Abstract

The seeming incurable status of HIV/AIDS and its associated virus infectivity had continuously led to series of scientific research, geared towards the amelioration of the increasing trend of the deadly disease. In this paper, a system of ordinary differential equations was used for the formulation of a 4-Dimensional mathematical dynamic HIV-pathogen model. The model was presented as optimal control problem, which accounted for the methodological pair chemotherapy treatment, with treatment factors clinically sandwiched in two temporal time-delay immunity chambers. The methodology of the model involved dual state infectious variables, pair treatment factors - reverse transcriptase inhibitors and protease inhibitors (RTI and PI), with immune system cells as vectors. The study explored numerical methods with analysis conducted using classical Pontryagin’s Maximum Principle. We proved that the model variables have non-negative solutions and as well, established the existence and uniqueness of the optimal control strategy, which led to the derivation of the model optimal dynamic solution. Numerical computations of the model explored Runge-Kutter of order of precision 4 in a Mathcad environment. The result demonstrated novel precision, which not only agreeing with known existing models but also showed that the higher the amount of optimal weight factor, the earlier, efficient, faster and less amount of chemotherapies required for the maximization of healthy CD4⁺ T cell count concentration. Furthermore, sustainability of declined infectivity and significant minimization of optimal cost was a function of prolong treatment schedule with drug validity period accounted for. Therefore, the study which could be readily adopted for other infectious diseases, suggests further investigations with more interplay of multiple control functions.

1. Introduction

The continuous insurmountable challenges posed by lack of précised cure for the human immunodeficiency virus (HIV), vice averse the acquired immunodeficiency syndrome (AIDS) has led to series of mathematical and optimal control models, not just on the dynamics of its infection but of diverse interest, are the dynamics of varying applications of chemotherapies and its associated treatments proceedings.

Following new diverse multi-facets of HIV and its associated infections, a number of models on the best practice of chemotherapy application have been formulated. Immunotherapy administration during the early stage of the disease progression is most beneficial for raising CD4⁺ T cell count ¹. However, understanding the interactions of the multi-facets infections on host CD4⁺ T cells can best be outlined by estimation of the parameters associated with the progression of the immunotherapies. The quantitative method by which we analyze this whole lot of progression is visible through application of optimal control theory.

Mathematical models of ^{2, 3, 4}, studied optimal control problems of HIV infection in varying modeling approach, using single treatment with similar objective functional. The model ⁴, investigated the optimal therapeutic control modeling, using linear optimal state evaluator in a feedback therapy to minimize the effect of measurement error, with the aid of immune system response model as explored in ^{5, 6}.

In ², immunotherapy of HIV-1 infection with the application of cytokine interleukin-2 (IL-2), was investigated with model using immune boosting response to fight HIV infection. Optimal control strategy was deployed in the analysis of the mechanism of the immune viral dynamics. The model observed that loss rate of plasma virus is dependent on the CD4⁺ T cell count, which represent the capacity of immune system to clear the virus. Survey on early optimal control application to biological problems was studied by ⁷; while the depiction of optimal control as an organizing principle for natural immune system behavior and its application on HIV treatment was studied by ^{8, 9}.

The studies ^{10, 11, 12}, investigated body biological systems in the defense against pathogen via the introduction of inflammation and vasodilatation, causing blood coagulation that slow the spread of infection to other parts of body, as well as raising alarm for more complete response. Examined by ¹³, was the dynamics of infection by HIV, as well as therapies that minimize viral load, restore adaptive immunity and utilization of minimal dosage of anti-HIV drugs. The study observed the never cured infection status and therefore, requires continuous treatment to keep the condition in remission.

Other relevant optimal control mathematical models on the dynamics of interactions of chemotherapies on viral load and the immune system could be found in ^{14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24}. The model ²⁵, applied stochastic optimal control theory to develop protocol for the treatment of human diseases and which also accounted for time dependent uncertainties of the model.

In this present model, motivated by ^{1, 2, 3}, we formulate, using ordinary differential equations, a 4-Dimensional dynamic HIV-pathogen model, presented as an optimal control problem. The novelty of the model, accounts for the methodological pair treatment of dual HIV-pathogen infectivity with treatment factors clinically sandwiched in a two temporal time-delay immunity chambers. The methodology of the model is the constituent of two infectious state variables (viral load and parasitoid-pathogen); two chemotherapies – reverse transcriptase inhibitors (RTI), as immune boosting and protease inhibitors (PI), as immune suppressing therapy; and the host – immune system cells. The method explores the use of numerical methods; skew to optimal control strategy with analysis conducted using Pontryagin’s Maximum Principle.

The entire model is composed of the introductory aspect as in section 1. Definition of model mathematical statement and design of the optimal control problem are absorbed in section 2. Section 3, is devoted to the existence and uniqueness of the optimal control pair treatment in the presence of latent chambers. Numerical simulations as well as model illustrations and discussion were designated for section 4. Finally, in section 5, we summarized the work, following the observatory conclusion and remarks base on model formulation, extensive analysis and illustrations thereof.

2. Material and Methods

We shall devote this section to the complete definition of the mathematical statement and model formulation of the study; as well as the design for the optimal control problem of the model.

With intending simplicity, we presuppose the formulation of the present model as constituted by host victim – CD4⁺ T cells having interaction with two infectious variables – viral load and parasitoid-pathogen, in the presence of two distorting treatment factors – reverse transcriptase inhibitors (RTI) and protease inhibitors (PI) under two ameliorating time-delay latent chambers. The definitions of the model dynamics is aided by the schematic structure as presented by Figure 1, below:

From Figure 1 below, we see that the model is defined around four subgroups leading to a 4 - Dimensional dual HIV dynamic model. Suppose we let the concentration of the subgroups represent the population understudy with unit volume,, then, by the inscription of Figure 1, we derive the following definitions: - uninfected CD4⁺ T cell count, - infected CD4⁺ T cells (by both viruses - viral load and parasitoid-pathogen), - infected CD4⁺ T cells receiving treatment and - full-blown AIDS population, represent the biological interaction of the variables with , sandwiched in between and , as two latent time-delay immunity chambers.

The dynamic physiology of the model is then represented by the following differential derivations:

(2.1)

where, . Then, equation (2.1) can be written as

(2.2)

As the T-cells becomes infected, the epidemiological interaction is given by

(2.3)

where and with known, equation (2.3), is derived as:

(2.4)

Applying , as treatment chamber to the infected T-cells, we have,

(2.5)

Finally, acute blown AIDS due to lack of treatment application and gradual maturity from cells exposed to treatment lead to the derivation,

(2.6)

with all initial conditions given as:

and satisfying the biological variables and parameters values as define in Table 1 below:

Therefore, the basic model equation derived as in equations (2.2), (2.4), and (2.5) and (2.6), are as summarized in Table 2 below:

Equations (2.2), (2.4), (2.5) and (2.6), constitute the model equations and its state space is given by

Thus, equations (2.2), (2.4), (2.5) and (2.6), with the condition (2.5’), is the standard model for the diagnosis of two infectious variables with multiple (two) treatment factors, enhanced by two latency time-delay immunity chambers. These chambers are the passive and active temporal delay reservoirs and not epidemiological transmission outlets. It serves as the temporal delay treatment compartments adequately enough to accommodate the asymptomatic stage of viruses infectivity, children yet to develop active infectious stage (at passive chamber, ); enable study of viremia levels, early stigmatization and the impoverished society (at the active chamber, ), ^{26, 27}.

The epidemiological description of equations (2.2), (2.4), (2.5) and (2.6), are viewed from the fact that, in equation (2.2), is the source/proliferation of uninfected CD4⁺ T cells, , is the proliferation of latently infected CD4⁺ T cells at asymptomatic stage under immune boosting chemotherapy and having limiting factors , is the gross loss by infection due to viruses. The death rate here is the natural death given by.

In equation (2.4), the proliferated uninfected CD4⁺ T cells become the source for latently infected T-cells following its interaction with viruses and are given by. The sum of infection transmission, death rate due to infection and natural loss rate are given by. The term is the probability at which progressing latently infected CD4⁺ T cells that become actively infectious T-cells are subjected to suppressing chemotherapy treatment and having as rate of treatment while becoming actively infectious.

Similarly, in equation (2.5), the first term accounts for the amount of actively infectious CD4⁺ T cells that are been subjected to suppressing chemotherapy, occasioned by external source of infection, and having treatment rate of as the third term. The last term defined the sum transmission rate of active infected CD4⁺ T-cells to full blown AIDS, infection clearance rate due to chronism and occasioned natural death rate.

Finally, from equation (2.6), the first term, denote progression of latent infected T-cells that transmit to full-blown AIDS with only immune boosting chemotherapy treatment, while the second term - accounts for the actively infected T-cells that were exposed to dual treatment but gradually transmitted to full-blown AIDS. The model life-cycle terminates following death rate due to chronic infection as well as natural death rate designated by.

If we decide to take into account the epidemiological dynamics of infections, then the optimal values of the control variables (optimal control treatments), can be estimated by transmutation of the system model (2.2), (2.4), (2.5) and (2.6), as an optimal control problem. This goal is achieved by the introduction of objective functional, which maximize the control system define by

(2.7)

where represents the optimal “weight factors” on the optimal cost ascertained by the optimal benefit based on the CD4⁺ T cells count concentration and which determines the toxicity of treatment control functions, respectively ⁹. Thus, equation (2.7), is said to be characterized by the optimal control pair, define by the limit

We at once observe from the above expression, that

where,

is considered admissibility limit of the model measurable function. Therefore, infection is bound to be endemic if, due to insufficient chemotherapy and if , representing maximal use of chemotherapy, infection is bound to be under control. Thus, maximal cost of chemotherapy is given by and the following proposition holds.

Proposition 1.

Assume there exist drug hazardous side effect, then, the inequality of the introduced optimal weight factors such that holds.

Next, we show using the following theorem, that the model variables, which represent the human immune system are all compactible and non-negative integers.

Theorem 1.

Given the model equations (2.2), (2.4), (2.5) and (2.6), let

Then, the positivity of the solution of the system (2.2), (2.4), (2.5) and (2.6), are guarantee and are compactible for all ²⁸.

Proof.

Consider the model equations (2.2), (2.4), (2.5) and (2.6), we differentiate each equation with respect to their variables, i.e.

From equation (2.2), we have

Differentiating with respect to , we have

The inequality here, means that is finite and positive or, since, (from the theorem).

Finding the integrating factor,

Multiplying by the integrating factor, we have

Rewrite the left hand side of the equation

Integrating both sides leads to

Divide through by the integrating factor, we have

Applying the initial conditions: at

When .

Therefore,

From equation (2.4), we set

Integrating both sides, we have

Taking integrating factor, we, have

Applying initial conditions: at

When .

Taking on equation (2.5), we have,

Therefore,

Finally, from equation (2.6), we have,

Therefore,

Hence, model variables are all non-negative and this complete the proof.

3. Existence and Uniqueness of Optimal Control Pair Treatment with Latent Chambers

In section two, we have proved that the variables of model are non-negative and compactible. Next, we justify the existence and uniqueness of optimal control pair chemotherapy treatment for the model in the presence of two temporal latency chambers. Here lies the novelty of the present model formulation.

The existence of the model system consisting of two infectious variables under optimal control pair and clinically administered via two control latent chambers can be prove by juxtaposing the boundedness of the solution of the system, given a finite time interval. Since we consider the boundedness of the solution, then there exists super-solution satisfying

(2.8)

which are bounded on a finite time interval. The existence is therefore established by applying results from ( ¹, Th. 2.1, p. 4-5; ²⁹, Th. 4.1, p. 68-69).

Theorem 2 Consider the optimal control problem with system equations (2.2), (2.4), (2.5) and (2.6). Then using proposition 1, there exists an optimal control pair such that

maximizes the objective functional.

Proof. We prove by showing that the following conditions hold:

i. All the initial conditions with the controls andsuch that, i=1,2 is a Lebesgue-integrable function on with values in the admissible control set , such that the state system is satisfied and not empty.

ii. The admissible control set, is closed and convex.

iii. The right hand side (RHS) of the state system is continuous, is bounded above by a sum of the bounded control and the state variables; and can be written as a linear function of , i = 1, 2 with coefficients depending on assumption 1 and on the state variables.

iv. The integrand of the objective functional is concave on the admissible, of the control set.

v. There exists consistence and , such that the integrand of the objective functional satisfies

To verifying these conditions, we invoke an existence result of [ ³⁰, Thm. 9.2.1], for the existence of the state system (2.2), (2.4), (2.5) and (2.6), with bounded coefficients, which gives condition (i). We note that the solutions are bounded. By definition, our control set is closed and convex, and then, satisfies condition (ii). Since our state system is bilinear in , the RHS of (2.2), (2.4), (2.5) and (2.6), satisfies condition (iii), using the boundedness of the solutions. In addition, the integrand of the objective functional is concave on the admissible control set. Finally, the completeness of the existence of solution of the optimal control is the fact that

where depends on the upper bound on , and since . Then this completes the proof.

Now, since we have shown that solution of optimal control problem of the system exist, we then have to derive the optimality control strategy of the system.

In sub-section 3.1, we had established the existence of the model optimality control pair, aimed at the maximization of the objective functional (2.7), subject to model equations (2.2), (2.4), (2.5) and (2.6). Then, we now define the system optimality control strategy. We proceed by driving the necessary conditions for the optimal control strategy using the Pontryagin’s Maximum Principle define by the Lagrangian:

where is penalty multiplier satisfying at the optimal and at the optimal

Theorem 3.

Let be the optimal control and having the solutions, for the corresponding basic model (2.2), (2.4), (2.5) and (2.6). Then, there exist adjoint variables satisfying

and having , as transversality conditions. In addition,

Proof.

From the above given Lagrangian, we differentiate w.r.t. , for the using the Pontryagin’s Maximum Principle ³¹, to obtain the standard adjoint equations and the transversality conditions. The preceding adjoint variable of the system is obtained as:

(2.9)

(2.10)

(2.11)

(2.12)

The result that follows is the optimality equation of the system ³² derived as:

Therefore, the optimality control are obtain as

(2.13)

(2.14)

It thus follows from the point of the boundedness on the control that,

i.e. is compactly notated as

In a similar approach,

with compact form as:

By definition, optimization control system is a constituents of the state system couple with the adjoint system with the initial and transversality conditions together with the characterization of the optimal control pair,

(2.15)

(2.16)

Hence, a complete representation of the optimality system is derived by substituting equations (2.15) and (2.16) into the original model equations (2.2), (2.4), (2.5) and (2.6), as seen below:

(2.17)

where, and ,, and

We had started our proof with the showing of the positivity of the model variables, followed by the existence of the model optimal control strategy. Then, we need to show that the control strategy is unique. Lemma 1 below, leads to the explicit proof of the theorem, which ascribes the uniqueness of the control strategy.

Lemma 1. The function is Lipschitz continuous in, where are some fixed positive constants.

Theorem 4. Given the final time sufficiently small, then, the bounded solutions of the optimality system are unique ^{1, 3}.

Proof. If and are two separate solutions of the optimization system (2.17), then, we define as follows: let ,, , , , ,, and ,, , , , ,, where is to be chosen. Furthermore, we let

and

If we consider each of the terms of the equations (2.17), such that we substitute each of the corresponding system solution i.e. into the first equation and differentiate, we have,

(2.18)

Similarly, for and putting also, the expressions for into their respective ODE’s, of equation (2.17), we have as follows:

(2.19)

(2.20)

(2.21)

(2.22)

(2.23)

(2.24)

(2.25)

We then proceed to carefully subtract solutions of from and progress same for the rest of other state solutions. The obtained results for each are multiplied by appropriate difference of functions and integrated from to . Finally, we collate all the eight integral equations and by estimation approach, we derive the uniqueness of the model.

Using lemma 1, the first result is obtain as

and

As an illustration of the estimation process, we consider a particular solution of the optimal control i.e. using estimate for the case of. We thus have,

where, the constants and are coefficient dependent and on bounds of the state and adjoints. The results of the combination of the eight estimates are as follows:

For a more simplify result, we let, and the expression becomes:

Clearly, it is observed from the above expression that

where are functions dependent on the coefficients and bounds of.

Hence, for any value of (), such that and for all, then . Therefore, the solution is unique for sufficiently small time.

We refer to ^{1, 3}, for related uniqueness proofs under relatively small time interval. The result here affirm most existing results on the basis that small time interval is often a two-point boundary value problem, following from system opposite time orientation and state initial condition with adjoint equations exhibiting final time conditions. More importantly, we deduce from the proof of Thm. 4 that for a model of pair chemotherapy treatment with time-delay immunity chamber, if for all such that then, infection is eliminated. On the other hand, if such that for then, infection is endemic and global.

4. Numerical Simulation and Discussion

In line with our theoretical derivations of the system model typically represented by the dynamic optimal control equations (2.17), we illustrate the model using RK4 in Mathcad software application in the following subsection.

In simulating our model, we consider the parameter values as in Table 1, above, amid predominant parameters such as: ,and under the transversality conditions ( ). We also regulated the system control function at and, with each having a normalizing optimal weight factor and, for all the various compartments of the initial simulations as represented by Figure 2(a-e) below. Here, we present the result analysis, without figures of the conditions for brevity.

From Figure 2 (a), we visibly observed sharp increase in the susceptible CD4⁺ T cells to an apex of , at the first two months, following consistent population natural source and consequence of dual application of chemotherapy treatment under dual temporal-delay immunity chambers . A slight decline was observed from the third month, with attainment of stability i.e. , from the seventh month and through duration of 30 months of chemotherapy regulation period.

From Figure 2 (b), at asymptomatic stage, the latently infected CD4⁺ T cells exhibit some increase at the third month. Consistent application of boosting chemotherapy (RTI), and the effect of the administration of suppressing chemotherapy (PI), saw accelerated decline of infection, which gradually stabilized after 22 months of drug schedule to as low as .

Figure 2 (c), represents simulation of acute infectious T-cells population, which is observed to exhibit initial spurring increase under first application of only immune boosting chemotherapy. The inclusion of suppressing treatment schedule under chamber, leads to the control of the growth of the infectious population stabilizing at.

It is seen from Figure 2 (d), that acute full-blown AIDS population exhibited sharp initial growth due to proliferation from latently infected that were not exposed to immune suppressing chemotherapy. Here, AIDS population was highest at the third month. Consistent subjection of the infectious T-cells at the third compartment to suppressing chemotherapy (PI), had some tremendous control effect on acute AIDS population, as infected T-cell decline to as low as, after 20 months of drug schedule.

Figure 2 (e), simulated the objective functional, capturing the corresponding optimal control cost as justified by the optimal benefit exhibited by the maximization of the healthy CD4⁺ T cells in Figure 2 (a). We observed an optimal cost, which maintained a steady insignificantly slight increase in treatment cost with , from treatment set point to treatment terminal schedule.

We further intensify model treatment schedule in order to ascertain our initial numerical illustration. Some notable variations of model predominant parameters were further considered for,, and having constant transversality conditions ( ). Of interest in this second investigation, are the enhanced variations of the system control function, regulated at, and each normalized by optimal weight factor of and respectively, as represented by Figure 3(a-e) below.

From Figure 3 (a), we observed the maximization of healthy CD4+ T cells, following the enhancement of consistent treatment chemotherapy schedule as inscribed by model predominant parameters and the toxicity of suppressing chemotherapy. Optimal benefits is seen to increase with apex at, as against, of Figure 2 (a) above. Continuous administration of treatment schedule within validity of 30 months treatment period saw stabilization of uninfected T-cells at , when compared with the stable level of as in Figure 2 (a).

Furthermore, at the asymptomatic stage, when infectivity by viruses on CD4⁺ T cells are at latent level, Figure 3 (b), indicates high concentration of viruses (viral load and parasitoid-pathogen) invasion of the immune system with apex at , at the third month. Active immune boosting chemotherapy and commencement of suppressing immune chemotherapy as latently infected T-cells progresses to active infectious stage, saw a drastic decline of latently infected T-cells group to as low as , within the duration of treatment schedule. Again, an improved result is achieved here, when compare with that of Figure 2 (b) above.

From Figure 3 (c) above, the transmutation of latently infected T-cells, which are basically exposed to immune boosting chemotherapy, become the source of active infectious T-cells. The population of infectious T-cells exhibit diagonal elevation at the early stage, with increasing number of viruses’ infectivity. The measure of the effect of treatment schedule in this compartment is seen by the decline of full-blown AIDS patients as in Figure 3 (d) above.

From Figure 3 (d) above, we again experience initial concentration of full-blown AIDS victims to as high as, at the third month, following early proliferation of latently infected T-cells with very little or no preventive and suppressive measures. However, with intense exposure to HAART treatment schedule at the infectious compartment, acute AIDS infected compartment gradually exhibit accelerated declination with stability tendency at , after 20 months of treatment schedule. The outcome showed an improvement as against , of Figure 2 (d).

With enhanced optimal control functions balanced by optimal weight factors ofand respectively, Figure 3 (e) above indicated a tremendous reduction in systemic cost as seen from Figure 2 (e) with to as low as in Figure 3 (e). The aim of the investigation was justified as minimization of optimal cost yields maximization of uninfected CD4⁺ T cells concentration, evident by Figure 3 (a) above.

Here, we have formulated an optimal pair control mathematical strategy, which accounted for the maximization of the optimal benefits based on the optimal cost analyses. The model was basically conducted as 4-Dimensional dynamic dual HIV-pathogen infectivity model, enhanced by dual temporal delay-immunity treatment chambers. Consisting the model population understudy, were four subgroups of CD4⁺ T cell counts, which played host to viral load and parasitoid-pathogen as visitors. Dual suppressing and preventive treatment chemotherapies were allowed in the interplay of the epidemiological cycle.

The method explore was an optimal control problem with optimal solution established. The existence and uniqueness of model solution was critically viewed using well known technique from numerical methods – the Pontryagin’s Maximum Principle and optimization control system defined by the state system coupled with the adjoint system with the initial and transversality conditions together with the characterization of the optimal control pair was derived. Thus, optimality control system is a two-point boundary value problem due to state initial data and adjoint system final time data. Concrete illustrative examples were simulated for which model investigated the methodological progression of chemotherapy treatment trend at set-point (considered as latent infection) and at active stage, known as infectious CD4⁺ T cells.

Two optimal control functions, was explored and clinically balanced by optimal weight factors , with each associated to upperbounds and, respectively. Analyses of the results indicated that with predominant parameters such as ; having control functions and weight factors ofand for all, infectivity (viral load and parasitoid-pathogen) were under control. Latent infected T-cells (Figure 2b) declined as low as, from initial apex of, while acute infectious T-cells,attained controllable stability. The effects of declination of viruses infectivity are evident by the maximization of healthy CD4⁺ T cells concentration (Figure 2a), with, in the first three months and attaining growth stability of, through rest treatment validity period. Acute full-blown AIDS infected CD4⁺ T cells ascertained the positivity of the investigation as overall declination of AIDS victims reduced drastically from to. The systemic cost showed impressive stability value with somewhat insignificant increase of.

Further intensification of the experiment as analyzed by Figure 3 (a-e), showed further enhancement of maximization of healthy CD4⁺ T cells concentration up to, in the first three months with attained stability of, through rest treatment period. Latent infected T-cells and active infectious CD4⁺ T cells were under immune boosting and suppressive chemotherapy measures, as justified by the reduction in acute full-blown AIDS infected immune system with. The corresponding treatment schedule indicated credible minimization of optimal cost at, as in Figure 3(e) when compared with of Figure 2(e) above.

The implications were that the administration of chemotherapies requires step-wise approach as in time-delay chambers and with drug toxicity as a function of optimal weight factorsand. The initialization of treatment with high drug intensity at set-point indicated more rewarding treatment outcome whereas, the positivity of treatment outcome are independent of treatment duration. On the other hand, optimal cost is a function of treatment duration i.e. minimization of systemic cost was better achieved with prolonged treatment schedule (see Figure 2 (e) & Figure 3 (e)) above.

5. Conclusion

The present model have been formulated as a 4-Dimensional dynamic HIV-pathogen mathematical model, presented as an optimal control problem strategy for the investigation of the methodological application of dual treatment factors, systematically sandwiched in a two time-delay immunity treatment chambers in dual infectious system. The method explored well-known numerical methods – Pontryagin’s Maximum Principle in the analysis of the model. Computation of resulting outcome indicated that, for a step-wise combination of dual chemotherapy, enhanced by optimal control functions with normalizing optimal weight factors, maximization of healthy immune system concentration was best achieved with the administration of significantly high toxicity chemotherapies at infection set-point. Furthermore, sustainability of declined infectivity and significant minimization of optimal cost is primordial to sequential prolong treatment schedule and within drug validity period. The outcome of the study were therefore, not only in consonant with the investigations of ^{1, 2, 3}, but an enhancement to those existing results, studied with one control function and without temporal time-delay chambers. The study thus, suggests further investigations with more lengthy interplay of multiple control functions and could be readily adopted for other infectious diseases.

References