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In this present paper, we proposed and formulated a quantitative approach to parametric identifiability of dual HIV-parasitoid-pathogen infectivity in a novel 5-dimensional algebraic identifiability HIV dynamic model, as against popular 3-dimensional HIV/AIDS models. In this study, ordinary differential equations were explored with analysis conducted via two improved developed techniques—the method of higher-order derivatives (MHOD) and method of mul
tiple time point (MMTP), with the later proven to be more compatible and less intensive. Identifiability function was introduced to these techniques, which led to the derivation of the model identifiability equations. The derived model consists of twelve identifiable parameters from two observable state variables (viral load and parasitoid-pathogen), as against popular six identifiable parameters from single variable; also, the minimal number of measurements required for the determination of the complete identifiable parameters was established. Analysis of the model indicated that, of the twelve parameters, ten are independently identifiable, while only the products of two pairs of the remaining parameters (

The idea to continuously research into modalities of best informed approach towards tackling the seeming insurmountable deadly infection, globally known as human immunodeficiency virus―HIV, and its associated infectivity is primordial by the incurable status of the disease. Suppressitivity and preventability of HIV infection has become the dwelling options by which respite can be afforded by victims of this scourging infection. Achieving the above succor requires the understanding of the virus dynamics and the methodological application of affordable chemotherapy treatment [

Based on the above premise, the present study proposes and by extension of earlier study [

Therefore, the novelty of the present model is the application of identifiability approach on two variable infectious diseases as compared to [

The concept of identifiability of nonlinear systems in mathematical modeling has been studied and applied in different contexts. The model [^{+} T cells, which degenerated to large error derivatives. Other notable application of identifiability in dynamic system could be found in [

The scope of this study is subdivided into 5 main sections, of which the introductory aspect occupies Section 1. In Section 2, we introduce the subject under investigation-identifiability of HIV-pathogen induced dynamic model. Section 3 is devoted to the derivation of the model identifiability function, which allows the introduction of two techniques―method of higher-order derivative (MHOD) and the method of multiple time point (MMTP). The validation of the parameter identifiability model and discussion form Section 4, of the work. Lastly, Section 5 presents the concluding part and recommendation arising from the study.

The dual infectivity considered here, consists of two infectious variables (HI-viral load and parasitoid-pathogen). In attempt to formulate this present model, we shall revoke the following two vital studies. First, the model [^{+} T cell count. The study assumed that virus infected CD4^{+} T cells were imperatively difficult to estimate due to the indistinguishable nature of the uninfected cells from the infected cells, refer [^{+} T cells.

In our present model, referencing [^{+} T cells,^{+} T cells and^{+} T cells , the biological and epidemiological model is governed by

satisfying the conditions

Let

surements for V and P are available [

such that

and

hold on

The realization of quantitative approach in identification model is in the ability to eliminate all unobserved state variables from the original system. The process of which involves computation of the higher order derivatives of the output variables. Therefore, form model (2.1); transforming the fourth and fifth equations by substituting the second and third equations, we derive the 2^{nd} order derivatives as follows:

Substituting for

and

From Equation (2.3), infection of CD4^{+} T cells by viruses are said to be simultaneous, therefore, summing Equations (2.4) and (2.5), we have,

Taking the third derivative of Equation (2.6), we obtain

Substituting the corresponding equations from model (2.1) and Equations (2.4) and (2.5), into Equation (2.7), we obtain:

Simplifying, we have

Now, from Equations (2.4) and (2.5), we see that

implying that

From Equation (2.8), solving for

Or

Or

Similarly,

If we then substitute Equations (2.9) and (2.10) into (2.8), we derive

Equation (2.11), is an equation that does not depend on the unobservable (latent) state variables

Therefore, the parametric estimation of the model parameters, as a major breakthrough for the evaluation of impact of chemotherapy on dual HIV infectivity, ultimately requires the reparameterization of system parameters. Achieving this, we let

Then, there exist a one-to-one mapping between

If we denote the right-hand side of Equation (2.11) by

The contribution of each of the parameters to the system can be explain from the partial derivative of Equation (2.12), with respect to each of the reparametized parameters, i.e.

Thus, we are disposed with the option of identifying the twelve parameters, i.e.

We shall derive using Equations (2.11) and (2.12), the identification functions as solution of the system, based on the following two approaches―method of higher-order derivative (MHOD) and method of multiple time point (MMTP).

The construction and derivation of the identifiability of a 5-dimensional differential pathogenic induced HIV infection via method of higher-order derivative, is an extension of the method initiated by [

In this present study, taking into account the above mentioned procedure, we construct the higher-order identifiable function for a 5D-basic model (2.1), as:

Then, ipso facto

The identifiability of

So we can essentially identify the twelve parameters of our model. The identifiability function, ^{th} order derivatives of

from [

Then, it is difficult to evaluate the rank of such whole matrix of order 5 × 5. Thus, it is evidently much more difficult taking any element of a matrix with

Overcoming this cancerous complexity in evaluation of such higher-order derivatives, we introduce to the model, an alternative method in constructing the desired identification function

Suppose we create the environment for which the quantities

we derive from Equations (2.11) and (2.12),

If

then by the implicit function theorem, there is a unique solution of

where,

So, we see that as much as

in order to formulate the twelve identification equations from Equation (2.15), fifteen measurements of V and P are required. The case here, is that the model is locally identifiable, on the account that ∑, contain some unknown parameters. Therefore, the perceptive approach of MMTP is said to be consistent with that of MHOD. The outstanding positive side of MMTP when compared with MHOD, is the less computational intensity and much more compatible and implementable due to its ≤3, lower-order derivatives for V and P.

Furthermore, the matrix ∑, of Equation (2.16), cannot be ascertained as a full rank matrix by mere observability or first-hand algebraic operation. A known practical approach is the application of numerical simulation, which quantitatively computes the rank of ∑. We initiate this approach by first simulating the output variables of

On the basis of the above, we provide as in

Now, for a dynamic system with unobservable state variables such as our 5D HIV-pathogen model, having unknown state variables, we avoid the use of the original Equation (2.1), in the evaluation of the unknown identifiable parameters. Here, we necessarily deploy the identifiability equations in obtaining the parameter esti-

Fixed parameters | Identifiability parameters | Model | Variables & mini. # of measurements reqd. |
---|---|---|---|

5D | |||

5D | |||

3D | |||

3D | |||

None | 3D | ||

None | 3D |

mates. Achieving this, we revoke Equation (2.11), which is with only viral load and parasitoid-pathogen as observable variables to impress on the problem of model (2.1).

Therefore, we rewrite Equation (2.11) as:

implying that

Equation (2.21) is the desired equation, which is solvable if the initial values of the output variables (observables) V and P are known. That is, the identifiable parameters can be evaluated without the need for the unobservable state variables

In real situation, we further simplify Equation (2.21), on the account that initial values of

Thus, we see that our basic model Equation (2.1), which contain both unknown (unobservable) state variables

In this section, following the administration of RTI treatment schedule, we validate the outcome of our quantitative analysis by the application of our constructed model Equation (2.22), in performing a number of numerical simulations with the aid of Runge-Kutter of order of precision 4, in a Mathcad surface. This is followed by a succinct discussion of the entire quantitative analysis and the outcome of computed illustrations.

In order to accomplish the desired task, we generate both our observable state variables and hypothetical initial parameter values base on previous HIV-Pathogen dynamic model by [

From Equation (2.22), the twelve parameter can be verified in the following six simulations and having a general representation with the seventh simulation.

Case 1:

From Figures 1(a)-(c), we observe that if source of infection rate is unknown but kept at zero when rate of replication is controlled at 50%, then after 30 months of chemotherapy schedule, viral load (1a), and decline in a diagonal trajectory to

Variable/ml | Parameter values (/mm^{3}d^{−1}) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.01 | 0.01 | 0.02 | 0.4 | 0.5 | 0.5 | 0.02 | 0.02 | 5 | 5 | 0.1 | 0.2 |

value after 24 months of chemotherapy. The overall impact of source of birth rate on dual infectivity (1c), indicates that infection decline to

Case 2: ^{+} T cells is at zero value, while rate of viruses replication are controlled at 50%, we simulate the system as represented in Figures 2(a)-(c).

Results from Figures 2(a)-(c), shows that, in 2(a), viral load exhibit inclinatory resilience at early onset of infection having its apex at 2 - 3 months, before declining to V = 0.141 ml, after 30 months of drug administration. For the parasitoid-pathogen as in 2(b), with zero natural death rate, infection decline rapidly and terminating after 24 months of chemotherapy treatment. Overall decline of viruses infected T-cells is put at N_{p} = 0.143 ml.

Case 3: ^{+} T cell becoming infected by both viruses are unchecked, such that

From Figures 3(a)-(c), we observe a diagonal trajectory decline of viral load with that of parasitoid pathogen slightly skew from 0.2 ml, to

Case 4: ^{+} T cell are unknown, such that there assume zero values, then infections are bound to decline following the application of chemotherapy treatment. The simulations of these parameters are as in

From the above simulation, the non-continuity of rate of infection avail us with the identifiable impact of the parameters (

Case 5:

From the Figures 5(a)-(c), the zero values of these identifiable parameters indicate that no death rate incurred by both infected CD4^{+} T cells and replications by viruses in infected cell were equally unknown. The implication is that infected population by both viruses remains stagnant throughout the 30 months period, de-

spite drug administration, i.e.

If only

Case 6:

From the

Case 7: All parameters known. With the identifiability of the parameters and its impacts on viruses’ transmission and treatment schedule completely specified as in

We observe from

By extension, 5-dimensional algebraic identifiability model for parameter estimation of dual HIV parasitoid- pathogen was formulated with the introduction of novel identifiability function and its associated identification equation. The study deployed implicit function theorem from numerical methods in the derivation and analysis of two identification function methods. Twelve parameters of the model were established with ten independently identifiable, while only the product of the rest two pairs can be identified. Numerical computations of the model were conducted using existing known values of only observable state variables and compatible initial parameter

values from published works. Simulations were stratified into six main cases.

To be able to assimilate the results of the outcome of the identifiability computations, we consider the stratification of the parameter identifiability as follows: -identifiably predominant, if parameter is unknown (i.e. having zero value) and infection remained stagnant or characterized by insignificant decline of infection.; identified with weak predominant magnitude, if unknown, infection declined slightly with value not less than half the original value; identifiable with no predominant magnitude, if unknown, infection declined significantly with value near zero; and unidentifiable but having predominant impact, if unknown, infection exhibit insignificant decline.

In the aforementioned pattern, cases 1 and 2, exhibit identifiability properties with no predominant magnitude. On the other hand, cases 3 and 4, were characterized by identifiability status having weak predominant magnitude. Furthermore, in case 5, we experienced some special behavior. The sum of the parameters is identifiable with very strong predominant magnitude but as well, could exhibit the status of identifiability with no predominant magnitude when

The parameters of case 6, are independently unidentifiable but exhibits product identifiable predominant magnitude. This character affirmed the indistinguishable nature of uninfected CD4^{+} T cells from the infected T-cells. Finally, with all known identifiable parameters, their magnitude and impacts on the viruses and treatment dynamics were illustrated in case 7. Thus, it can be viewed that the key dominant parameters of the system lies in case 5 and 6, respectively.

In this paper, a novel differential 5-dimensional algebraic identifiability model for the estimation of parameters of dual HIV-parasitoid pathogen was formulated with the introduction of identifiability function, which led to the derivation of the model identifiability equations. Two observable state variables and twelve identifiable parameters constituted the derived model of the system. Analysis of the derived model was conducted via established novel techniques―MHOD and MMTP, with the later technique proven to be more practicable and less cumbersome in implementation. Validations of the outcome of model analyses were stratified into six strata, which simplified the trend of parameter identifiability predominant magnitude in a 5-dimensional dual HIV-pa- thogen dynamic model. Results show that cases 5 and 6, constituted by the rate of replication of viruses, rate at which viruses attacked the immune system and the rate at which the immune system became infected by viruses formed the core dominant identifiable parameters; when significantly controlled, eradication of dual infectivity can be achieved. Furthermore, the study was in agreement with existing results on the indistinguishable nature of uninfected CD4^{+} T cells from infected CD4^{+} T cells. Therefore, simulated outcome ascertained the theoretical analysis of the study, which can as well, be adapted conveniently to any nonlinear system of other related infectious diseases. Nonetheless, we acknowledge any further studies that may add novelty to the methodology and possible affirmation of the inputs of this work.

Bassey E. Bassey,Lebedev K. Andreyevich, (2016) On Quantitative Approach to Parametric Identifiability of Dual HIV-Parasitoid Infectivity Model. Open Access Library Journal,03,1-14. doi: 10.4236/oalib.1102931