A note on the impact of late diagnosis on HIV/AIDS dynamics: a mathematical modelling approach.

OBJECTIVES: The global incidence of HIV infection is not significantly decreasing, especially in sub-Saharan African countries. Though there is availability and accessibility of free HIV services, people are not being diagnosed early for HIV, and hence HIV-related mortality remains significantly high. We formulate a mathematical model for the spread of HIV using non linear ordinary differential equations in order to investigate the impact of late diagnosis of HIV on the spread of HIV.
RESULTS: The results suggest the need to encourage early initiation into HIV treatment as well as promoting HIV self-testing programs that enable more undiagnosed people to know their HIV status in order to curtail the continued spread of HIV.

Introduction

Antiretroviral therapy (ART) has successfully transformed human immunodeficiency virus (HIV) infection from a fatal to a manageable chronic disease [1]. Nonetheless, there remains critical factors to be addressed along with the roll out of effective ART regimens in order to eradicate HIV. We seek to investigate the impact of late diagnosis on the transmission dynamics of HIV. Mathematical modeling of HIV dynamics is quite advanced, see for instance the following works on HIV and the references therein [2-9].

We extend a more recent HIV/AIDS mathematical model developed by Omondi et al. [8] to investigate the impact of late diagnosis on the spread and control of HIV. In their work, Omondi et al. [8] proposed a five state deterministic compartmental model for the time evolution of population states to study the trend of HIV infection in Kenya. The model was premised on dividing the infected classes according to CD4+ T cell counts in the blood. For more information about the description of parameters and model analysis, readers are referred to Omondi et al. [8].

The paper is arranged as follows; in "Main text" section, we formulate and establish the basic properties of the model. The model is analysed for stability in this section. In "Results and discussion" section, we carry out some numerical simulations. Parameter estimation and numerical results are also presented in this section. The paper is concluded in "Conclusions" section.

Main text

The model

We propose a five state compartmental model for HIV that takes into account untimely initiation of HIV positive individuals into ART. The human population comprises classes; S(t), , , and . The class S(t) represents the population at high risk of HIV infection. Upon acquiring HIV infection, susceptible individuals move to infection class which is divided into two stages according to CD4+ T cell count in the blood. The infectives class comprise of individuals with CD4+ T cell count /μL. Individuals in class are assumed to be having a lower viral load and hence are considered to be the new infections. Individuals in class progress to the second stage of infection at a rate given by . This class consists of individuals with CD4+ T cell count in the range /μL. Individuals in this stage are assumed to be having high viral load. Individuals in class are initiated into ART treatment at a rate given by . In this paper, we develop a mathematical model that takes into account the effect of late initiation into ART treatment of HIV positive patients. We define initiation of HIV positive individuals in stage into ART treatment by the expressionHere, represent the maximum treatment uptake per unit of time for individuals in class and r measures the extent of the effect of late initiation into ART treatment. Firstly, observe that for small , . Secondly, observe that for large , . Finally, when , we obtain , which is the case considered in Omondi et al. [8]. Individuals in class move to the class through a deteriorative process at a rate given by whereas individuals in class move to the class through an ameliorative process at a rate given by . In this model, we exclude the class of full blown AIDS patients as these are usually hospitalised and/or sexually inactive and hence their contribution to new HIV infections is negligible [8]. The total human population is thus given bySusceptible humans are recruited into the system through births or immigration at a constant rate . Susceptible individuals acquire new HIV infections at a rate given bywhere , , and denote the HIV transmission rates between susceptible individuals and infectious individuals. We assume that individuals in each compartment are indistinguishable and there is homogeneous mixing. Individuals in classes and experience disease related death at rates given respectively by and . The natural death rate of the general population is represented by . The differential equations for the model are given as follows;with the initial conditions:where we assume that all the model parameters are positive.

Analysis of the model

Positivity of solutions

The following theorem (Theorem 1) entails that all the state variables remain non-negative and the solutions of system (3) with positive initial conditions will remain positive for all .

Theorem 1

Given that the initial conditions of system (3) are, , , and. There existswhich solve system (3).

For more details on the proof of Theorem 1, we refer the reader to [8].

Invariant region

The feasible region for system (3) is given byResults to verify that the region is positively invariant with respect to system (3) can be obtained as given in [8].

Disease-free equilibrium and the basic reproduction number

The model has a disease-free equilibrium given bya scenario depicting a disease-free state in the community or society. The basic reproduction number of the model, is defined herein as the average number of people infected by each HIV infected individual during his/her infectious period in a population of completely susceptible individuals. The determination of is done using the next generation matrix approach [10]. It works out that, the basic reproduction number of system (3) is given by:Here, the four sub-reproduction numbers , , and represent the contributions of individuals in compartments , , and on the spread of HIV infection respectively. We can clearly note that is non-negative as which implies that .

Local stability of the disease-free steady state

The following theorem follows from van den Driessche and Watmough [10] (Theorem 2).

Theorem 2

The disease-free equilibrium pointof model system equations (3) is locally asymptotically stable ifand is unstable if.

Endemic equilibrium

The endemic equilibrium denoted by satisfiesFrom the first, third, fourth and fifth equation of (6), we have expressed in terms of as followsSubstituting expressions (7) into the second equation of (6) leads to the following fourth order polynomial equationSolving (8) gives which corresponds to the disease-free equilibrium orwhere the coefficients , are given in (10).We can clearly note that, and . We now determine the number of possible positive real zeros of the polynomial (10) using the Descartes Rule of Signs. The possibilities can be presented as shown below. Here, the number of possible positive real zeros is denoted by .

Backward bifurcation

Theorem 4.1 proven in Castillo-Chavez and Song [11] will be useful. We show that system (3) undergoes a backward bifurcation. Let us make the following change of variables:

, so that . We now use the vector notation . Then, system (3) can be written in the form

, whereWe now definewith signifying that the chance of acquiring HIV infection upon contact with individuals in class or upon contact with individuals in classes , and is the same, signifying a reduced chance of acquiring HIV infection upon contact with individuals in classes , and as compared to individuals in class , signifies an increased rate of acquiring HIV infection upon contact with individuals in classes , and as compared to individuals in class .

Let be the bifurcation parameter, corresponds toThe Jacobian matrix of model system (3) at when is given bywhere , , and are defined as before.

Model system (11), with has a simple eigenvalue, hence the center manifold theory can be used to analyse the dynamics of model system (3) near . It can be shown that , has a right eigenvector given by , whereHere, we note that and . Further, the left eigenvector of , associated with the zero eigenvalue at is given by , whereHere, take note that , accordingly as and , accordingly as . Also, and .

The computations of a and b are necessary in order to apply Theorem 4.1 in Castillo-Chavez and Song [11]. For system (11), the associated non-zero partial derivatives of F at the disease-free equilibrium are given in (14).It thus follows thatwhereNote that if , then and if then . Lastly,We thus have the following result

Theorem 3

If, then system (3) has a backward bifurcation at. Otherwise, ifthe endemic equilibrium is locally asymptotically stable forbut close to one.

We show the existence of a backward bifurcation through numerical example by creating bifurcation diagram around (Fig. 1). To draw a bifurcation curve (the graph of as a function of ), we fix the following parameters for illustrative purposes: .

Fig. 1

The figure showing a backward bifurcation. The solid lines denote stable states and the dotted lines denote unstable states

Remark

Epidemiologically, when a model exhibits backward bifurcation, this entails that it is not enough to only reduce the basic reproductive number to less than one in order to eliminate the disease.

Results and discussion

Numerical simulations

We carry out numerical simulations to support our theoretical findings.

Estimation of parameters

Parameter values used for numerical simulations are given in Table 1.

Table 1

Parameter values used in numerical simulations

Parameter	Definition	Range	Value	Source
β_1	Contact for individuals in S with those in I_1	0-1	0.912	[8]
β_2	Contact for individuals in S with those in I_2	0-1	0.894	[8]
β_3	Contact for individuals in S with those in I_A1	0-1	0.095	[8]
β_4	Contact for individuals in S with those in I_A2	0-1	0.091	[8]
σ_1	Progression from I_1 to I_A1	0.01-1	0.084	[8]
σ_2	Progression from I_2 to I_A2	0-1	0.1	Assumed
δ	Progression from I_1 to I_2	0.01-1	1.0	[8]
γ_1	Progression from I_A1 to I_A2	0.01-1	0.096	[8]
γ_2	Progression from I_A2 to I_A1	0.1-1	0.112	[8]
r	Effect of late initiation into ART	0-1	0.45	Assumed
ω_1	Disease related death of individuals in I_2	0-1	0.089	[7]
ω_2	Disease related death of individuals in I_A2	0-1	0.095	[7]
Λ	Recruitment rate into S	0-1	0.0239	[12–14]
μ	Natural death rate	0-1	0.0172	[14]

Numerical results

Figure 2 illustrates the effect of varying the parameter r on the prevalence of HIV. We note that increasing the parameter r results in an increase in the prevalence of HIV. In particular, increasing r from 0.1 up to 1.0 increases the prevalence rate of HIV with a level of approximately . This is a reflection that late diagnosis of HIV contributes to an increase in HIV infections. Thus, more effort should be directed towards encouraging individuals to get tested for HIV and ensuring those who are positive are timely initiated into ART treatment.

Fig. 2

Effect of varying parameter r on the prevalence of HIV, starting from 0.1 up to 1.0 with a step size of 0.01

Conclusions

A mathematical model that describes the dynamics of HIV/AIDS has been formulated using nonlinear ordinary differential equations. The model takes into account the impact of late diagnosis on HIV/AIDS transmission dynamics. Initiation into ART treatment of individuals with a CD4+ T cell count in the range 200–350\μ L has been described by the function (1). The model developed in this paper fits well with settings in most underdeveloped countries where stigma of HIV remains prevalent. Inclusion of the treatment function (1) increases the realism of the model developed by [8] and leads to some interesting dynamical aspects such as the occurrence of backward bifurcation.

In this study, it has been shown that the classical —threshold is not the key to control the spread of HIV infection within a population. In fact HIV infection may persist in the population even with subthreshold values of . Our results suggest that considerable effort should be directed towards encouraging early initiation into ART in order to reduce HIV prevalence. For instance, strategies such as the implementation of HIV self-testing programs would be of great help in the fight against HIV.

Limitations

Like in any model development, the model is not without limitations. The model can be extended to include the contribution of pre-exposure prophylaxis (PrEP) and other control measures not considered in the work.

ξ_3>0
ξ_2>0					ξ_2<0
ξ_1>0			ξ_1<0		ξ_1>0		ξ_1<0
ξ_0>0		ξ_0<0	ξ_0>0	ξ_0<0	ξ_0>0	ξ_0<0	ξ_0>0	ξ_0<0
i^∗	0	1	2	1	2	3	2	1