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Zoonotic Visceral Leishmania: Modeling and Control | OMICS International
ISSN: 2168-9679
Journal of Applied & Computational Mathematics
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Zoonotic Visceral Leishmania: Modeling and Control

Zamir Muhammad1* and Rahmat Ali2

1Department of Mathematics, University of Science and Technology Bannu, Khyber Pakhtunkhwa, Pakistan

2Department of Mathematics, University of Malakand Chakdara Lower Dir, Khyber Pakhtunkhwa, Pakistan

*Corresponding Author:
Zamir M
Department of Mathematics, University of Science and Technology Bannu
Khyber Pakhtunkhwa, Pakistan
Tel: 0092331917093
E-mail: [email protected]

Received May 17, 2015; Accepted July 17, 2015; Published July 24, 2015

Citation: Muhammad Z, Ali R (2015) Zoonotic Visceral Leishmania: Modeling and Control. J Appl Computat Math 4: 238. doi:10.4172/2168-9679.1000238

Copyright: © 2015 Muhammad Z, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

In this work we focus on the transmission dynamics of Visceral strains of leishmania, using mathematical model with two latent compartments in human. From the governing differential equations of the model, we find the reproductive number R0; the number of secondary infection and its biological interpretation. Using Routh- Hurwitz criteria on upper bound matrix, the threshold condition, for stability of the Disease Free State, is calculated. Finally we show that the disease free equilibrium is globally asymptotically stable if R0<<ξ 11.

Keywords

Leishmaniasis; Basic reproductive number; Mathematical model; Local and global stability

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 R0. Using upper bound matrix AI(X) of the matrix AI(X), of the infected classes, the threshold number is found. Comparing R0 and we find three values for R0. 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, Sh; E1; I1; P2; R1; E12. Here Sh represent the class of susceptible human, E1 is the V1 infected class, E12 is the class recovered from Vl and exposed to PKDL. P2 is the human class with PKDL and R1 is the human recovered class, I1 is the human class infectious with V1l,

The total human population Nh is

Nh = Sh+ E1 + I1 + E12 + P2 + R1

The vector population is divided into two sub-classes Sv(t) and Iv(t), also the reservoir class is divided into Sr(t) and Ir(t).

Nv(t) = Sv(t) + Iv(t); Nr(t) = Sr(t) + Ir(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 Sv(t), bite the infectious person, the vector moves from susceptible compartment to the infectious compartment Iv(t) [11].

The interaction of human, reservoir and vector population is represented in the flowchart as shown in Figure 1.

computational-mathematics-interaction-human-reservoir

Figure 1: The interaction of human, reservoir and vector population.

The dynamical system for human, reservoir and vector population is given by

equation (1)

The description of the parameters is given in Table 1.

Notation Parameter definition Value Resource
c2 Progression rate of VL in sand y(from human) 0.22 [14]
a Sandflies biting rate 0.2856day-1 [14]
Γh Recruitment rate of human 0.0015875day-1 [15]
Γv Recruitment rate of sandfly 0.299day-1 [16]
Γr Recruitment rate of reservoir 0.073day-1 Assumed
Γh Natural mortality rate of human 0.00004day-1 [16]
µv Natural mortality rate of Sandflies 0.189day-1 [16]
µr Natural mortality rate of Reservoirs 0.000274day-1 Assumed
µ2 PKDL recovery rate after treatment 0.033day-1 [17]
1−α1 Developing PKDL rate after treatment 0.36day-1 [17]
β1 PKDL natural healing rate 0.00556day-1 [17]
c Progression rate of Vl in sandfly (from reseroir) Variable Variable
b Progression rate of Vl in reservoir(from sandfly) Variable Variable
γ1 Treatment rate of VL variable Assumed
δ1 VL induced death rate 0.011day-1 [18]
k2 1/k2is Incubation period of vl 0.006555day-1 [19]
k3 1/k3 is Incubation period of PKDL 0.004925925day-1 [2,20]
δ2 PKDL induced death rate assumed Assumed
b2 Progression rate of VL in human (from sandfly) 0.0714day-1 [21]

Table 1: Description of the parameters.

The terms of interaction λh, λr and λvare as under equation is the average rate of infection rate of human with Vl, from infectious sandfly.

λr is the average rate of infection of susceptible reservoir by infected sandfly.

equation; b is transmission probability of V l to reservoir from sandfly.

equation, is the average rate of infection of sandfly with Vl strain from human or reservoir. Where c2 is the transmission probability of Vl from human in stage I1 and P2 to sandfly

Mathematical Analysis of the Model

In this section, we discuss invariant region, the disease free equilibrium point and reproductive number R0, 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.

equation (2)

equation (3)

equation (4)

If the human population is disease free, i.e. I1 = P2 = 0, then equation (2) reduces to the form;

equation (5)

Equilibrium in this case is

equation (6)

From equation (2) and the fact that equation, we have

equation (7)

The lower bond for equation (7) is given by

equation (8)

The equilibrium of equation (8) is

equation (9)

With the initial condition

Nh(0) = N0. (10)

If Nu, and Nl, denote the solution of equation (5) and equation (8), then any solution of equation (2), satisfy

N1 ≤ Nh ≤ Nu.(11)

Consider the biological feasible region Ω given by:

equation

From equation (2), using standard comparison theorem, we have

equation

So

equation as equation

Similarly

equation and equation

Hence is positively invariant domain, and the model is epidemiologically and mathematically well posed.

Let us de ne a new region G as

equation

where

equation

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:

equation

Reproductive number

The number of secondary infections occurring in completely susceptible population by introducing an infectious individual to the population is called reproductive number R0 [12]. In order to find the basic reproductive number, we use next generation method for R0= (-FV-1), [13]. Where is spectral radius? And

equation

Here

equation

equation

with

equation

equation

After simplification, we get reproduction number

equation

We can further simplify to get equation where equation

equation

equation

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 R2 indicates the transmission of Vl from reservoir to sand y. So the term Ra indicate the transmission of Vl between sandfly and reservoir. Similarly the term Rb indicates the transmission of Vl between human and sand fly. The term Ra and Rb 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 V1 to the next victim. So R0 is biologically sensible.

Stability Analysis

In this section, we discuss the relation between additional threshold number and basic ξ reproductive number R0, 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 R0<1 and unstable if R0>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 1: An m × m matrix, for m>2 is called irreducible if for any proper sub-set I of {1, 2,…,m}, ∃ , equation and equation such that equation

Definition 3: The compact set M ⊂ Ω 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 tx,y>0, such that every trajectory initiated at x, belongs to Y for t>tx,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.

Proof: Let

equation (12)

equation (13)

be initial conditions associated with equation (3) and equation (4). And for equation , equation be defined as;

equation

where

equation

Since the collection equation is a complete neighborhood system of the compact set G. So X and Y as discussed in above definitions, also belong to this collection.

Consider an arbitrary equation. The points equation are globally asymptotically stable equilibria of the dynamical system defined by equation (3), equation (4), equation (5), and equation (8) on (0, ∞).

For any initial point of the model (1), equation Hence there exists equation so that for any equation we have

equation

equation

equation

equation

Thus G is global attractor.

Next to show that G is stable

On the basis of monotonicity of equation, we have

equation

Thus we have shown that any solution of the model (1), starting from a point in equation, remains in equation. So G is stable. Thus G is globally asymptotically stable. Hence we can now study the system (1) on G, instead of Ω .

Theorem: Let a positive system be defined on set equation and let ε ⊂ Ω be globally asymptotically stable. Let M be the largest invariant sub set ofε . Then M is globally asymptotically stable on Ω . Particularly if M = {x*}where x* is equilibrium point of the system with basin of attraction containingε . Then x* is GAS for the system onΩ .

Proof: For the proof of the theorem verify the reference [9] theorem (5). To prove the global stability of the disease free equilibrium, we use theorem (4.3) of [10].

For this let

equation

Now for global asymptotic stability of the disease free equilibrium of the system(1) on smaller set G, we decompose X as, Xs and XI of noninfected and infected, humans reservoirs and sandies, such that

equation

equation

So the model can now be written as

equation

where

equation

equation

And the matrix AI(X) is given by

equation

We restrict the domain of the system (1) from G to G, to ensure the irre-ducibility of AI(X), such that equation.

The set equation is positively invariant because only the initial point of any trajectory can have Xs = 0, Putting equation , in the system (1), we have equation .

So all of the diagonal entries of AI (X) are nonnegative, hence AI (X) is metzler and irreducible equation.

Since diagonal entries of AS are negative. So we state the following result

Proposition: Let equation be the non-infected class of the total population, then

equation

is globally asymptotically stable equilibrium point of the system (1) reduced to the sub-domain equation

Corollary: The system (14) is globally asymptotically stable if there exist a matrix equation such that

equation (15)

and if

equation for some equation then equation (16)

equation (17)

Where α is stability modulus or the largest real part of the eigen values of equation

Proof:

Since

equation

So the upper bond of equation denoted by equation is given by

equation

and

Clearly equation as equation

And

equation only if; equation

Thus H4 of theorem (4.3) holds [10], equivalently equation (15) and equation (16), hold.

To show that H5 or equation (18) holds, we state the following theorem.

Theorem: The metzler matrix satisfy the axiom equation if the basic reproductive number R0 satisfy the inequality; R0 ≤ξ , where ξ, is the additional threshold number given by

equation

equation

Proof: We decompose the matrix equation in the blocks such that

equation

where L, M, P,Q are 3× 3 sub-matrices. The matrix equation is stable if S and equation are metzler stable. Here S is metzler stable, because all its off diagonal entries are nonnegative, and all the eigen values are negative.

Let

equation

Thenequation is stable if Y is stable.

And Y is stable if det(Y) ≥ 0

This means thatα equation only if

equation

where

equation

At the disease free equilibrium,

equation

equation

By putting these values in above equation, we have

equation

We take this value as ξ. Thus H5 or equation (17) holds, if ξ ≥1 . Also R0 <ξ . So using theorem (4.3) of [10], we claim the following result.

Theorem: If the parameters of the model satisfy the condition equation , then the disease free equilibrium of the system (1) is globally asymptotically stable.

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 γ12=0:023, a=0:2856 (normal); and α1=0:064. The graph shows that it takes long time to eradicate the diseases.

computational-mathematics-population-behavior-graph-1

Figure 2: Population behavior graph 1.

In Figure 3 we have increased the treatment rate for both Vl and PKDL and also a proper arrangement for vector control. Taking γ1=0:5, γ2=0:4, biting of sandfly a=0.1856 medicine effectiveness α1=0:74. The graph shows that with in short time the disease can be eradicated.

computational-mathematics-population-behavior-graph-2

Figure 3: Population behavior graph 2.

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 R0 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.

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