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In this paper, we propose a determinist mathematical model for the co-circulating into two circulating recombinants forms (CRFs) Of HIV-disease in Mali. We divide the sexually active population within three compartments (susceptible, CRF-1 infected and CRF-2 or CRF-12 infected) and study the dynamical behavior of this model. Then, we define a basic reproduction number of the CRF-2 or CRF-12 infected individuals R0 and shown that the CRF-2 or CRF-12 infected-free equilibrium is locally-asymptotically stable if R0 < 1 (thus the CRF-2 or CRF-12 infected becomes extinct in population) and unstable if R0 > 1 (thus the CRF-2 or CRF-12 infected invade in the population). Fur-thermore, we prove that under certain conditions on the parameters of the model the controllability of CRF-2 or CRF-12 infected with regard to the CRF-1 infected. Numerical simulations are given to illustrate the results.

AIDS is one of the most deadly diseases caused by a humain immunodeficiency virus (HIV). The virus destroys all the immune system and leaves individuals susceptible to any other infections. The lymphocites (in particular the lymphocites T-CD4) multiplies insade those lymphocites and finally destroy them. When the lymphocytes are reduced to a certain numbers, the immune system stops functioning correctly. Therefore, the individual can catch any kind of disease that might kill him easily because of the failure of the immune system. However, there exist drugs that can slow down the evolution of the virus. HIV is usually transmitted in three different ways: sexual contacts, blood transfusion, and exchange between mother and child during pregnancy, childbirth and breastfeeding.

Humain immunodfiency virus (HIV), the causative agent of AIDS, is classified into types, groups, subtypes and sub-subtypes according to its genetic diversity [

We propose in this work a mathematical model which describes the cocirculation of two circulating recombinants forms (CRF-1 and CRF-2) of the HIV-1 in Mali. We suppose that the CRF-1 is not resistant in antiretrovirals and it is in an endemic state in the population, whereas the CRF-2 and CRF-12 which is the recombination of the CRF-1 and CRF-2 resist antiretrovirals.

The population is divided into tree compartments (_{1} and the CRF-2 or CRF-12 infected (infected individuals by CRF-2 or CRF-12) represented by Y_{2}.

Our Model is based on the model proposed in [_{1}), CRF-2 or CRF-12 infected (denoted by Y_{2}).

Our model is given by the following system of ODEs:

where the parameters are defined in

To analyse the model (7), we introduce the following variables:

Then,

After normalization of the initial data, we obtain

and

The variables of the model (2) are defined in

By definition, the variables in

We shall compute the basic reproduction number as

where

We shall prove by rigorous mathematical analysis that if

In

In both figures, the initial population size is

Recruitment rate. | |
---|---|

Mortality rate. | |

Transmission rate for CRF-1 infected. | |

Transmission rate for CRF-2 or CRF-12 infected. | |

Probability of reinfection. |

Proportion of susceptible individuals. | |
---|---|

Proportion of CRF-1 infected individuals. | |

Proportion of CRF-2 or CRF-12 infected individuals. |

zontal axis is measured in years.

In

The paper is organized as follows: preliminary technical results on our model are given in Section 3. In Section 4, the basic reproduction number R_{0} is introduced and is used to determine the local extinction of the CRF-2 or CRF-12 infective population when

In this section, we establish the invariance of the first quadrant,

and the plane

Lemma 1. Let P(t) and Q(t) be n X n matrices of bounded measurable functions on

Proof. Indeed, this follows from the integrated form of the differential equation,

Lemma 2. The following identities hold:

Proof. Adding all the Equations of (2), we obtain

and recalling Equation (3), the assertion 9 follows.

Lemma 3. The following inequalities hold:

Proof. From (2), we have:

so that, by Lemma 1,

In this section, we define a basic reproduction number

We consider the system of Equations (2)

The following matrix will play a fundamental role in the sequel:

The point

is the CRF-2 or CRF-12 disease-free equilibrium point of system Equations (10).

R_{0} is defined by Equation (5). Note that the average infectious period of a single CRF-2 or CRF-12 is

Hence,

We note that if

Theorem 1. If

Proof. Let A the Jacobian matrix of system of Equations (10)

Let us assess A at the CRF-2 or CRF-12 disease-free equilibrium point,

The eigenvalues of the matrix A are:

Thus all the eigenvalues of the matrix A have their real part strictly negative if

So if

The aim of this section is to provide simple conditions for the parameters of the msystem of Equations (2) that makes possible to control the CRF-2 or CRF-12 infected individuals, by using the notion of the exterior contin-

gent cone to a convex subset

According to Equation (8), the susceptible compartment

thus the system of Equations (2) is reduced to:

The question we address is: does there exist parameters which allow the system of Equations (11) to evolve toward a fixed region

We begin by given the definition of the contingent cone.

Definition 2. The contingent cone to

where

When a point

Lemma 4. Let

Theorem 3. If the parameters of the system of Equations (11) verify:

then the vector defined by

Furthermore, for any initial condition

Proof. Before starting the proof of theorem, we give the following result ([

Lemma 5. The exterior contingent cone to

where

From the definition of the exterior contingent cone

By using the fact that

If the condition (13) is satisfied then,

Fix

Biologically the condition (13) characterizes the improvement of the efficiency of antiretroviral treatment.

Let us end this section with numerical examples. The system of Equations (11) is discretized with a Runge-Kutta’s method (ODE45). By using available data from Mali in 2011 in the system (11). Population sexually active in Mali is taken to be

The cone

In

In

In this paper, we showed theoretically and numerically that if the basic reproduction number

MahamadouAlassane,AmadouMahamane,OuaténiDiallo,JérômePousin, (2015) Mathematical Model of HIV-1 Circulating Recombinants Forms in Mali. Open Journal of Modelling and Simulation,03,137-145. doi: 10.4236/ojmsi.2015.34015