In our presupposition to study the compatibility of optimal control strategy for the treatment of dual HIV―pa- thogen induced infection; we bring to relation, ordinary differential equation in mathematical modeling, structured as problem statement solvable possibly by discretization method.

We construct our model from a considered population density consisting of five different subpopulations, giving rise to a set of five ordinary differential equations captured from the pictorial representation of Figure 1.

Physiologically, we let denote the uninfected CD4⁺ T cells, -HIV virus (viral load), -parasitoid- pathogen; then for virus-infected CD4⁺ T cells and pathogen-infected CD4⁺ T cells, we shall denote by and respectively. The biological description of the parameters with which these variables interact and its corresponding ODE’s are defined as in Table 1, thereof:

The differential equation of the model is derived as follows:

and satisfying all the variables and parameters as defined in Table 2.

Biologically, Equations (2.1) assumed a relatively steady viral level during the asymptomatic stage of HIV and pathogen infection known as “set-point”. At this initial set-point, the body develops an immune system called, the innate immune system, which act as antibodies against HIV-infection and pathogen barrier preventing mechanism. However, the replication of viral load and the rapid adaptivity of pathogen make it impossible for easy detection and subsequently neutralize this innate immune system, which then leads to gradual full blown AIDS [1] [15] [16] . It is known that for most HIV patients, and, which also holds for and [4] .

Furthermore, it can be shown mathematically from Equation (2.1), that the amount of and in the set-point are given by the equilibrium of virus and parasitoid-pathogen depicted

i.e.

Thus, model (2.1) adequately reflected the disease progression from initial infection to an asymptomatic stage [5] .

The discretization method which is aimed at estimating all the parameters of HIV and pathogen as involved in our basic model (2.1) is applied here. The method affords us the opportunity to transform and study the compatibility of equations as in model (2.1) into solvable discrete form. Clearly, with discretization method, we aim to estimate (or measure) all the twelve parameters in model (2.1). None-the-less, we shall deliberately omit the va-

riables and at the set-point (initial stage) of our parameter estimation in view of the following limitations:

1) The microscopically indistinguishable nature of infected cells from the uninfected cells which leads to development of state estimator is a factor [7] .

2) Extreme high cost of quantification of these infected cells at the set-point is another factor [1] .

3) The number of infected CD4⁺ T cells at the set-point is found to be too small (negligible) compared to the number of healthy CD4⁺ T cells [4] .

4) At the set-point, treatments are certainly not administered from the first hour/day or even weeks of initial infection. Therefore, estimation of parameters starts with setting and, to be able to obtain desired interval for drugs administration.

Therefore, we see from Equation (2.1) that the first and second derivatives becomes

which is the equation representing the progression of infection at the asymptomatic stage.

Equations (2.3) accounts for the model parameters without and. Biologically, these clearly indicate that infection of CD4⁺ T cells by viral load and parasitoid pathogen (and) at set point through asymptomatic stage are indistinguishable from healthy CD4⁺ T cells and are insignificant. Hence, the death rates (and) of these viruses at set point and in the short period after chemotherapy treatment of the CD4⁺ T-cell count does not change significantly (see assumptions i, iii & iv). Therefore, the conditions and, given above is justified, [4] [5] . Surpassing the asymptomatic stage (or at symptomatic stage/chronic level) with, Equation (2.1), using Equation (2.3), becomes:

Equation (2.4) is a nonlinear optimal control problem (NOCP), with uncertain parameters, which necessarily need to be transform into a new problem in the form of calculus of variations from which we can apply nonlinear programming (NLP) approach. This step is obvious in order to simplify the seemingly complex biological equations (containing many variables and model parameters) into a few numbers of indicators without loss of originality.

Achieving this, we rewrite Equation (2.4) by introducing new variables i.e. let, and, and such that the twelve parameters in model (2.1) are denoted by the coefficients. Then we have,

This is to say that the coefficients of the original system can be expressed through the new coefficients as follows:

Therefore, by discretization of Equation (2.5) and substitution of approximate values of the first derivative of, for CD4⁺ T cells; the first and second derivatives of and for viral load and pathogen; as well as the first derivative of and for infected CD4⁺ T cells by viruses respectively, we can investigate the measurement of the variables at different time intervals. That is, having the experimental dependence of, Equation (2.5) can be written as the system of algebraic equations relative to the vector, i.e.

Similarly, for a matrices, we have

, (2.7)

, (2.8)

In matrix form, taking Equation (2.6), we have,

Matricizing Equations (2.7), we derive as follows:

Similarly, for Equation (2.8), we have,

Therefore, the basic model (2.5) which has been transformed to the matrix equations (2.9)-(2.11), satisfies the vector properties and each can be written as a vector form

Equation (2.12), justify that our model is described by quantifiable magnitude and has direction of purpose. Furthermore, it is observed that Equation (2.5) contains twelve unknown parameters, i.e. the variables, , with which all the parameters of the basic model (2.1) can be calculated. A case study of some of the parameter estimation as in basic model (2.1) can be found in [4] . Therefore, to determine these parameters, it becomes necessary to generate a minimum of 12 equations based on Equation (2.5). This can be achieved by differentiating Equation (2.5), more times, resulting in derivatives of and, up to the order of measurement . In coping with these orders of derivatives, we need at least

and,

(where the superscripts denote sample numbers) measurements for the complete determination of all the HIV/AIDS parameters in the five-dimensional model (2.1), [1] [4] [5] .

Then, from Equation (2.5), the identifiability of, means that all the parameters of our model (2.1) can be determined from the output of the CD4⁺ T cell count, viral load, parasitoid pathogen, HI-virus infected CD4⁺ T cells and pathogen infected CD4⁺ T cells. Thus, we establish (as in Table 3) below, the available measurement for the count of CD4⁺ T cells, viral load and pathogens, HI-virus infected CD4⁺ T cells and pathogen infected CD4⁺ T cells at varying time intervals:

Then using a number of these measurements, we investigate if the matrix A, of Equation (2.12) is nonsingular (not equal to zero); a condition for unique solution for the coefficients and hence, a prime modality for estimation of the model parameters Furthermore, this investiga-

tion is prompted by the fact that, at long asymptomatic stage (set point) and at the short period after administration of chemotherapy, when either of the, is constant, the matrix A, can never be nonsingular for any choice of number of measurements. Therefore, a complete determination of the model parameters is imperatively impossible due to these two periods of time [5] . To compute the values for the parameters of our model system, we need to compute each of the determinants of Equations (2.9)-(2.11). We see that the determinants of Equations (2.9)-(2.11) are not equal to zero, but very small, in the range of Therefore, applying Equation (2.12), that is, we need from Equation (2.12), to solve Equations (2.9)-(2.11), for the parameters. The following results were obtain

Here, the solution, , implies

So we see that as a result of the significantly small determinant, the computation of the coefficients, all yields large error derivative when compared with the parameter values of our optimal control as in Table 2, above. Whence, compatibility of optimal control strategy via discretization for a 5-D mathematical model is seemingly elusive. Non-the-less, other possible options include:

a) Application of derived formula for high order of accuracy;

b) Carrying out interpolation followed by computation of the derivative of the interpolating polynomials;

c) Since the regions of the coefficients are all non-negative, we account for the estimation of the model parameters by stepwise variation of the parameter values in order to study their respective behavior to viral load replication and pathogen resistivity.

In general, for options (a) and (b), we need small time interval range of 3 - 4 years and require calculating the derivative in the middle of the steps, resulting to complex procedures. In this case, patients are likely to die without waiting for simulation results. Therefore, option (c), is convenient for the estimation of model parameters. Thus by option (c), we return to Equation (2.1), from which we define the coefficients of our set-point. This criteria is of essence for the simple fact that, it a process to overcome indistinguishability nature of the infected cells and the uninfected cells immediately after asymptomatic stage. It also aid in the definition of treatment time limits and as an overall check to our earlier assumptions. Furthermore, chemotherapy has a certain designated time for allowable treatment, since HIV is able to build up resistance after finite time frame due to its mutation ability and its potential hazardous side effects [17] .

Keeping the parameters values of Table 2 in view, we determine the solution of the model (2.1) for the set-point, with the initial values of the coefficients generated as:, , , , , using well-known numerical method like RK4 in a Mathcad environment, we presents the graphical simulations as in Figures 2 (a)-(e).

Figures 2(a)-(e) shows decrease in the numbers of susceptible CD4⁺ T cells. That is, at, is the minimum count of CD4⁺ T cells. So, the treatment interval is months, [18] - [20] . This is evident by

the initial sharp inclination of HI-virus infected CD4⁺ T cells (i.e.) and pathogenic-infected

CD4⁺ T cells (i.e.) and then experience steady decline in population due to the influence of the viruses. Also observed, are the decrease in the numbers of free viruses which has become replicated in the blood plasma (CD4⁺ T cells), as indicated by and respectively. This gives a lead way in our investigation following the initiation of chemotherapy.