^{1}

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Multiplicity of new cases of HIV/AIDS and its allied infectious diseases daunted by lack of proper parametric estimation necessitated this present work. Formulated using ordinary differential equation was a five-dimensional (5D) differential mathematical model with which compatibility of optimal control strategy for dual (viral load and parasitoid-pathogen) infectivity in the blood plasma was investigated. Discretization method indicated the incompatibility of the model due to large error derivatives. The study using numerical method established treatment set point with which we explored the variation of predominant model parameters and thereof investigated the maximization of uninfected healthy CD4
^{ }
T cell count as well as the de-replication of viruses following the consistent administration of reverse transcriptase inhibitor from set point. Presented was a series of numerical calculations obtained using well-known Runge-Kutter of order of precision 4, in Mathcad platform. Analysis of simulated parameters showed that distortion of replication viruses and de-transmutation of susceptible CD4
^{ }
T cells by viruses via chemotherapy led to restoration and gradual increase of healthy blood plasma, with near zero declination of both viral load and parasitoid-pathogen within chemotherapy validity time frame. The model was worthy in the study of treatment analysis of dual HIV—pathogen infection and thereof recommended for other related dual infectious diseases.

Still a daunting hurdle for the scientists of infectious diseases is the unfounded clear medical literature for the absolute eradication of the human immune deficiency virus (HIV), a primary route of the dreaded disease called Acquired Immune Deficiency Syndrome (AIDS). In the circumstance, suppression and prevention of HI-virus and its associated infectious diseases have become an inevitable remedy in the annals of study into the cure for HIV/AIDS. Further hindrance to the achievable goals from both clinical trials and scientific researches is the multiplicity of new cases of HIV/AIDS and its affiliated infectious diseases.

The formulation and analysis of HI-virus and its allies significantly revolve round the parameters with which the models are formulated. Therefore, evaluation (or estimation) of the parameters is of paramount importance. Attempts in this direction by a number of researchers [^{+} T cells from the uninfected CD4^{+} T cells [

In this present paper, we presuppose two infectious parasitoid-pathogenic induced HIV infections. The study is formulated as a 5-Dimensional (5D)―ODE model aimed at investigating the compatibility of optimal control strategy for the treatment of HIV and its allies of infections. Use as treatment factor is reverse transcriptase inhibitor (RTI) with the blood plasma (CD4^{+} T cells) as the prime host. Unlike several other studies conducted using 3-Dimensional differential equations, on a single HIV infection, the novelty of this present paper lies in the enhance formulation of 5-Dimensional mathematical model, propose to investigate the parameter estimation of dual HIV―pathogen induced infection. Thus, the objective is in the investigation of the compatibility of optimal control strategy for the estimation of model parameters of dual HIV―pathogen infection. Therefore, the present work does not only accounts to establish the compatibility of application of optimal control in parameter estimation of dual infectious diseases but also, accounts for viral load de-replication and de-transmutation of pathogen resistivity through varying of model parameters. The model explores numerical method via discretization techniques (method), with numerical illustrations using Range-Kutter of order of precision 4, in Mathcad platform.

Exceptional works on parameter estimations include: virus clearance rate and death rates of infected CD4^{+} T cells [^{+} T cells measurement and viral load count, using reverse transcriptase inhibitor (RTI) as single treatment. Other notable models involving discretization methods for parameter estimations could be found in [

The scope of this work is characterized by four subsections, which includes: introductory aspect as in Section 1. The material and methods of the model, which includes: System modalities as a problem statement, discretization technique and model parameter variation constitute Section 2. Section 3 is covered by a number of numerical illustrations and discussion, while the last Section 4 is devoted to conclusion and recommended remarks. The study is anticipated to throw more insight to compatibility of optimal control strategy in 5D-model in the treatment of dual infectivity.

We present in this section, the statement of the problem and model formulation followed by the discretization technique used, as well as model parameter variation of the system.

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

Physiologically, we let ^{+} T cells, ^{+} T cells and pathogen-infected CD4^{+} T cells, we shall denote by

The differential equation of the model is derived as follows:

and satisfying all the variables and parameters as defined in

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 [

Biological description | Interaction | Reaction rate | Translation to ODE |
---|---|---|---|

CD4^{+} T cells production. | |||

CD4^{+} T cells natural death. | |||

CD4^{+} T cells become infected by virus. Rate at which virus attack CD4^{+} T cells. | |||

CD4^{+} T cells invaded by pathogen. Rate at which pathogen invade CD4^{+} T cells. | |||

Death of virus infected CD4^{+} T cells. | |||

Virus replication in infected CD4^{+} T cells. | |||

Virus natural death. | |||

Death of pathogen infected CD4^{+} T cells. | |||

Pathogen replication in infected CD4^{+} T cells. | |||

Elimination of pathogens. |

Dependent variables | Initial values |
---|---|

^{+} T cells population. ^{+} T cells. | 0.8/mm^{3} 0.01/mm^{3} 0.01/mm^{3} |

0.08/ml | |

0.07/ml | |

Parameters and Constants | Values |

^{+} T cells natural source production. | |

^{+} T cells. | |

^{+} T cells. | |

^{+} T cells. | |

^{+} T cells invaded by pathogen. | |

^{+} T cells become infected by virus. |

Furthermore, it can be shown mathematically from Equation (2.1), that the amount of

i.e.

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

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

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

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

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) 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

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 ^{+} T cells by viral load and parasitoid pathogen (^{+} T cells and are insignificant. Hence, the death rates (^{+} T-cell count does not change significantly (see assumptions i, iii & iv). Therefore, the conditions

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

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^{+ }T cells; the first and second derivatives of ^{+} T cells by viruses respectively, we can investigate the measurement of the variables at different time intervals. That is, having the experimental dependence of

Similarly, for a

, (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

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

Then, from Equation (2.5), the identifiability of^{+} T cell count, viral load, parasitoid pathogen, HI-virus infected CD4^{+} T cells and pathogen infected CD4^{+} T cells. Thus, we establish (as in ^{+} T cells, viral load and pathogens, HI-virus infected CD4^{+} T cells and pathogen infected CD4^{+} T cells at varying time intervals:

Time (t) | CD4^{+} T cell count (y_{1}) | Viral load (y_{2}) | Pathogen (y_{3}) | Infected CD4^{+} T cell by HIV (y_{4}) | Infected CD4^{+} T cell by HIV (y_{5}) |
---|---|---|---|---|---|

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

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

Here, the solution,

So we see that as a result of the significantly small determinant, the computation of the coefficients

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 [

Keeping the parameters values of

Figures 2(a)-(e) shows decrease in the numbers of susceptible CD4^{+} T cells. That is, at^{+} T cells. So, the treatment interval is

the initial sharp inclination of HI-virus infected CD4^{+} T cells (i.e.

CD4^{+} T cells (i.e.^{+} T cells), as indicated by

Here, the decision for RTI as the chemotherapy follows its dual characteristics tailored on viral load and the activation of the adaptive immune system, which act against parasitoid-pathogen. Specifically, RTI is responsible for the prevention of uninfected lymphocyte cells from infection by viral load and as well, the elimination of infected pathogen cells [^{+ }T cells becoming infected by viruses diminish in rate (i.e. decrease in ^{+} T cells.

Illustratively, keeping in view other parameter values as in

We see from Figures 3(a)-(e), that with intensive commencement of chemotherapy for

crease in healthy CD4^{+} T cells is evident by the drastic decline in the rate of HI-virus infected CD4^{+} T cells from ^{+} T cells from

^{+} T cells, adversely attribute to sharp decline (suppression) of viral load in blood plasma from

Furthermore, observing the same model coefficients as in Figures 3(a)-(e), but with reduced rate of CD4^{+} T cells becoming infected by both viral load and pathogen, following adherent administration of chemotherapy (i.e.

Analysis from ^{+} T cells was experience at ^{+} T cells and parasitoid-pathogen infected CD4^{+} T cells.

to near zero after 24 months; while from

In this paper, nonlinear 5-Dimensional mathematical models had been formulated with which the compatibility of optimal control strategy for parameter estimation of dual infectivity (HI-virus and parasitoid-pathogen) was investigated. Using discretization technique, it was established that optimization control strategy were incompatible with the particular model, following the insignificant non-singularities of the model coefficients, which led to varying error derivatives. The study further explored predominant parameters to investigate the maximization of healthy blood plasma and the trend of the viruses, following coherent chemotherapy. Time limit for chemotherapy was established with which simulation was conducted. Analysis of results showed that restoration and increase of healthy blood plasma were achieved with the administration of chemotherapy from set point. Furthermore, with the distortion of viruses’ replication and de-transmutation of healthy blood plasma by viruses from the point of chemotherapy application, eradication of dual HI-virus and parasitoid-pathogen were achieved within the ambit of chemotherapy time validity. The study therefore suggested the extension of model in the evaluation of other related dual infectious diseases. Furthermore, a more improved 5-Dimensioanl model compatible with the application of optimal control strategy is thereof recommended.

The authors (Bassey B. E, Lebedev, K. A), acknowledge with thanks, the support of the Department of Math and Computer Science; and Galina Govorova-Head of International Relation, Kuban State University, Krasnodar, Russia, for their immense contributions.

Bassey E. Bassey,Lebedev K. Andreyevich, (2016) On Analysis of Parameter Estimation Model for the Treatment of Pathogen-Induced HIV Infectivity. Open Access Library Journal,03,1-13. doi: 10.4236/oalib.1102603