }_a=0.00027$ and $a_1=5,b_1=3$. The initial conditions are given by $S_e\left(0\right)=500,E_e\left(0\right)=250,I_e\left(0\right)=150,S_a\left(0\right)=1000,E_a\left(0\right)=500,I_a\left(0\right)=1000,R_a\left(0\right)=2000,S_v\left(0\right)=10,000,E_v\left(0\right)=4000,I_v\left(0\right)=2000,\mathrm{and}L\left(0\right)=15,000$. We obtain $\kappa =40.6015$ and ${\mathcal{R}}_0=1.7127>1$. (a) Exposed mosquitoes and (b) infectious mosquitoes.

Figure 6. Distribution of infected non-immune with $a_1=3,b_1=2.5,d_l=4,d_v=7.5,b=90,{\gamma }_e={\gamma }_a=0$. The initial conditions are given by $S_e\left(0\right)=500,E_e\left(0\right)=500,I_e\left(0\right)=1000,S_a\left(0\right)=1000,E_a\left(0\right)=500,I_a\left(0\right)=1000,R_a\left(0\right)=2000,S_v\left(0\right)=10,000,E_v\left(0\right)=8000,I_v\left(0\right)=4000$ and $L\left(0\right)=15,000$. We obtain $\kappa =9.47$ and ${\mathcal{R}}_0=0.3828<1$. (a) Exposed non-immune and (a) infectious non-immune.

Figure 7. Distribution of semi-immune infected with $a_1=3,b_1=2.5,d_l=4,d_v=7.5,b=90,{\gamma }_e={\gamma }_a=0$. The initial conditions are given by $S_e\left(0\right)=500,E_e\left(0\right)=500,I_e\left(0\right)=1000,S_a\left(0\right)=1000,E_a\left(0\right)=500,I_a\left(0\right)=1000,R_a\left(0\right)=2000,S_v\left(0\right)=10,000,E_v\left(0\right)=8000,I_v\left(0\right)=4000$ and $L\left(0\right)=15,000$. We obtain $\kappa =9.47$ and ${\mathcal{R}}_0=0.3828<1$. (a) Exposed semi-immune and (b) infectious semi-immune.

Figure 8. Distribution of infected mosquitoes with $a_1=3,b_1=2.5,d_l=4,d_v=7.5,b=90,{\gamma }_e={\gamma }_a=0$. The initial conditions are given by $S_e\left(0\right)=500,E_e\left(0\right)=500,I_e\left(0\right)=1000,S_a\left(0\right)=1000,E_a\left(0\right)=500,I_a\left(0\right)=1000,R_a\left(0\right)=2000,S_v\left(0\right)=10,000,E_v\left(0\right)=8000,I_v\left(0\right)=4000$ and $L\left(0\right)=15,000$. We obtain $\kappa =9.47$ and ${\mathcal{R}}_0=0.3828<1$. (a) Exposed mosquitoes and (b) infectious mosquitoes.

Figure 9. Distribution of infected non-immune and semi-immune with $d_l=6.5,r_v=7,d_v=7.5,b=20,{\gamma }_e={\gamma }_a=0$ and $a_1=5,b_1=3$. The initial conditions are given by $S_e\left(0\right)=500,E_e\left(0\right)=250,I_e\left(0\right)=150,S_a\left(0\right)=1000,E_a\left(0\right)=500,I_a\left(0\right)=1000,R_a\left(0\right)=2000,S_v\left(0\right)=10,000,E_v\left(0\right)=4000,I_v\left(0\right)=2000$ and $L\left(0\right)=15,000$. We get $\kappa =1.3827$ and ${\mathcal{R}}_0=0.2139<1$. (a) Infectious non-immune and (b) infectious semi-immune.

Figure 10. Distribution of infected mosquitoes and immune humans with $d_l=6.5,r_v=7,d_v=7.5,b=20,{\gamma }_e={\gamma }_a=0$ and $a_1=5,b_1=3$. The initial conditions are given by $S_e\left(0\right)=500,E_e\left(0\right)=250,I_e\left(0\right)=150,S_a\left(0\right)=1000,E_a\left(0\right)=500,I_a\left(0\right)=1000,R_a\left(0\right)=2000,S_v\left(0\right)=10,000,E_v\left(0\right)=4000,I_v\left(0\right)=2000$ and $L\left(0\right)=15,000$. We get $\kappa =1.3827$ and ${\mathcal{R}}_0=0.2139<1$. (a) Infectious mosquitoes and (b) immune humans.

Figure 11. Distribution of infected non-immune and semi-immune with $d_l=4,r_v=15,d_v=3.5,b=180,{\gamma }_e={\gamma }_a=0$ and $a_1=5,b_1=3$. The initial conditions are given by $S_e\left(0\right)=500,E_e\left(0\right)=250,I_e\left(0\right)=150,S_a\left(0\right)=1000,E_a\left(0\right)=500,I_a\left(0\right)=1000,R_a\left(0\right)=2000,S_v\left(0\right)=10,000,E_v\left(0\right)=4000,I_v\left(0\right)=2000$ and $L\left(0\right)=15,000$. We get $\kappa =1.3827$ and ${\mathcal{R}}_0=0.2126=1.4173\times 0.15<1$. (a) Infectious non-immune and (b) infectious semi-immune.

Figure 12. Distribution of infected mosquitoes and immune humans with $d_l=4,r_v=15,d_v=3.5,b=180,{\gamma }_e={\gamma }_a=0$ and $a_1=5,b_1=3$. The initial conditions are given by $S_e\left(0\right)=500,E_e\left(0\right)=250,I_e\left(0\right)=150,S_a\left(0\right)=1000,E_a\left(0\right)=500,I_a\left(0\right)=1000,R_a\left(0\right)=2000,S_v\left(0\right)=10,000,E_v\left(0\right)=4000,I_v\left(0\right)=2000$ and $L\left(0\right)=15,000$. We get $\kappa =1.3827$ and ${\mathcal{R}}_0=0.2126=1.4173\times 0.15<1$. (a) Infectious mosquitoes and (b) immune humans.

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