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

^{2}Department 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.

**Visit for more related articles at** Journal of Applied & Computational Mathematics

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.

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

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.

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 Sh represent the class of susceptible human, E_{1} is the V_{1} 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 V_{1}l,

The total human population N_{h} is

N_{h} = S_{h}+ E_{1} + I_{1} + E_{12} + P_{2} + R_{1}

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 Sv(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

(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] |

µ_{v} |
Natural mortality rate of Sandflies | 0.189day^{-1} |
[16] |

µ_{r} |
Natural mortality rate of Reservoirs | 0.000274day^{-1} |
Assumed |

µ_{2} |
PKDL recovery rate after treatment | 0.033day^{-1} |
[17] |

1−α1 | Developing PKDL rate after treatment | 0.36day^{-1} |
[17] |

β_{1} |
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 |

γ_{1} |
Treatment rate of VL | variable | Assumed |

δ_{1} |
VL induced death rate | 0.011day^{-1} |
[18] |

k_{2} |
1/k_{2}is Incubation period of vl |
0.006555day^{-1} |
[19] |

k_{3} |
1/k_{3} is Incubation period of PKDL |
0.004925925day^{-1} |
[2,20] |

δ_{2} |
PKDL induced death rate | assumed | Assumed |

b_{2} |
Progression rate of VL in human (from sandfly) | 0.0714day^{-1} |
[21] |

**Table 1: **Description of the parameters.

The terms of interaction λ_{h}, λ_{r} and λ_{v}are as under is the average rate of infection rate of human with Vl, from infectious sandfly.

λ_{r} is the average rate of infection of susceptible reservoir by infected sandfly.

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

, is the average rate of infection of sandfly with Vl strain from human or reservoir. Where c_{2} is the transmission probability of Vl from human in stage I_{1} and P_{2} to sandfly

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.

(2)

(3)

(4)

If the human population is disease free, i.e. I_{1} = P_{2} = 0, then equation (2) reduces to the form;

(5)

Equilibrium in this case is

(6)

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

(7)

The lower bond for equation (7) is given by

(8)

The equilibrium of equation (8) is

(9)

With the initial condition

N_{h}(0) = N_{0}. (10)

If N_{u}, and N_{l}, denote the solution of equation (5) and equation (8), then any solution of equation (2), satisfy

N_{1} ≤ N_{h} ≤ N_{u}.(11)

Consider the biological feasible region Ω given by:

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

So

as

Similarly

and

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

Let us de ne a new region G as

where

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

Here

with

After simplification, we get reproduction number

We can further simplify to get where

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 Vl from reservoir to sand y. So the term R_{a} indicate the transmission of Vl 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_{1} to the next victim. So R_{0} is biologically sensible.

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 1:** An m × m matrix, for m>2 is called irreducible if for any proper sub-set I of {1, 2,…,m}, ∃ , and such that

**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>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.

**Proof: **Let

(12)

(13)

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

where

Since the collection 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 . The points 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), Hence there exists so that for any we have

Thus G is global attractor.

Next to show that G is stable

On the basis of monotonicity of , we have

Thus we have shown that any solution of the model (1), starting from a point in , remains in . 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 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

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

So the model can now be written as

where

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

So all of the diagonal entries of A_{I} (X) are nonnegative, hence AI (X) is metzler and irreducible .

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

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

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

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

(15)

and if

for some then (16)

(17)

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

**Proof:**

Since

So the upper bond of denoted by is given by

and

Clearly as

And

only if;

Thus H_{4} of theorem (4.3) holds [10], equivalently equation (15) and equation (16), hold.

To show that H_{5} or equation (18) holds, we state the following theorem.

**Theorem:** The metzler matrix satisfy the axiom if the basic reproductive number R_{0} satisfy the inequality; R_{0} ≤ξ , where ξ, is the additional threshold number given by

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

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

Let

Then is stable if Y is stable.

And Y is stable if det(Y) ≥ 0

This means thatα only if

where

At the disease free equilibrium,

By putting these values in above equation, we have

We take this value as ξ. Thus H5 or equation (17) holds, if ξ ≥1 . Also R_{0} <ξ . So using theorem (4.3) of [10], we claim the following result.

**Theorem: **If the parameters of the model satisfy the condition , 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 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 α1=0:74. The graph shows that with in short time the disease can be eradicated.

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. R_{0} 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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