Treatment of leishmaniasis with chemotherapy and vaccine: a mathematical model

Figures & data

Table 1. List of variables of the model.

Variables	Descriptions	Variables	Descriptions
D	Density of dendritic cells	$M_1$	Density of pro-inflammatory
$M_2$	Density of anti-inflammatory macrophages	T	Density of combined CD4${}^{+}$ and CD8${}^{+}$ T cells
$P_1$	Density of Leishmania in M1 macrophages	$P_2$	Density of Leishmania in M2 macrophages
$I_2$	Concentration of IL-2	$I_{10}$	Concentration of IL-10; IL-10 is combined with IL-13
$I_{12}$	Concentration of IL-12	$I_{\gamma }$	Concentration of IFN-γ
NO	Concentration of Nitric Oxide	$T_{\alpha }$	Concentration of TNF-α
$T_{\beta }$	Concentration of TGF-β
A	Concentration of meglumine antimoniate (MA)	S	Concentration of sodium antimony gluconate (SAG)
V	Concentration of vaccine LEISH-F1+MPL-SE

Figure 1. Effects of vaccine LEISH-F1+MPL-SE (V), and drugs sodium antimony gluconate (SAG, S) and meglumine antimoniate (MA, A) on the immune response to Leishmania infection.

Table

$D(0)=4\times {10}^{-4}$	$M_1(0)=1.1\times {10}^{-3}$	$M_2=4.65\times {10}^{-4}$	$T(0)=3.5\times {10}^{-4}$
$P_1(0)=4.65\times {10}^{-5}$	$P_2(0)=3.5\times {10}^{-5}$	$I_2(0)=7.5\times {10}^{-11}$	$I_{10}(0)=2.4\times {10}^{-12}$
$I_{12}(0)=6.4\times {10}^{-12}$	$I_{\gamma }(0)=2.3\times {10}^{-11}$	$NO(0)=4.73\times {10}^{-8}$	$T_{\alpha }(0)=1.7\times {10}^{-11}$
$T_{\beta }(0)=2.4\times {10}^{-12}$	$A(0)=0$	$S(0)=0$	$V(0)=0$

Figure 2. No self-healing; ${\lambda }_{NO{P}_1}={\lambda }_{I_{\gamma }{P}_1}={\lambda }_{I_{\gamma }{P}_2}={\lambda }_{T_{\alpha }{P}_1}={\lambda }_{T_{\alpha }{P}_2}=0.062$, ${\lambda }_{NO{P}_2}=0.56$. (a) All variables with no self-healing and (b) all macrophages and parasites with no self-healing.

Figure 3. Self-healing; (factor of 3) ${\lambda }_{NO{P}_1}={\lambda }_{NO{P}_2}={\lambda }_{I_{\gamma }{P}_1}={\lambda }_{I_{\gamma }{P}_2}={\lambda }_{T_{\alpha }{P}_1}={\lambda }_{T_{\alpha }{P}_2}=0.186$, ${\lambda }_{NO{P}_2}=1.67$. (a) All variables with self-healing and (b) all macrophages and parasites with self-healing.

Table 2. Descriptions, values, and sources of the model parameters.

Parameter	Description	Value/range	Reference
$K_{I_2}$	Half-saturation of IL-2	$14\times {10}^{-11}$ g/cm${}^3$	[8]est.
$K_{I_{10}}$	Half-saturation of IL-10	$1.53\times {10}^{-12}$ g/cm${}^3$	[26]est.
$K_{I_{12}}$	Half-saturation of IL-12	$9\times {10}^{-12}$ g/cm${}^3$	[3]est.
$K_{I_{\gamma }}$	Half-saturation of IFN-γ	$4.37\times {10}^{-11}$ g/cm${}^3$	[26]est.
$K_{T_{\alpha }}$	Half-saturation of TNF-α	$21.83\times {10}^{-12}$ g/cm${}^3$	[48]est.
$K_{T_{\beta }}$	Half-saturation of TGF-β	$1.53\times {10}^{-12}$ g/cm${}^3$	fitted
$K_{NO}$	Half-saturation of NO	$5.36\times {10}^{-8}$ g/cm${}^3$	[4, 30]est.
$K_D$	Half-saturation of DCs	$7\times {10}^{-4}$ g/cm${}^3$	[79]
$K_{M_1}$	Half-saturation of M1 macrophages	$1.2\times {10}^{-3}$ g/cm${}^3$	[79, 80]est.
$K_{M_2}$	Half-saturation of M2 macrophages	$3\times {10}^{-4}$ g/cm${}^3$	[79, 80]est.
$K_P$	Half-saturation of parasites	$5\times {10}^{-4}$ g/cm${}^3$	[79, 80]est.
$K_T$	Half-saturation of T cells	$6.5\times {10}^{-4}$ g/cm${}^3$	[49]est.
${\lambda }_{M_2{M}_1{I}_{\gamma }}$	IFN-γ-induced transition rate from $M_2$ to $M_1$	0.69 d${}^{-1}$	[39, 67]
${\lambda }_{M_2{M}_1{T}_{\alpha }}$	TNF-α-induced transition rate from $M_2$ to $M_1$	0.69 d${}^{-1}$	est.
${\lambda }_{M_1{M}_2{I}_{10}}$	$I_{10}$-induced transition rate from $M_1$ to $M_2$	0.27 d${}^{-1}$	fitted
${\lambda }_{M_1{M}_2{T}_{\beta }}$	$T_{\beta }$-induced transition rate from $M_1$ to $M_2$	0.27 d${}^{-1}$	fitted
${\lambda }_{T{I}_2}$	Production rate of $I_2$ by T cells	$7.1\times {10}^{-5}$ d${}^{-1}$	est.
${\lambda }_{M_1{I}_{12}}$	Production rate of IL-12 by $M_1$	$2.1\times {10}^{-8}$ d${}^{-1}$	est.
${\lambda }_{M_2{I}_{10}}$	Production rate of IL-10 by $M_2$ macrophages	$7.4\times {10}^{-8}$ d${}^{-1}$	est.
${\lambda }_{M_1{T}_{\alpha }}$	Production rate of TNF-α by M1 macrophages	$2.35\times {10}^{-6}$ d${}^{-1}$	est.
${\lambda }_{T{T}_{\alpha }}$	Production rate of TNF-α by T cells	$2.35\times {10}^{-6}$ d${}^{-1}$	est.
${\lambda }_{T{I}_{\gamma }}$	Production rate of IFN-γ by T cells	$2.22\times {10}^{-6}$ d${}^{-1}$	est.
${\lambda }_{M_2{T}_{\beta }}$	production rate of TGF-β by M2 macrophages	$2.04\times {10}^{-6}$ d${}^{-1}$	est.
${\lambda }_{MNO}$	Production rate of NO by macrophages	$2.4\times {10}^{-3}$ d${}^{-1}$	est.

Figure 4. All macrophages and parasites with no healing: (factor of 1.7) ${\lambda }_{NO{P}_1}={\lambda }_{I_{\gamma }{P}_1}={\lambda }_{I_{\gamma }{P}_2}={\lambda }_{T_{\alpha }{P}_1}={\lambda }_{T_{\alpha }{P}_2}=0.122$, ${\lambda }_{NO{P}_2}=1.106$.

Figure 5. Schedule for drug administration. As in [65], shots of the vaccine LEISH-F1+MPL-SE adjuvant are given at Days 0, 28 and 56 for the LEISH-F1 and the MPL-SE adjuvant; meglumine antimoniate is administered daily, in cycles of 21 days, for the first 10 days followed by 11 days of rest.

Figure 6. Treatment of leishmaniasis with MA (A), vaccine LEISH-F1+MPL-SE (V) and MPL-SE adjuvant as in [65]. The numbers in parentheses represent the recovery rates.

Figure 7. Treatment of leishmaniasis with MA (A), vaccine LEISH-F1+MPL-SE (V) and MPL-SE adjuvant as in [65]. Profiles of total parasites load with alternative doses of V . The numbers in parentheses represent the recovery rates.

Figure 8. Treatment of leishmaniasis with SAG, vaccine LEISH-F1+MPL-SE (V) and MPL-SE adjuvant. The numbers in parentheses represent the recovery rates.

Figure 9. Standard protocol treatment of leishmaniasis with MA and vaccine LEISH-F1+MPL-SE. The vaccine is given at Days 0 and 28 and 56, and the chemotherapy is administered at Days 0–20, daily. The numbers in parentheses are the recovery rates at Day 84.

Figure 10. Standard protocol treatment of leishmaniasis with MA and vaccine LEISH-F1+MPL-SE. The vaccine is given at (a) Days 0 and 28 and (b) Day 0; the chemotherapy is administered at Days 0–20, daily. The numbers in parentheses are the recovery rates at Day 84. (a) 2 vaccine dose and (b) 1 vaccine dose.

Figure 11. Long-run total parasite load ($P_1+P_2$ at day 84), $P_1+P_2$, combination of (a) MA with LEISH-F1+MPL-SE and (b) SAG with LEISH-F1+MPL-SE. The parasite loads are in units of g/cm${}^3$.

Figure 12. Parameter sensitivity analysis for the total parasite load after 84 days. The p-values are less than .01.

Table 3. Descriptions, values, and sources of the model parameters.

Parameter	Description	Value/range	Reference
${\lambda }_{NO{P}_1}$	Strength of NO-effect on parasites $P_1$	0.062	Fitted
${\lambda }_{NO{P}_2}$	Strength of NO-effect on parasites $P_2$	0.56	Fitted
${\lambda }_{I_{\gamma }NO}$	Strength of $I_{\gamma }$-effect on NO production by $M_1$	2	Est.
${\lambda }_{I_{\gamma }{P}_1}$	Strength of $I_{\gamma }$-effect on parasites $P_1$	0.062	Fitted
${\lambda }_{I_{\gamma }{P}_2}$	Strength of $I_{\gamma }$-effect on parasites $P_2$	0.062	Fitted
${\lambda }_{T_{\alpha }{P}_1}$	Strength of $T_{\alpha }$-effect on parasites $P_1$	0.062	Fitted
${\lambda }_{T_{\alpha }{P}_2}$	Strength of $T_{\alpha }$-effect on parasites $P_2$	0.062	Fitted
${\lambda }_{A{I}_{10}}$	Effect of SAG on IL-10 secretion	$5\times {10}^{10}$	Fitted
${\lambda }_{S{I}_2}$	Effect of SAG on IL-2 secretion	$2\times {10}^4$	Fitted
${\lambda }_{V{I}_2}$	Effect of LEISH-F1+MPL-SE on IL-2 secretion	${10}^{10}$ cm${}^3$/g	Fitted
${\lambda }_{S{I}_{12}}$	Effect of SAG on IL-12 secretion	$4\times {10}^2$	Fitted
${\lambda }_{V{I}_{\gamma }}$	Effect of LEISH-F1+MPL-SE on IFN-γ secretion	${10}^{10}$ cm${}^3$/g	Fitted
${\lambda }_{A{T}_{\alpha }}$	Effect of meglumine antimoniate on TNF-α secretion	${10}^{10}$ cm${}^3$/g	Fitted
${\lambda }_{V{T}_{\alpha }}$	Effect of LEISH-F1+MPL-SE on TNF-α secretion	${10}^{10}$ cm${}^3$/g	Fitted
${\lambda }_{VT}$	Transition rate of T cells to memory cells	$8\times {10}^3$ cm${}^3$/g d${}^{-1}$	Fitted
${\lambda }_{SNO}$	Effect of SAG on NO secretion	$1.7\times {10}^3$	Fitted
${\lambda }_{ANO}$	Effect of meglumine antimoniate on NO secretion	${10}^7$ cm${}^3$/g	Fitted
${\lambda }_{M_1}$	Basal production rate of M1 macrophages	$4.94\times {10}^{-5}$ g/cm${}^3$ d${}^{-1}$	Fitted
${\lambda }_{M_2}$	Basal production rate of M2 macrophages	$2.625\times {10}^{-5}$ g/cm${}^3$ d${}^{-1}$	Fitted
${\lambda }_{P_1}$	Intrinsic growth rate of parasites within $M_1$	2.21 d${}^{-1}$	[18, 29, 79]
${\lambda }_{P_2}$	Intrinsic growth rate of parasites within $M_2$	3.82 d${}^{-1}$	[18, 29, 79]
${\lambda }_T$	$I_2$-induced rate of proliferation of T cells	$1.9\times {10}^{-4}$ d${}^{-1}$	Est.
${\lambda }_D$	Activation rate of DCs	0.66 d${}^{-1}$	Est.
${\lambda }_{DT{T}_0}$	Activation rate of T cells by naive $T_0$ cells	$3.42\times {10}^{-4}$ g/cm${}^3$ d${}^{-1}$	Est.
${\lambda }_{MT}$	Activation rate of T cells by M1 macrophages	$3.8\times {10}^{-4}$ d${}^{-1}$	Est.
${\beta }_A,{\beta }_S,{\beta }_V$	PK exponents of A, S and V	0.016, 0.01, 0.01	Fitted
θ	Fraction of parasites phagocyted by $M_1$ after burst	0.8	Fitted
N	Average number of parasites for macrophage burst	50 (30–50)	[18]
$N_1$	Average number of parasites in M1 macrophages	2	[18]
$N_2$	Average number of parasites in M2 macrophages	20	[18]
$N_{\ast }$	Maximum number of parasites in macrophages	50	[18, 79]
$c_{I_{\gamma }\text{adj}}$	Augmented rate of IFN-γ secretion by MPL-SE adjuvant	0.35	Fitted
${\mu }_D$	Death rate of DCs	0.1 d${}^{-1}$	[39]
${\mu }_{NO}$	Decay rate of NO	134.6 d${}^{-1}$	[36]Est.
${\mu }_{M_1}$	Death rate of M1 macrophages	0.056 d${}^{-1}$	[79, 91]
${\mu }_{M_2}$	Death rate of M2 macrophages	0.014 d${}^{-1}$	[79, 91]
${\mu }_{P_1}$	Death rate of $P_1$ parasites	1.85 d${}^{-1}$	[33, 79, 87]
${\mu }_{P_2}$	Death rate of $P_2$ parasites	2.22 d${}^{-1}$	[33, 79, 87]
${\mu }_T$	Death rate of T cells	0.197 d${}^{-1}$	[20, 40, 93]
${\mu }_{I_2}$	Decay rate of IL-2	330 d${}^{-1}$	[24]
${\mu }_{I_{10}}$	Decay rate of IL-10	14.5 d${}^{-1}$	[45, 51, 63]
${\mu }_{I_{12}}$	Decay rate of IL-12	1.39 d${}^{-1}$	[5]
${\mu }_{I_{\gamma }}$	Decay rate of IFN-γ	33 d${}^{-1}$	[44]
${\mu }_{T_{\alpha }}$	Decay rate of TNF-α	199 d${}^{-1}$	[77, 81]
${\mu }_{T_{\beta }}$	Decay rate of TGF-β	399.25 d${}^{-1}$	[78, 85]
${\mu }_A$	Decay rate of meglumine antimoniate	4.95 d${}^{-1}$	[35]Est.
${\mu }_S$	Decay rate of SAG	1.65 d${}^{-1}$	[57]Est.
${\mu }_V$	Decay rate of vaccine LEISH-F1+MPL-SE	${10}^{-3}$ d${}^{-1}$	Fitted
${\tilde{\mu }}_{M_2}$	Bursting rate of $M_2$	0.17 d${}^{-1}$	[79, 83]
${\tilde{\mu }}_{M_1}$	Bursting rate of $M_1$	0.017 d${}^{-1}$	[72, 79]
${\mu }_{MA}$	Depletion rate of meglumine antimoniate by M1 macrophages	${10}^{1}0$ cm${}^3$/g d${}^{-}1$	Fitted
${\mu }_{SM}$	Depletion rate of SAG by M1 and M2	1 cm${}^3$/g d${}^{-1}$	Fitted
${\mu }_{ST}$	Depletion rate of SAG by T cells	1 cm${}^3$/g d${}^{-1}$	Fitted
${\mu }_{VT}$	Depletion rate of vaccine LEISH-F1+MPL-SE by T cells	100 cm${}^3$/g d${}^{-}1$	Fitted
$D_0$	Density of DC in healthy	$0.7\times {10}^{-3}$ g/cm${}^3$	[43, 79]

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Data availability statement

All relevant data are within the manuscript and its Supporting information files.