Control strategies and sensitivity analysis of anthroponotic visceral leishmaniasis model

Figures & data

Figure 1. Flow chart representing the dynamics of the disease. The blue lines represent the interaction between susceptible and infected populations.

Table

	Table of parameters
Notation	Parameter definition	Value	Resource
${\mathrm{\Gamma }}_{\mathrm{h}}$	Humans' recruitment rate	0.0015875 day^−1	[9]
${\mathrm{\Gamma }}_{\mathrm{v}}$	Sandflies' recruitment rate	0.299 day^−1	[13]
${\mu }_{\mathrm{h}}$	Humans' natural mortality rate	0.00004 day^−1	[13]
${\mu }_{\mathrm{v}}$	Sandflies' natural mortality rate	0.189 day^−1	[13]
${\xi }_1$	Inverse of sojourn time at $E_1$	0.0166666	[11]
${\xi }_2$	Inverse of sojourn period at $E_2$	0.0833333	[23]
${\xi }_3$	Inverse of sojourn period at $I_1$,	1	[23]
${\tau }_1$	Inverse of first-line treatment period	0.0333333 day^−1	[10]
${\tau }_2$	Inverse of second-line treatment period	0.0333333 day^−1	[10]
${\tau }_3$	Inverse of treatment period of PKDL	0.0055555 day^−1	[20]
${\alpha }_2$	Inverse of relapse period to PKDL	0.0015873 day^−1	[20, 27]
${\alpha }_3$	Inverse of sojourn period at $R_1$	0.013513	[23]
${\alpha }_1$	Inverse of sojourn period at $R_2$	0.003257	[23]
${\alpha }_4$	Inverse of sojourn period at $R_3$	0.013513	[23]
$p_1$	Fraction not responding to first-line treatment	0.05	[10]
$p_2$	Fraction developing PKDL after first-line treatment	0.03	[23]
$p_3$	Fraction of VL recovering with first-line treatment	0.92	[23]
$p_4$	Fraction developing PKDL after second-line treatment	0.03	[23]
$p_5$	Fraction recovering from VL with second-line treatment	0.97	[23]
${\sigma }_{\mathrm{v}}$	Inverse of incubation period at $E_{\mathrm{v}}$	0.2	[22]
a	Sandflies biting rate	0.2856 day^−1	[7]
${\mu }_{T1}$	Excess in mortality due to first-line treatment toxicity	0.00167	[23]
${\mu }_{T2}$	Excess in mortality due to second-line treatment toxicity	0.00167	[23]
${\mu }_k$	VL-induced death rate	0.011 day^−1	[24]
${\nu }_1$	Effect of control of sandfly biting	(0,1)	Free to choose
${\nu }_2$	Effect of ‘control on sandfly population’	(0,1)	Free to choose
$c_1$	Transmission probability of Vl to sandfly from class $E_1$	0.0125	[23]
$c_2$	Transmission probability of Vl to sandfly from class $E_2$	0.025	[23]
$c_3$	Transmission probability of Vl to sandfly from states $I_1,I_2,I_3$	1	[23]
$c_4$	Transmission probability of Vl to sandfly from class $I_4$	1	[23]
b	Transmission probability of Vl in humans	1	[23]
${\omega }_1$	Fraction of $E_2$, entering state $I_1$	0.0033	[23]
${\omega }_2$	Fraction of $E_2$, entering state $E_3$	0.0001	[23]
${\omega }_3$	Fraction of $E_2$, entering state $R_1$	0.0066	[23]
β	Inverse of feeding cycle period of sandfly	0.25	[23]
${\beta }_1$	PKDL natural healing rate	0.00556 day^−1	[8]
${\beta }_2$	Fraction recovering from PKDL due to treatment	0.981	[17]

Table 1. The sensitivity indices of parameters.

Parameter	Index	Parameter	Index
a	+1	b	+0.5
$c_1$	+0.0000088	$c_2$	+0.499
$c_4$	+0.00002	${\sigma }_{\mathrm{v}}$	+0.49
${\xi }_1$	+0.00003	${\xi }_2$	−0.997
${\omega }_1$	−0.65	${\omega }_2$	−0.004
${\omega }_3$	−0.314	${\alpha }_2$	0.0000006
${\beta }_2$	−0.0000007	${\mu }_{\mathrm{h}}$	0.498
${\mu }_{\mathrm{v}}$	−1	${\mathrm{\Gamma }}_{\mathrm{v}}$	+0.5
${\mathrm{\Gamma }}_{\mathrm{h}}$	−0.5	${\nu }_1$	−0.42
${\nu }_2$	−0.17	β	1

Table 2. Strategies based on treatment and vector-related interventions.

Strategy	$\frac{1}{{\tau }_1}$	$\frac{1}{{\tau }_2}$	$\frac{1}{{\tau }_3}$	$\frac{1}{{\xi }_3}$	${\nu }_1$	${\nu }_2$	Π
1	30 days	30 days	180 days	5 days	0.09	0.06	0.0416965936
2	15 days	15 days	90 days	1 day	0.09	0.06	0.0416965936
3	30 days	30 days	180 days	5 days	0.5	0.3	0.00896962611
4	15 days	15 days	90 days	1 day	0.5	0.09	0.01204364265

Figure 2. The time spent in elimination of infectious classes, using control strategy No. 1.

Figure 3. The time spent in elimination of infectious classes, using control strategy No. 2.

Figure 4. The time spent in elimination of infectious classes, using control strategy No. 3.

Figure 5. The time spent in elimination of infectious classes, using control strategy No. 4.

Table 3. Results of control strategies.

Strategy	$E_1,\mathrm{TS}$	$E_2,\mathrm{TS}$	$E_3,\mathrm{TS}$	$I_2,\mathrm{TS}$	$I_3,\mathrm{TS}$	$I_4,\mathrm{TS}$	$I_{\mathrm{v}},\mathrm{TS}$	$E_{\mathrm{v}},\mathrm{TS}$
1	580	500	2400	200	130	2600	65	57
2	460	457	2400	130	70	2400	65	57
3	350	320	2700	180	130	2700	25	15
4	580	570	2400	180	140	2700	55	65

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