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A mathematical model of HIV transmission dynamics is proposed and
analysed. The population is partitioned into five compartments of susceptible *S*(*t*),
Infected *I*(*t*), Removed *R*(t), Prevented *U*(*t*) and the Controlled *W*(*t*).
Each of the compartments comprises of cohort of individuals. Five systems of
nonlinear equations are derived to represent each of the compartments. The
general stability of the disease free equilibrium (DFE) and the endemic
equilibrium states of the linearized model are established using the linear
stability analysis (Routh-Hurwitz) method which is found to be locally asymptotically
stable when the infected individuals receive ART and use the condom. The
reproduction number is also derived using the idea of Diekmann and is found to
be strictly less than one. This means that the epidemic will die out.

One of the greatest threats to human survival in contemporary times is the Human Immune deficiency Virus (HIV) pandemic. The Human Immune deficiency Virus is understood to be the causative agent of the deadly disease called the Acquired Immune Deficiency Syndrome (AIDS), Centers for Disease Control and Prevention, (1989) [

The main aim of this research is to propose a modified mathematical model of HIV transmission dynamics that will be used to check for the spread of the epidemic and to check for the effectiveness of Antiretroviral Therapy (ART) and the use of a preventive measure (condom) is in controlling the spread of HIV/AIDS. In general, these are some of the objectives of the research:

To investigate the extent to which, Antiretroviral Therapy and the use of a preventive measure (condom) will reduce the spread of the virus;

To provide information to guide advocacy and future program planning;

Institutionalizing best practices in care and support for people living with HIV/AIDS;

And to stimulate more researches on HIV/AIDS.

To derive the modified model and to establish the general stability of the free equilibrium states of the model, three methods would be used. First we would use the method of Kimbir et al., (2008) [

We partition the population into five compartments of Susceptible S(t), Infected I(t), Removed R(t), Prevented U(t) and the Controlled W(t). Asusceptible is an individual that is yet to be infected, but is open to infection as he or she interacts with members of the I-class. An infected individual is one who has HIV and is at some stage of infection. A received individual is one that is confirmed to be HIV positive, counselled, and is receiving Antiretroviral Therapy. A prevented individual is one who is confirmed to be HIV positive, counselled, and is using a preventive measure. A controlled individual is one that is confirmed HIV positive, counselled, and is both receiving ART & using condom. Eventually, we will establish the stability of the equilibrium states of the model. See table 3.1 for details of the definition of the model parameters.

From the model assumptions, assuming that the birth rate b is proportional to the total population N, there is a natural death rate µ in all compartments, and there are disease-induced death rates in the (I, R, U, W) sub-classes denoted by α_{0}, α_{1}, α_{2}, α respectively. We obtain the following model equations;

where

And

Removed, prevented and controlled at time t respectively.

Susceptible are individuals that are not infected with the disease yet, infected are individuals that are infected with the disease and can transmit it to susceptible, Removed are the individuals that are infected and are receiving ART, Prevented are infected individuals using condom, and Controlled are infected individuals receiving Antiretroviral therapy and using condom.

And b is natural death rate, B is the incidence rate,

The incidence rate

Here we would establish the general stability of the disease free equilibrium (DFE) states, by considering the model parameters and using the model equations. Since we have five systems of nonlinear equations, we know that it is almost impossible to obtain an analytical solution to these systems; therefore, we use the idea of Kimbir et al., (2008) [

The sub model equations are:

The equilibrium points of the above equations are given by thus:

condition. The Jacobian matrix is:

If

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We would have for stability

Only when

conditions for stability of the DFE state. Thus the DFE is not stable. Hence there is an endemic equilibrium (EE)

state

would persist in the absent of ART and Condom.

The sub model equations are thus:

The equilibrium points are

The community matrix corresponding to this state is

This implies that the entire expression would be non-negative, that is, if the infected individuals received ART the DFE state would be stable provided these threshold conditions hold. Please observe, also the trace (trace of a matrix is the sum of the diagonal of the matrix) of the matrix

The resulting submodel equations are as follows:

The DFE state is

The Jacobian matrix corresponding to this state is

This means that the trace of the contact matrix would be negative, that is, if the infected individuals used condom the DFE state would be stable provided these threshold conditions hold.

The resulting equations are:

Please observe that here the entire model equations are involved, therefore, we use the basic reproduction ratio of the infective to discuss the stability of the general model to avoid algebraic complexity involved in the computation of the Routh-Hurwitz stability method. In adopting this approach we use the idea of the next generation operator by Muhammad, (2010) [

Then we have thus:

and

Letting^{*} > 0, D^{*} > 0 is a diagonal matrix,

Then

The basic reproduction number of the infective is defined as the spectral radius (dominant eigenvalues) of the matrix M^{*}D^{*‒1}.

Thus,

for stability of the DFE state, we have;

and the disease would not persist, it would die out, if the threshold parameters satisfy this condition, while for

The main result of the study is found in section three, where threshold conditions are given for the stability of the disease-free and the endemic equilibrium states of the model. Whereas the condition

is strictly less than 1. From the right hand side of the expression for R_{0}, we see that increasing the value of σ, λ, ρ and π (i.e. increasing the treatment rates) reduces the value of R below 1. Similarly, reducing the value of c_{0} and _{0} < 1, we see that the minimum proportion of infected individuals to receive ART/condom is