Zoonotic Visceral Leishmania: Modeling and Control

Visceral leishmaniasis is a vector-borne disease of humans and other mammals. This disease is caused by parasites of the Leishmania donovani complex. There are two main forms of visceral leishmania: (1) zoonotic visceral leishmaniasis (ZVL), which affects mainly young children and the domestic dog as its principal reservoir and (2) anthroponotic visceral leishmaniasis (AVL), this affects people of all ages, and infectious sand y transmit it from human to human via biting [1]. Visceral leishmaniasis (Vl) is severe and fatal. The average incubation period is 2-6 months; however it may vary from 10 days to one year [2,3]. Some of the patients recovered from V l, develops Post kala-Azar dermal leishmania with in the interval of 6 months to 3 years [4]. The vector latent period is assumed roughly to be 3 to 7 days [5,6].


Introduction
Visceral leishmaniasis is a vector-borne disease of humans and other mammals. This disease is caused by parasites of the Leishmania donovani complex. There are two main forms of visceral leishmania: (1) zoonotic visceral leishmaniasis (ZVL), which affects mainly young children and the domestic dog as its principal reservoir and (2) anthroponotic visceral leishmaniasis (AVL), this affects people of all ages, and infectious sand y transmit it from human to human via biting [1]. Visceral leishmaniasis (Vl) is severe and fatal. The average incubation period is 2-6 months; however it may vary from 10 days to one year [2,3]. Some of the patients recovered from V l, develops Post kala-Azar dermal leishmania with in the interval of 6 months to 3 years [4]. The vector latent period is assumed roughly to be 3 to 7 days [5,6].
No doubt leishmania control is challenging because the control of both sandflies and the reservoir is di cult. The failure rate of treatment is high due the two factors. Clinical structure of disease, the response of human immune system and the drug resistance acquired by the species [7].
Motivated from Hashim et al. [8] and Shillor et al. [9], the authors did not consider Homogenous population. We in our work have considered the homogenous mixing of the population. The Reproductive number so calculated, depends upon the densities of humans, reservoirs and vectors, which highlights the importance of homogenous mixing. Also we have applied new concept for calculating threshold condition, for disease free state as developed by Kamgang and Sallet [10].
In this paper, we present a mathematical model for the transmission dynamic of leishmaniasis. The model of 10 compartments includes 2 exposed classes of human infected with visceral leishmaniasis and PKDL. These exposed classes were not considered previously in the models. We find positive invariant region and use next generation matrix method to find the basic reproduction number R 0 . Using upper bound matrix A I (X) of the matrix A I (X), of the infected classes, the threshold number is found. Comparing R 0 and we find three values for R 0 . On the basis of these values, we discuss the dynamical behavior of the model. Finally we show the global stability of the disease free equilibrium, and the existence of endemic equilibrium.

Model Formulation
In this section we present the formulation of the model.
We divide the compartmental model of human, reservoir and vector populations into different classes. The human population consist of sub-classes, S h ; E 1 ; I 1 ; P 2 ; R 1 ; E 12 . Here S h represent the class of susceptible human, E 1 is the Vl infected class, E 12 is the class recovered from Vl and exposed to PKDL. P 2 is the human class with PKDL and R 1 is the human recovered class, I 1 is the human class infectious with Vl, The total human population N h is The vector population is divided into two sub-classes S v (t) and I v (t), also the reservoir class is divided into S r (t) and I r (t).
N v (t) = S v (t) + I v (t); N r (t) = S r (t) + I r (t): After susceptible person, being bitten by infectious vector, he/she can't transmit leishmania virus immediately. We call this person as infected (exposed). When a susceptible vector S v (t), bite the infectious person, the vector moves from susceptible compartment to the infectious compartment I v (t) [11].
The interaction of human, reservoir and vector population is represented in the flowchart as shown in Figure 1.
The dynamical system for human, reservoir and vector population is given by The description of the parameters is given in Table 1. ; b is transmission probability of V l to reservoir from sandfly.

Mathematical Analysis of the Model
In this section, we discuss invariant region, the disease free equilibrium point and reproductive number R 0 , of the system (1).

Invariant region
We have assumed all the parameters as nonnegative. Since the model is concerned with living population, therefore the state variables are assumed to be nonnegative at t=0. The dynamic of overall population is given by the following differential equations.
If the human population is disease free, i.e. I 1 = P 2 = 0, then equation (2) reduces to the form; Equilibrium in this case is From equation (2) and the fact that 1 2 The lower bond for equation (7) is given by The equilibrium of equation (8) is With the initial condition If N u , and N l , denote the solution of equation (5) and equation (8), then any solution of equation (2), satisfy Consider the biological feasible region Ω given by: [ ] From equation (2), using standard comparison theorem, we have Similarly Hence is positively invariant domain, and the model is epidemiologically and mathematically well posed.

Let us de ne a new region G as
Clearly G is the sub region of Ω. In light of equation (3), equation (4) and equation (11), it is reasonable to work on G instead of Ω.

Disease free equilibrium
The disease free equilibrium of the model (1) is given by:

Reproductive number
The number of secondary infections occurring in completely susceptible population by introducing an infectious individual to the population is called reproductive number R 0 [12]. In order to find the basic reproductive number, we use next generation method for R 0 = (-FV -1 ), [13]. Where is spectral radius? And After simplification, we get reproduction number We can further simplify to get The term R1 indicates that if sandfly is infectious and the reservoir is susceptible, the contact would result the transmission of Vl from sand y to reservoir. The term R 2 indicates the transmission of V l from reservoir to sand y. So the term R a indicate the transmission of V l between sandfly and reservoir. Similarly the term R b indicates the transmission of Vl between human and sand fly. The term R a and R b both denote the transmission of visceral strains of leishmania. There is no term representing the transmission of PKDL because it is the silent complication of V l. When a susceptible vector bites human/reservoir infected with PKDL, the vector does not transmit PKDL but transmit V l to the next victim. So R 0 is biologically sensible.

Stability Analysis
In this section, we discuss the relation between additional threshold number and basic ξ reproductive number R 0 , to find the global stability of the disease free equilibrium, and existence of endemic equilibrium of the system (1).

Proposition:
The disease free equilibrium is locally asymptotically stable if R 0 <1 and unstable if R 0 >1.

Proof:
For the proof of this result verify the reference [13].

Global stability of the disease free equilibrium
To find the global stability of the disease free equilibrium of the system (1), we state some definitions [9,10].

Definition 3: The compact set
⊂ Ω M is called stable for the dynamical system defined on Ω if for every trajectory initiated from a point in U is in W, for all t≥0. Here U and W are neighborhoods of M.

Definition 4: A compact set
⊂ N D is called an attractor for a dynamical system defined on D if there exist a nbhds X and Y of N such that for every point ∈ X x , there exists a time t x,y >0, such that every trajectory initiated at x, belongs to Y for t>t x,y . The largest set X is called a bassin of attraction.
If X=D the set N is then called global attractor. A set N which is both stable and a global attractor is called globally asymptotically stable.

Theorem:
The set G is globally asymptotically stable for the dynamical system (1) defined on Ω.
For any initial point of the model (1), Thus G is global attractor.

Next to show that G is stable
On the basis of monotonicity of , , , Thus we have shown that any solution of the model (1), starting from a point in . So G is stable. Thus G is globally asymptotically stable. Hence we can now study the system (1) on G, instead of Ω .  So the model can now be written as And the matrix A I (X) is given by We restrict the domain of the system (1) from G to G, to ensure the irre-ducibility of A I (X), such that The set G is positively invariant because only the initial point of any trajectory can have X s = 0, Putting is globally asymptotically stable equilibrium point of the system (1) reduced to the sub-domain { ; 0} ∈ = I X G X .

Corollary:
The system (14) is globally asymptotically stable if there exist a matrix I A such that and if Where α is stability modulus or the largest real part of the eigen values of I A So the upper bond of ( ) I A X denoted by I A is given by  We take this value as ξ. Thus H 5 or equation (17)

Simulation results of the model
In the Figure 2 below, we have reduced the treatment rate of both Vl infected and PKDL infected humans, in the sense that we have used drugs other than sodium stibogluconate (expensive medicine) or that the hospital is far away or that the case is not properly diagnosed leading to wrong treatment. No mass awareness program is lunched for vector control. Taking γ 1 =γ 2 =0:023, a=0:2856 (normal); and α 1 =0:064.  The graph shows that it takes long time to eradicate the diseases.
In Figure 3 we have increased the treatment rate for both Vl and PKDL and also a proper arrangement for vector control. Taking γ 1 =0:5, γ 2 =0:4, biting of sandfly a=0.1856 medicine effectiveness α 1 =0:74. The graph shows that with in short time the disease can be eradicated.

Conclusion
In this work a mathematical model of leishmania transmission was presented. The novelty of the model is, the homogenous mixing of human, reservoir and vector. The basic reproduction number R 0 so calculated, depends upon the density of human, vectors and reservoirs, which highlights the importance of homogenous mixing. R0 is most sensitive to a; b and c and can have value greater than 1 (endemic state), if a; sand y biting rate, b; transmission probability of either strain in reservoir from sand y and c, transmission probability of either strain in sand y from reservoir, were not controlled. For this, different measures to control phlebotomine sandfies, like residual spraying of dwellings and animal shelters, insecticide treated nets; application of repellents/ insecticides to skin or to fabrics and impregnated dog collars may be taken. Sand y is susceptible to all the major insecticidal groups. In ZVL foci, where dogs are the unique domestic reservoir, a reduction in Leishmania transmission would be expected if we could combine an effective mass treatment of infected dogs with a protection of both healthy and infected dogs from the sand y bites. Since sand y can y up to the range of 1km, so leishmania transmission in dogs can be controlled, if they were kept away at least by 1km, from villages and cities. The disease can be controlled in human within a short time, however in reservoir class; the disease control takes long time. It is suggested to cull PCR+ dogs; this strategy gives imminent results in disease control.