Modeling malaria transmission in Nepal: impact of imported cases through cross-border mobility

Figures & data

Figure 1. Malaria transmission dynamics with cross-border mobility. The upper SI-SIRS (inside red dashed line) and lower SI-SIRS (inside blue dashed line) represent the dynamics of malaria abroad and in the home country. The solid arrows represent the transfer of populations, and the dotted arrows represent the interaction between the susceptible human and infectious female Anopheles mosquitoes and infectious humans with susceptible female Anopheles mosquitoes. Here, subscripts H, A, and M refer to home, abroad, and migrant, respectively, and the subscript h and v refer to human and vector (mosquito), respectively.

Figure 2. Model fitting to the data. (a) Solution of the fitted model along with the data of indigenous, imported, and total malaria incidences in Nepal, and (b) Model prediction of cumulative indigenous, cumulative imported, and cumulative total cases in Nepal.

Figure 3. Endemic equilibriums. (a) Graphs of $F_L\left(y\right)$ and $F_R\left(y\right)$ with possible three intersections corresponding to three endemic equilibrium points. (b) Graphs of $F_L\left(y\right)$ and $F_R\left(y\right)$ with exactly two endemic equilibrium points $y_1^{\ast }=y_2^{\ast }$ and $y_3^{\ast }$. Decreasing the slope of $F_L\left(y\right)$ further gives only one equilibrium point $y_3^{\ast }$ (a high epidemic level). (c) Graphs of $F_L\left(y\right)$ and $F_R\left(y\right)$ with exactly two endemic equilibrium points $y_1^{\ast }$ and $y_2^{\ast }=y_3^{\ast }$. Increasing the slope of $F_L\left(y\right)$ further gives only one equilibrium point $y_1^{\ast }$ (a low epidemic level).

Table 1. Base value of demographic variables of malaria in Nepal.

Description	State variables	Base Value	Reference
Risk humans population in Nepal	$N_{hH}\left(0\right)$	9,756,426	Calculated
Susceptible humans in Nepal	$S_{hH}\left(0\right)$	9,754,000	Calculated
Infectious humans in Nepal	$I_{hH}\left(0\right)$	2000	Assumed
Recovered humans in Nepal	$R_{hH}\left(0\right)$	426	Assumed
Susceptible Migrants in India	$S_{hM}\left(0\right)$	1,898,300	[29]
Infectious Migrants in India	$I_{hM}\left(0\right)$	1400	Assumed
Recovered Migrants in India	$R_{hM}\left(0\right)$	300	Assumed
Susceptible mosquitoes in Nepal	$S_{vH}\left(0\right)$	9,754,176	Assumed
Infectious mosquitoes in Nepal	$I_{vH}\left(0\right)$	2250	Assumed

Table 2. Model parameters of incidence of malaria in Nepal.

Description	Parameters	Base value and unit	Reference
Probability of malaria transmission from an infectious mosquito (per bite) to a susceptible human	${\alpha }_{vh}$	0.0195	[1,13]
Probability of malaria transmission from an infectious human to a susceptible mosquito (per bite)	${\alpha }_{hv}$	0.63	[5,9,13]
Humans recruitment rate	Λ	287,460 Number $\times {\mathrm{y}\mathrm{r}}^{-1}$	Calculated
Per capita recovery rate of humans	${\gamma }_h$	1.85 ${\mathrm{y}\mathrm{r}}^{-1}$	[1,13]
Per capita natural birth and death rate of mosquitoes	$d_v$	27.9113 ${\mathrm{y}\mathrm{r}}^{-1}$	[1,13]
Per capita disease induced death rate of humans	${\delta }_h$	0.00171${\mathrm{y}\mathrm{r}}^{-1}$	Calculated
Per capita natural death rate of humans	$d_h$	0.0149${\mathrm{y}\mathrm{r}}^{-1}$	Calculated
Per capita mobility rate of healthy humans	η	0.0183${\mathrm{y}\mathrm{r}}^{-1}$	Calculated
Per capita disease import rate of humans	θ	0.98 ${\mathrm{y}\mathrm{r}}^{-1}$	Estimated
		[0.4172, 1.5428]
Per capita disease export rate of humans	p	$0{\mathrm{y}\mathrm{r}}^{-1}$	Assumed
Per capita rate of Immunity loss of humans	q	4 ${\mathrm{y}\mathrm{r}}^{-1}$	[15]
Parameter related with incidence rate in India	ζ	$0.00105{\mathrm{y}\mathrm{r}}^{-1}$	Estimated
		[0.0005, 0.0016]
Per capita biting rate of mosquitoes at home country up to 2014	$b_1$	48 ${\mathrm{y}\mathrm{r}}^{-1}$	Estimated
		[39.6537, 66.3463]
Per capita biting rate of mosquitoes at home country after 2014	$b_2$	39.5 ${\mathrm{y}\mathrm{r}}^{-1}$	Estimated
		[29.6318, 49.3682]

Figure 4. Model prediction of the malaria epidemic in Nepal. (a) The model prediction of the annual incidence of indigenous, imported, and total malaria cases from 2020 to 2026; and (b) the model prediction of the cumulative cases of indigenous, imported and total malaria infection from 2020 to 2026.

Figure 5. Impact of the API of India. Reduction of total malaria incidence at the year 2026 (Left) and reduction of cumulative malaria cases from 2020–2026 (Right) with Annual Parasitic Incidence (API) of India taking its value 0.1 for the base year 2020.

Figure 6. Condition for malaria elimination in Nepal. Threshold indices $R_0$, $R_1$, $R_2$, $a_{10}$, $a_{11}$, $a_{12}$ as a function of controls ${\varphi }_{ITN},{\varphi }_{IRS},{\varphi }_{BSI}$, and ${\varphi }_{MR}$ for a low (first row) and high (second row) mosquito biting conditions. Note that the malaria is eliminated if $R_0<1,a_{10}>0$, $R_1<1,a_{11}>0$, and $R_2<1,a_{12}>0$, respectively, where $a_{10},a_{11}$, and $a_{12}$ are corresponding values of $a_1$ for case I (absence of cross-border mobility), case II (full protection of transmission abroad), and case III (strict border screening and isolation), respectively.

Figure 7. Effects of control strategies on the annual incidence rate. The model-predicted annual incidence rate in the year 2026 for various levels of ITN, IRS, BSI, and MR control in a low biting rate scenario (first row) and a high biting rate scenario (second row).

Figure 8. Effects of control strategies on the cumulative cases. The model-predicted cumulative cases for 2020–2026 for various levels of ITN, IRS, BSI, and MR control in a low biting rate scenario (first row) and a high biting rate scenario (second row).

Figure 9. Sensitivity of coverage and efficacy of ITN. The model-predicted annual incidence rate in 2026 for various levels of efficacy and coverage of ITN in a low biting rate scenario (a) and a high biting rate scenario (b). The model-predicted cumulative cases for 2020–2026 for various levels of efficacy and coverage of ITN in a low biting rate scenario (c) and a high biting rate scenario (d).

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