Epidemic Model of HIV/AIDS Transmission Dynamics with Different Latent Stages Based on Treatment
Ram Singh^{1}, Shoket Ali^{1}, Madhu Jain^{2}, Rakhee^{3}
^{1}Department of Mathematical Sciences, Baba Ghulam Shah Badshah, University, Rajouri, Jammu and Kashmir, India
^{2}Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttrakhand, India
^{3}Department of Mathematics, Birla Institute of Technology and Science, Pilani, Rajasthan, India
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To cite this article:
Ram Singh, Shoket Ali, Madhu Jain, Rakhee. Epidemic Model of HIV/AIDS Transmission Dynamics with Different Latent Stages Based on Treatment. American Journal of Applied Mathematics. Vol. 4, No. 5, 2016, pp. 222-234. doi: 10.11648/j.ajam.20160405.14
Received: August 30, 2016; Accepted: September 23, 2016; Published: October 14, 2016
Abstract: The mathematical model for analyzing the transmission dynamics of HIV/AIDS epidemic with treatment is studied by considering the three latent compartments for slow, medium and fast progresses of developing the AIDS. By constructing the system of differential equations for the different population groups namely susceptible, three types of latent individuals, symptomatic stage group and full blown AIDS individuals, the mathematical analysis is carried out in order to understand the dynamics of disease spread. By determining the basic reproduction number (R_{0}), the model examines the two equilibrium points (i) the disease free equilibrium and (ii) the endemic equilibrium. It is established that if the disease free equilibrium is locally and globally asymptotically stable. The stability of endemic equilibrium has also been discussed.
Keywords: Transmission Dynamic, HIV/AIDS, Latent Compartments, Reproduction Number, Stability
1. Introduction
HIV/AIDS is one of the fatal diseases, which causes millions of death in both developed and developing nations. More than 35 million people approximately in the worldwide are living with HIV. In 2013, the newly infected persons were reported around 2.1 million which is 38% less from 2001. HIV infections among children are also declined by 58% since 2001. AIDS related deaths have also fallen by 35% since the peak in 2005, which is a result of availability of Antiretroviral Therapy (ART). Still tuberculosis related deaths in people living with HIV remain the leading cause of death. Since epidemic modeling initially studied by May and Anderson [1], many advance researches have taken place and several important aspects were incorporated by different authors during last three decades. Several epidemiological investigations related to HIV/AIDS infection have been conducted via mathematical models by many researchers [2-6]. An epidemiological model with nonlinear incident rate was discussed by several authors [7, 8]. Cai et al. [9] established the ordinary differential equation (ODE) model with two infective stages before transition to AIDS. They considered by all sort of treatment methods by considering that some individuals transformed into asymptomatic individuals from symptomatic individuals. The capacity of human immune system can be reduced by chronic diseases, such as diabetes and tuberculosis and can reduce. To analyze this behavior, Huo and Feng [10] developed a model with slow and fast latent compartments. Okosun et al. [11] studied the treatment of HIV/AIDS and screening of unaware infective on the transmission dynamics of disease in a homogeneous population. Defeng and Wang [12] proposed a time delayed mathematical model to analyze the effect of vaccination and ART (Antiretroviral Therapy) on HIV/AIDS. They considered two types of individuals one who are aware of their infected stage and other individuals are unaware about their infected stage. Bhunu and Mushayabasa [13] formulated a mathematical model of the co-dynamics of hepatitis C virus and HIV/AIDS in order to assess their impact on the dynamics of each disease in the presence of treatment. Cai et al. [14] investigated an HIV/AIDS treatment model with multiple infection stages and treatment where infection was assumed to be of density dependent form. Kaur et al. [15] also proposed a nonlinear model for studying the transmission dynamics of HIV/AIDS epidemic with emphasis on the role of female sex workers. Elaiw and Almuallem [16] to investigate the qualitative behaviors of three HIV dynamical models with two types of co circulating target cells. Wang et al. [17] has been studied the global stability of HIV viral infection model with continuous age-structure using the direct lyapunov method. Shen et al. [18] to analyze the mathematical model of global dynamics with two lyapunov functions are constructed to prove the global stability of disease free and endemic equilibria. Treatment class of HIV/AIDS epidemic model is introduced by Huo et al. [19].
In this paper, the mathematical model have been proposed for analyzing the transmission dynamics of HIV/AIDS epidemic with treatment by considering the three latent compartments for slow, medium and fast progresses of developing the AIDS. To study the dynamics of the spread of HIV, the basic reproduction number under disease free equilibrium is analyzed. The global stability of the endemic equilibrium for some special cases is discussed. The remaining contents of the paper are organized into different sections as follows. Section 2 presents the model description, system equations and some basic properties. Stability analysis is done in the section 3. Numerical results to support the proposed model are provided in section 4. Section 5 summaries the important findings and scope of the future works.
2. Model Description
The mathematical model to analyze the transmission dynamic of HIV/AIDS epidemic is developed by dividing the total population into six compartments, namely the susceptible compartment (S), slow latent compartment (I_{1}), medium compartment (I_{2}), fast latent compartment (I_{3}), symptomatic stage (J) and a full-blown AIDS (A) group. The model flow depicting the biological system is illustrated in fig. 1. The total number of population at time t is given by
For the mathematical formulation of the model, the following notations are used:
Recruitment rate of the population
Transmission coefficient of the fast latent compartment
Transmission coefficient of the symptomatic stage
The fraction of susceptible being infected by and entering into respectively;
The fraction of susceptible being infected by and entering into respectively;
Progression rate from latent compartmentto
Progression rate from latent compartmentto symptomatic compartment
Progression rate of disease from compartment to
Treatment rate from compartment to
Natural (disease induced) death rate
2.1. The Governing Equations
The following differential equations governing the model are constructed by considering the appropriate in-flow and out-flow rates of each compartment:
(1)
(2)
(3)
(4)
(5)
(6)
For brevity of notation, by using
The above set of equations (1)-(6) becomes
(7)
(8)
(9)
(10)
(11)
(12)
2.2. Basic Properties
In this sub-section, we discuss some preliminary concepts which will be needed further for the mathematical analysis of the concerned model. We will show that for all the solutions are positively invariant in some region.
2.2.1. Invariant Region
Since the model displays the changes in the human population, the variables and the parameters are assumed to be positive for all The system of equations (7)-(12) will therefore be analyzed in a suitable feasible region of biological interest. We have the following lemma related to feasible region for the system (7)-(12).
Lemma1: The feasible region defined by
with initial conditions is positively invariant for the system of equations (7)-(12).
Proof: Adding the system of equations (1)-(6), we obtain
(13)
Solving the above differential equation (13), we have
where represents the initial values of the total population. Thus
It implies that the region is a positively invariant set for the system (7)-(12). Now we consider the dynamics of system (7)-(12) on the region given by set.
2.2.2. Positivity of Solutions
It is important for the model described by the system (7)-(12) to prove that all the state variables are non-negative so that the solutions of the system with positive initial conditions remain positive for all t > 0. We thus state the following lemma.
Lemma 2: Given that the initial conditions of the system (7)-(12) by S(0) > 0, I_{1}(0) > 0, I_{2}(0) > 0, I_{3}(0) > 0, J(0) > 0, and A(0) > 0, the solutions and are non-negative for all t > 0.
Proof: Under the given initial conditions, it is easy to prove that the solutions of the system (7)-(12) are positive; if not, we assume a contradiction that:
(i) there exists a first time such that
(ii) there exists a
(iii) there exists a
(iv) there exists a
(v) there exists a
(vi) there exists a
In the first case (i), we have
which is a contradiction meaning that
In the second case (ii), we have
which is a contradiction meaning that. Similarly, from (iii)–(vi), it can be easily shown that I_{2}(0) > 0, I_{3}(0) > 0, J(0) > 0, and A(0) > 0, for all.Thus, the solutions and of system (7)-(12) remain positive for all t > 0.
3. The Analysis
In this section, we compute the equilibrium states, namely the disease free equilibrium (DFE) and the endemic equilibrium (EE) and provide stability analysis by determining the basic reproduction number.
3.1. Disease Free Equilibrium (DFE) and Basic Reproduction Number R0
Proposed model (7)-(12) has a disease free equilibrium given by
The basic reproduction number of the system (7)-(12) is obtained by considering the next generation matrices F and V defined for the appearance of new infection terms and the transfer of individuals out of latent compartment, respectively [20]. Let then the system (7)-(12) can be written as
(14)
where
(15)
and
(16)
Now we employ the linearization method. On taking partial derivatives, the associated matrices at DFE are obtained as,
(17)
and
(18)
Consider the following matrix
(19)
where
Thus the reproduction number , is obtained as
(20)
Following Theorem 2 of Van den Driessche and Watmough [17], we have the following result on the local stability of E_{0 }:
Theorem 1: The disease free equilibrium of the system (7)-(12) is locally asymptotically stable if and unstable if
3.2. Global Stability of Disease Free Equilibrium (DFE)
Here we analyze the global stability by using a comparison theorem [21, 22].
Theorem 2: The disease free equilibrium of the system (7)-(12) is globally asymptotically stable if and unstable if
Proof: The equation of the infected components in system (7)-(12) can be written as
(21)
where F and V are defined in (17) and (18) for all in .
Thus, we have
(22)
Since all eigen values of the matrix have negative real parts, the linearized differential inequality (22) is stable whenever Consequently as. Thus following the comparison theorem, we have and as.
Hence the DFE is globally asymptotically stable for
3.3. Endemic Equilibrium (EE)
The endemic equilibrium point of the system (7)-(12) is given
;
(23)
3.4. Global Stability of Endemic Equilibrium
In this sub section, we present the global stability of endemic equilibrium.
Theorem 3: If and , the endemic equilibrium is globally asymptotically stable.
Proof: To prove global stability of endemic equilibrium, we compute the following Lyapunov function:
(24)
The derivative of is
(25)
The system (7)-(12) satisfies the following relation at the equilibrium point:
(26)
(27)
(28)
(29)
(30)
Substituting the results from (26)-(30) in (25), we get
(31)
Using the following variables substitutions
(32)
Equation (31) reduces to
(33)
The only variable terms that appears in (33) with positive coefficients are and Making the coefficient of and equal to zero, we have
(34)
(35)
(36)
(37)
(38)
(39)
Solving equations (34), (38) and (39), we get
(40)
From equations (36) and (37), we have
and (41)
Thus, we get
(42)
Hence, equation (33) becomes
(43)
Thus, we have
(44)
Since the arithmetic mean is greater than or equal to the geometric mean, we have
(i) for and if and only if
(ii) for and if and only if
(iii) for and if and only if
(iv) for and if and only if
(v) for and if and only if
(vi) for and if and only if
(vii) for and if and only if
(viii) for and if and only if
(ix) for and if and only if
Therefore, for and if and only if The maximum invariant set of system (7)-(12) on the set is the singleton Thus for system (7)-(12), the endemic equilibrium is globally asymptotically stable if and by LaSalle [23].
4. Numerical Simulation
To validate the analytical results established in previous section, we conduct the numerical simulation by taking an example. The system (7)-(12) is simulated by fixing the default values of the parameters as
For the other parameters chosen as
figs. 2(a)-2(b) demonstrate that the reproduction number which indicates that the disease free equilibrium pointsis globally stable.
2(a)
2(b)
Figs. 2. Stability of the disease free equilibrium point .
For the figs. 3(a)-3(b), the parameters are set as and . It is clear from these figs. that the reproduction number which corresponds to the global stability of endemic equilibrium points
3(a)
3(b)
Figs. 3. Stability of the endemic equilibrium point.
Figs. 4(a)-4(b) depict the relation among (a) & and (b)& , respectively. From these figs., it is clear that when are small and is too much large then the basic reproduction numberof the system (7)-(12) is less than unity. Biologically, we can infer that the treatment for individuals in slow, medium and fast compartments has significant impact on the transmission dynamic of the disease.
4(a)
4(b)
Fig. 5 depicts the relationship among,and . It is seen that when the values of and are less than of the model for slow and fast compartments, our results are in good agreement with the results of Huo and Feng [10] for two latent compartments model for which the basic reproduction number (R_{1}) is given by
(45)
6(a)
6(b)
6(c)
6(d)
Figs. 6. Relationship among R_{2} and x_{1}for different values of (a) (b) µ (c) (d) .
For varying values of parameters and respectively, figs. 6(a)-6(d) exhibit the relation between and . The other parameters are set as and . Fig. 6(a) displays that increases when the value of goes on increasing but it decreases as increases. In fig. 6(b) reveals the trends of with for different values of It is observed that the value of decreases with the increment in Fig. 6(c) shows that increases by increasing Biologically it can be interpreted easily that the transmission coefficient of symptomatic stage has significant effect on for single latent stage. Fig. 6(d) reveals that the value of remains almost constant when the value of increases; this demonstrates that the transmission coefficient of fast latent compartment has also significant effect on for single latent stage.
5. Conclusion
In this paper, the proposed epidemic model has three latent stages to explore the dynamical behavior of HIV/AIDS. Based on numerical simulation, it is concluded that the disease free equilibrium is both locally and globally asymptotically stable whenever the basic reproduction number is less than unity. Further, the endemic equilibrium is globally asymptotically stable whenever the corresponding reproduction number is greater than unity. It is noticed that the treatment of infective compartment has positive impact on HIV/AIDS control. The present model can be extended to multi-latent compartments for infective population depending upon several stages due to treatment and socio-economic constraints.
References