### Introduction

*M*, into two compartments, inactive and active immune response cells. On one hand, this process of activation leads us to take into account that cytotoxic cells are always present in the body. On the other hand, inactive cytotoxic immune cells get activated through specific biochemical processes related to the presence of HIV.

### Materials and methods

### 2.1 Previous models

*V*(

*t*) denote the average viral particle concentration at time t assuming that when the initial time

*t*= 0, an initial viral load

*V*

_{0}> 0 enters into the body. In ideal conditions, this initial viral load is going to be eliminated from the body at a constant rate

*c*and this depends only on the virus ability to infect the immune cells. Therefore, as long as the infection has not been established, the viral load can be described bywhere

*V*(

*t*) =

*V*

_{0}

*e*

^{−ct}. Perelson and Nelson studied a similar situation to the one we just described but there is an unknown quantity (and to be determined) describing the creation of new viral particles [16]. To define this quantity, first, we consider that once the virus enters into the body and it will infect the CD4 T-cells, which are immunologically activated T-cells. Let

*T*=

*T*(

*t*) be the average concentration of

*healthy CD4 T-cells*at time

*t*at the constant recruitment rate

*σ*and death rate

*μ*. In this way

*T*without infection. In the absence of virus, the CD4 T-cells reaches the equilibrium level of

*σ/μ*cells per mm

^{3}. If we denote

*β*as the probability of a CD4 T-cell for to be infected by the HIV, then from the

*mass action principle*,

*βTV*represents the average number of CD4 T-cells per unit of time that getting infected at time

*t*. Thus the equation becomes

*infected CD4 T-cells*is represented by the variable

*cytopathic action*but later it was found other indirect destruction mechanisms such as induction of apoptosis through soluble viral proteins, secondary cellular death due to immunological hyper-activation, syncytia formation and progressive damage of the primary and secondary lymphoid organs [1,17]. Let

*δ*be the infected CD4 T-cells death rate, so

*t*. Hence, the equation describing the variation of infected CD4 T cells take the form,

*η*new viral particles, then the equation (1) becomes,Where

*t*. So far, from these equations (2–4), we have constructed the following system of ordinary differential equations (ODEs)

*M*=

*M*(

*t*), which corresponds to the average concentration of active cytotoxic immune response cells, i.e. cells capable of eliminating infected CD4 T cells through cytotoxic action with a probability

*γ*. Then, the term

### 2.2 HIV model with both active and inactive immune cells

*T*=

*T*(

*t*) and

*T*

^{∗}=

*T*

^{∗}(

*t*) describe the average concentration of healthy and infected CD4 T-cells, respectively.

*M*=

*M*(

*t*) and

*M*

^{∗}=

*M*

^{∗}(

*t*) correspond to the average concentration of inactive and active cytotoxic immune response cells.

*V*=

*V*(

*t*) is the average concentration of viral particles at time

*t*. The time variation for

*T*,

*T*

^{∗}and

*V*are modeled in similar manners as the ones described in the introduction. The active cytotoxic immune response cells,

*M*

^{∗}, kill infected cells by an average quantity

*γT*

^{∗}

*M*

^{∗}. The detection of infection in the body produces

*αT*

^{∗}

*M*

^{∗}active cytotoxic immune cells, therefore,

*α*is a cytotoxic immune response activation rate. It is biologically meaningful to consider

*γ*≥

*α*because it implies that immune response cells kill more cells than they replicate themselves by this process. The inactive immune response is self-produced at a constant rate of

*λ*. The infected cells stimulate the inactive immune response cells at a rate of

*ψ*. Hence

*ψT*

^{∗}

*M*indicate the number of inactive immune response cells that become active. The natural death rate of both inactive and active immune cells is denoted by

*ρ*. Hence, the mathematical model can be written as:

*ηT*

^{∗}. Within the system (7), the factor

*δ*is excluded because we want to consider all the cellular production of virus, not only those who are released by the cell when it dies out. In this model we neglect, as in [7,12,16], the loss of virus during the infection. Once the model is formulated, standard mathematical analysis is carried out including the stability analysis based on the basic reproduction number

*R*

_{0}(shown in Appendix).

### Results

*β*, namely,

*R*

_{0}> 1 and

*R*

_{0}< 1. First, in the case of

*R*

_{0}> 1, Figure 1 illustrates the HIV dynamics of a patient at an initial stage of infection with no drug treatment under four distinct values of

*β*. It is assumed the patient has a normal count of CD4

^{+}T-cells around 1000 cells per unit volume at the beginning. As we can see in the last panel of Figure 1, it is typical to observe the viral particles reaches the peak at around between day 20 and 40, then, they decrease and remain low for the rest of the simulation time duration. Also, it is clear that as greater the value of

*β*, greater the average of infected CD4 T cells and viral particles. Also, the peak timing of infected CD4 T cells and viral particles occurs earlier as the value of

*β*gets larger.

*β*, each one of results show different levels of intensity for active and inactive immune responses to help reduce infected virus cells. Next, Figure 2 displays the dynamics of infection of a patient at an initial stage of infection when

*R*

_{0}< 1. Reduction in

*β*(below a certain thresh hold number) leads to the reduction on the initial outbreak of infection. Therefore, when

*R*

_{0}< 1, infection would not establish in the body (the number of infected virus goes to zero shown in the last panel of Figure 2). More extensive and rigorous mathematical analysis and numerical simulations will be needed for clarifying the impact of active and inactive immune responses on the HIV dynamics.