Impact of Case Detection and Treatment on the Spread of HIV/AIDS: a Mathematical Study

Transcription

1 Malaysian Journal of Mathematical Sciences (3): (8) MALAYSIA JOURAL OF MATHEMATICAL SCIECES Journal homepage: Impact of Case Detection and Treatment on the Spread of HIV/AIDS: a Mathematical Study Athithan, S and Ghosh, M Department of Mathematics, SRM Institute of Science and Technology, Kattankulathur, India School of Advanced Sciences, Vellore Institute of Technology, Chennai Campus, India minighosh@vitacin Corresponding author Received: 9 September 7 Accepted: May 8 ABSTRACT Here a simple mathematical model for HIV/AIDS with standard incidence is formulated and analyzed It is assumed that only a fraction of total HIV and AIDS infected are detected So only this fraction of infectives are subjected to proper counseling and are not taking part in the hetero-sexual transmission of HIV/AIDS The model is first analyzed by considering the case detection parameters as constants The basic reproduction number R of the model is computed and its relation with the existence and stability of different equilibria of the model is investigated It is found that the disease-free equilibrium of the model is locally and globally asymptotically stable for R < When R >, the endemic equilibrium exists and is locally and globally asymptotically stable under some restriction on parameters ext the optimal control problem is formulated by considering case detection parameters as time dependent This problem is analyzed using Pontryagin s maximum principle umerical simulation is performed to show the impact of optimal control and it is found that optimal control strategy gives a better result

2 Athithan, S and Ghosh, M Keywords: HIV/AIDS, mathematical model, stability, optimal control Introduction Although, more than 35 years have passed since the first HIV case was identified in USA, still there is no cure for this disease As per the Global Health Observatory (GHO) data, an estimated 8% of the adult population of world aged between 5 to 49 years are living with HIV However the burden of this epidemic varies considerably between countries and regions The sub-saharan Africa is badly affected followed by Americas, South-East Asia, Europe, Eastern Mediterranean, Pacific countries WHO (5) It is an established fact that mathematical modeling helps in control and prediction of an infectious disease Although, there exist many mathematical models which describes the transmission dynamics of HIV/AIDS, but because the dynamics of HIV varies from one region to another depending upon infection prevalence, social and cultural environments, etc, the existing models do have limitations in applicability of their results and selection of model parameters At present, different types of incidence functions are being used to model different scenarios and the importance of different types of incidence functions for HIV models is discussed in Cai et al (4), Sharomi et al (7) Here in Sharomi et al (7), the vaccine-induced backward bifurcation is demonstrated and in Cai et al (4) the resulting incidence term incorporates both the bilinear and the standard incidence In the Athithan and Ghosh (5), we have presented the simple mass action model for HIV which is applicable for the region where HIV prevalence is very high ow, here in this paper, we are going to present an HIV model with standard incidence which is suitable for the region where infection prevalence of HIV/AIDS is moderate to low The HIV/AIDS models with standard incidence have been formulated and analyzed by some researchers (See Cremin et al (3), Lima et al (8), aresh et al (6), yabadza and Mukandavire (), Sharomi et al (8)) but not much emphasis has been given to case detection and screening and HIV models which incorporate screening of infectives followed a different modelling approach In this paper, first a simple mathematical model for HIV with standard incidence is formulated and analyzed by considering a constant rate of case detection Later, an optimal control problem is formulated and analyzed by considering the case detection parameter as the control parameter 34 Malaysian Journal of Mathematical Sciences

3 Impact of Case Detection and Treatment on HIV/AIDS This paper is organized as follows: Section describes the basic model and its formulation, Section 3 exhibits the existence of equilibria and stability analysis with subsections as Basic Reproduction umber R, Existence of Equilibria and Jacobian, Local Stability of Disease Free Equilibrium (DFE), Local Stability of Endemic Equilibrium (EE) and Global Stability of DFE In Section 35, we present numerical computations to support our analytical findings Section 4 deals with the optimal control problem which contains the optimal control model with subsections as Hamiltonian and Adjoint Equations, and Optimal Control Theorems Section 5 explores the numerical simulation for the optimal control model and finally the results of our model are discussed in detail in Section 6 Model Formulation The disease HIV/AIDS caused mainly by sexual transmission, so our population under consideration is only the adult population We first divide the whole adult population into three disjoint compartments: S(t), I (t) and I (t) with (t) = S(t) + I (t) + I (t) as total population Here S(t), I (t) and I (t) denote the susceptible individuals, the HIV infectious individuals and the AIDS infected individuals, respectively, at time t The total population is assumed to be variable and homogeneously mixed ie, all are equally likely to acquire infection by the infectious individuals in case they come into contact The fraction of total HIV and AIDS infectives who are detected are given by η and ν respectively Further it is assumed that the rate of transmission of HIV due to detected group of infected individuals will be less compared to rate of transmission due to undetected group of infected individuals Let τ be the rate of treatment in HIV class By these assumptions the individuals who are either undetected or untreated will progress to AIDS class fast compared to those HIV infectives who are opting for treatment Figure : Transfer diagram of the model () Malaysian Journal of Mathematical Sciences 35

4 Athithan, S and Ghosh, M The transfer diagram of our proposed model is presented in Fig Based on above assumptions, we formulate our mathematical model as follows: ds = Λ µs ( k I + k I ) S, di = ( k I + k I ) S (µ + µ )I k 3 I, () di = k 3 I (µ + µ )I, where k = α η + ψ ( η)], k = α ν + ψ ( ν)] and k 3 = σ τη + σ ( τ)η + ( η)] The parameters used in the model () are described in Table Table : Description of parameters Parameter Description Λ Recruitment rate µ atural death rate η Fraction of total HIV infectives, who are detected ν Fraction of total individuals with AIDS, who are detected α Rate of transmission(in HIV class) α Rate of transmission(in AIDS class) ψ, ψ Modification parameters τ Rate of treatment σ Rate of progression to AIDS(detected & treated) σ Rate of progression to AIDS(undetected & untreated) µ Death rate due to HIV infection µ Death rate due to AIDS As (t) = S(t) + I (t) + I (t), for the analysis purpose we consider the following system: d = Λ µ µ I µ I, di = ( k I + k I ) S (µ + µ )I k 3 I, () di = k 3 I (µ + µ )I, Here S, I, I Model systems () and () are involving human population, hence all the variables and parameters of the proposed model are 36 Malaysian Journal of Mathematical Sciences

5 Impact of Case Detection and Treatment on HIV/AIDS positive The region of attraction of the model is given by { Ω = (S, I, I ) R+ 3 : S + I + I = Λ } µ We now show the positive invariance of Ω We have (t) = S(t) + I (t) + I (t) The rate of change of the total population under consideration is given by adding all the equations in the () and is given by: d = Λ µ µ I µ I Clearly whenever > Λ µ, d < ote that d is bounded by Λ µ By using standard comparison theorem Lakshmikantham et al (989) it can be shown that (t) Λ µ ( e µt ) + ()e µt In particular, (t) Λ } µ if () Λ µ {(S, Hence, the region Ω = I, I ) R+ 3 : Λ µ is positively invariant for the system () ow we prove that all the variables of the model () are non-negative This confirms that the solution of the system with positive initial conditions remains positive for all t > The fact is described in the following lemma Lemma If S(), I (), I (), the solutions S(t), I (t) and I (t) of the system () are positive for all t > Proof Assume that (t) t > We prove this lemma by using a contradiction We assume that there exists a first time t such that: S(t ) =, ds (t ) <, I (t), I (t), t t (3) there exists a first time t such that: I (t ) =, di (t ) <, S(t), I (t), t t (4) there exists a first time t 3 such that: I (t 3 ) =, di (t 3) <, S(t), I (t), t t 3 (5) Malaysian Journal of Mathematical Sciences 37

6 Athithan, S and Ghosh, M From (3) ds (t ) = Λ >, which is a contradiction to our assumption ds (t ) < and meaning that ds (t), t From (4) di (t I ) = k (t )S(t ) I (t ) = k (t )S(t ) S(t )+I (t ) > and not negative, which is again a contradiction to our assumption di (t ) < and meaning that di (t), t From (5) di (t 3) = k 3 I (t 3 ) > which is again a contradiction to our assumption di (t 3) < and meaning that di (t), t Hence we conclude that I (t) for t Thus the solutions S(t), I (t), I (t) of the system () remain positive for all t > 3 Equilibria and Stability Analysis 3 Basic Reproduction umber R The basic reproduction number is defined as the average number of secondary cases generated by an infected individual in his/her whole infectious period in a fully susceptible population We follow the method described in Driessche and Watmough () and compute the basic reproduction number (R ) Using the same notation as in Driessche and Watmough () the matrices F and V are given by F = (( k I + k I ) ) ( I I ), V = ( ) (µ + µ + k 3 )I k 3 I + (µ + µ )I ow, the matrices F and V evaluated at disease-free equilibrium point are given by ( ) ( ) k k F = µ + µ + k, V = 3 k 3 µ + µ The ext Generation matrix F V is given by ( F V = k (µ+µ +k 3) + k k 3 (µ+µ )(µ+µ +k 3) ] k (µ+µ ) ) So, the reproduction number R which is the spectral radius of the matrix F V is given by R = k (µ + µ + k 3 ) + k k 3 (µ + µ )(µ + µ + k 3 ) 38 Malaysian Journal of Mathematical Sciences

7 Impact of Case Detection and Treatment on HIV/AIDS The reproduction number R gives the average number of infected humans generated by one typical infected human in a fully susceptible population 3 Existence of Equilibria and Jacobian The equilibria for our model are determined by setting right hand side of the model () to zero The system () has the following equilibria namely Disease Free Equilibrium (DFE) E, I, I = ( ) ( ) Λ µ,, and Endemic Equilibrium (EE) E (, I, I ), where = Λ µ I µ I ( ), I k3 = I = k 4 I µ µ + µ and I is given by I = = Λ (k + k k 4 ) (µ + µ + k 3 )] (k + k k 4 ) µ + µ (µ + µ )k 4 ] (µ + µ k 4 )(µ + µ + k 3 ) Λ(R ) (k + k k 4 ) (µ + µ k 4 ) >, as R > = (k + k k 4 ) > (µ + µ + k 3 ) ie (k + k k 4 ) > µ + µ + k 4 (µ + µ ) = µ + (µ + µ k 4 ) + µk 4 = (k + k k 4 ) > (µ + µ k 4 ) µ µ µ The Jacobian matrix for the system () is given by M = a a a 3 a 3 a 33 where I a = k + (k + k ) I I + k, I I ] ] ] I a = k k k I + (k + k )I (µ + µ + k 3 ), ] I a 3 = k k I I ] ] I k + (k + k )I, a 3 = k 3, a 33 = (µ + µ ) I Malaysian Journal of Mathematical Sciences 39

8 33 Local Stability of DFE Athithan, S and Ghosh, M Theorem 3 The Disease Free Equilibrium E is locally asymptotically stable provided R < Proof The Jacobian matrix for the system () at DFE E is given by µ µ µ M = k (µ + µ + k 3 ) k k 3 (µ + µ ) Clearly µ is one of the eigenvalue of M The other two eigenvalues are computed from the matrix ( ) k (µ + µ + k 3 ) k k 3 (µ + µ ) The characteristic polynomial of this matrix is given by λ + h λ + h =, where h = k (µ + µ + k 3 ) (µ + µ )] and h = (k (µ + µ + k 3 ))(µ + µ ) k k 3 By Routh-Hurwitz criteria the DFE E is locally asymptotically stable when h, h > ow ] k h = (µ + µ + k 3 ) + (µ + µ ) = (µ + µ + k 3 ) when R < Further, (µ + µ + k 3 ) + ( R + ) + k k 3 (µ + µ )(µ + µ + k 3 ) ] + (µ + µ ) > h = (k (µ + µ + k 3 ))(µ + µ ) k k 3 ] k = (µ + µ + k 3 ))(µ + µ ) (µ + µ + k 3 ) k k 3 (µ + µ )(µ + µ + k 3 ) + = (µ + µ + k 3 ))(µ + µ ) R ] > only when R < Hence the Disease Free Equilibrium E is locally asymptotically stable provided R < 33 Malaysian Journal of Mathematical Sciences

9 Impact of Case Detection and Treatment on HIV/AIDS 34 Local Stability of EE Theorem 3 The Endemic Equilibrium E is locally asymptotically stable when µ + µ > b, µk 3 b 3 + µ (µ + µ )b ] > k 3 µ b + µ(µ + µ )b ] and {µ + µ b } {µ b + µ(µ + µ )] (µ + µ )b k 3 b 3 } > {µk 3 b 3 + µ (µ + µ )b k 3 µ b µ(µ + µ )b } Proof The Jacobian matrix for the system () at EE E is given by µ µ µ M = I b b b 3 I b 3 b 33 where b = k k I b = k + (k + k ) I I I I (µ + µ + k 3 ), b 3 = k k I b 3 = k 3, b 33 = (µ + µ ) I + k, ] ] I k + (k + k )I I ] ] ] I I k + (k + k )I I ], The eigenvalues of this Jacobian matrix are given by the roots of the following cubic equation in λ: λ 3 + g λ + g λ + g 3 = which is the characteristic equation of the Jacobian matrix M, where g = µ + b + b 33 ] = µ + µ b > if b <, g = µ µ b b + b b 3 k 3 (µ + µ ) + µ µ (µ + µ ) = µ b + µ(µ + µ )] (µ + µ )b k 3 b 3 g 3 = { µ k µ 3 b b 3 (µ + µ ) µ µ } b b = µk 3 b 3 + µ (µ + µ )b k 3 µ b µ(µ + µ )b Malaysian Journal of Mathematical Sciences 33

10 Athithan, S and Ghosh, M By Routh-Hurwitz criteria the EE E is locally asymptotically stable when g, g 3 > and g g > g 3 ie The EE E is locally asymptotically stable when µ + µ > b, µk 3 b 3 + µ (µ + µ )b ] > k 3 µ b + µ(µ + µ )b ] and {µ + µ b } {µ b + µ(µ + µ )] (µ + µ )b k 3 b 3 } > {µk 3 b 3 + µ (µ + µ )b k 3 µ b µ(µ + µ )b } 35 Global Stability of DFE Theorem 33 The disease-free equilibrium E of the model () is globally asymptotically stable if R < Proof We can prove the global stability of the DFE using the comparison theorem(see Ref Lakshmikantham et al (989), p 3) Re-writing the equations for the infected compartments in (), we have di di = (F V ) ( I I ) (I +I ) (k I + k I ) where F and V are as defined in Section 3 Since I + I (k (I + k I ) for all t >, it follows that di di (F V ) ( I I ) Since the eigenvalues of the matrix F V all have negative real parts (this comes from the local stability results in Lemma in Driessche and Watmough ()), then system () is stable whenever R < So, (I, I ) (, ) as t By the comparison theorem, it follows that (I, I ) (, ) and S A d and as t Then (S, I, I ) E as t ie E is globally asymptotically stable for R < when β = In this section we visualized our analytic result through numerical simulation The system () is simulated for various set of parameters To see the dynamic behavior of the model for disease-free equilibria the system is integrated numerically by fourth order Runge-Kutta method We consider the parameter set given in Table Here all the parameters are per day 33 Malaysian Journal of Mathematical Sciences

11 Impact of Case Detection and Treatment on HIV/AIDS Table : Values of parameters Parameter Description value Ref Λ Recruitment rate Assumed µ atural death rate Okosun et al (3) η Fractn of total HIV infectives who are detected 5 Assumed ν Fractn of total AIDS infectives who are detected Assumed α Rate of transmission(in HIV class) Okosun et al (3) α Rate of transmission(in AIDS class) 5 Assumed ψ, ψ Modification parameters 9, 9 Assumed τ Rate of treatment 4 Okosun et al (3) σ Rate of progrsn to AIDS(detected & treated) Okosun et al (3) σ Rate of progrsn to AIDS(undetected & untreated) 5 Assumed µ Death rate due to HIV infection 3 Assumed µ Death rate due to AIDS 5 Assumed Malaysian Journal of Mathematical Sciences 333

12 Athithan, S and Ghosh, M For the parameter set given in the table, the system () has only the diseasefree equilibrium E and it is locally asymptotically stable (see Fig ) Again for the parameter set given in the table except for µ =, the system () has two equilibria the disease-free equilibrium and the endemic equilibrium Here the disease-free equilibrium E is unstable and the endemic equilibrium E is locally asymptotically stable (see Fig 3) This fact is more clear from the Fig 4, where phase portrait in S I I are shown for different set of initial conditions Figs 5-8 explore the relation between R with other parameters In Fig 5 we see the relation between R, η and ν From this figure it is clear that increase in detection parameters causes decrease in R In Fig 6 we see the relation between R, ψ and ψ So if these modification parameters are increasing then there is an increase in the value of basic reproduction number causing elimination of disease almost impossible In Fig 7 we see the relation between R, σ and σ Here too increase in σ and σ causes increase in basic reproduction number Fig 8 is reflecting the relation between R, α and α As expected increase in rates of transmission α and α causes increase in R So our aim should be to manipulate these parameters using different control strategies such that the value of the basic reproduction number R can be made less than one This will lead to DFE to be globally stable causing the complete elimination of the disease from the population 9 8 Population I I Time (in days) Figure : Variation of, I, I with time showing the stability of disease-free equilibrium (when R = 8956 < ) for the parameter values are given in Table 334 Malaysian Journal of Mathematical Sciences

13 Impact of Case Detection and Treatment on HIV/AIDS 9 8 I I 7 Population Time (in days) Figure 3: Variation of, I, I with time showing the stability of endemic equilibrium (when R = 478 > ) for the parameter values are given in Table except µ = 8 6 I I 4 6 Figure 4: I I phase plot showing the stability of the EE for R > 5 R 5 5 ν 4 η 6 8 Figure 5: Variation of R with ν and η showing the impact of η and ν on R Malaysian Journal of Mathematical Sciences 335

14 Athithan, S and Ghosh, M R ψ ψ Figure 6: Variation of R with ψ and ψ showing the impact of ψ and ψ on R 3 5 R σ 4 σ 6 8 Figure 7: Variation of R with σ and σ showing the impact of σ and σ on R 8 6 R 4 5 α 4 α 6 8 Figure 8: Variation of R with α and α showing the impact of α and α on R 336 Malaysian Journal of Mathematical Sciences

15 Impact of Case Detection and Treatment on HIV/AIDS 4 Optimal Control Problem In the preceding model with detection, we considered the fixed value of the detection parameter throughout the analysis But in reality these parameters may be dependent on time Therefore we used the detection parameters related to HIV and AIDS classes as time dependent parameters and we study the optimal control over the detection parameters Through this study we develop a strategy through the objective function for minimizing the cost as well as the number of infectives We use Pontryagin s Maximum Principle (see Bartl et al (), Kar et al (), Kar and Ghosh (), Zaman et al (8), etc) to accomplish our aim The optimal control system with the objective functional is given below: ds = Λ µs + ( k (t) I + k (t) I ) S, di = ( k (t) I + k (t) I ) S (µ + µ + k 3 (t))i, (6) di = k 3 (t)i (µ + µ )I, where = S + I + I, k (t) = α η(t) + ψ ( η(t))] = α ( ψ )η(t) + ψ ], k (t) = α ν(t) + ψ ( ν(t))] = α ( ψ )ν(t) + ψ ], k 3 (t) = {σ τη(t) + σ ( τ)η(t) + ( η(t))]} = σ τη(t) + σ τη(t)] = τη(t)σ σ ] + σ We formulate an optimal control problem with the objective (cost) functional given by T ( J = C I + C I + C 3η + C 4ν ) (7) subject to the state system given by (6) Our objective is to find a control η and ν such that J(η, ν ) = min J(η, ν), where Ω = {η, ν: measurable and η(t), ν(t) for t, t ]} is the set for the η, ν Ω controls Here, the value η(t) =, ν(t) = represents the maximal control of detection on HIV and AIDS class respectively The quantities C, C, C 3 and C 4 represent, respectively, the weight constants The terms C 3 η and C 4 ν describe the cost associated with detection control on HIV and AIDS class respectively Malaysian Journal of Mathematical Sciences 337

16 Athithan, S and Ghosh, M 4 Hamiltonian and Adjoint Equations The Lagrangian of this problem is given by L(S, I, I, η, ν) = C I + C I + C 3η + C 4ν (8) ext we form the Hamiltonian H for our problem as follows: ds H(S, I, I, η, ν) = L(S, I, I, η, ν) + λ + λ di + λ 3 di (9) where λ i, i =,, 3 are the adjoint variables or the co-state variables and can be determined by solving the following system of differential equations: ]} dλ = H (S+I +I = λ S {µ + (k I + k I ) ) S (S+I +I ) dλ λ (k I + k I ) = H = C I λ { k S λ {k S ] S+I (S+I +I ) ] I +I (S+I +I ) ] S+I (S+I +I ) k, + k ] I S (S+I +I ) ]} I S (S+I +I ) } (µ + µ + k 3 ) () dλ 3 λ 3 k 3, { = H = C I λ k λ { k ] I S (S+I +I ) ] I S (S+I +I ) k S + k S ]} S+I (S+I +I ) ]} S+I (S+I +I ) + λ 3 (µ + µ ), Let S, Ĩ, Ĩ and Ñ be the optimum value of S, I, I and { λ, λ, λ 3 } be the solutions of the system () Also let 4 Optimal Control Theorems We now state and prove the following theorem by following Lukes (98)and Zaman et al (8) Theorem 4 There exists optimal controls η, ν Ω such that J(η, ν ) = min η, ν Ω J(η, ν) subject to the system (6) Proof This theorem is proved using Lukes (98) It is easy to see that all the control and the state variables are nonnegative Also the necessary convexity 338 Malaysian Journal of Mathematical Sciences

17 Impact of Case Detection and Treatment on HIV/AIDS of our objective functional in η and ν is satisfied for this minimizing problem The control variable set η, ν Ω is also closed and convex by definition The boundedness of the optimal system determines the compactness required for the existence of the optimal control In addition, the integrand in the functional (7), C I + C I + C 3η + C 4ν is convex on the control set Ω and also the state variables are bounded This proves the theorem As there exists an optimal control for minimizing the functional subject to equations (6) and (), the Pontryagin s Maximum Principle is used to obtain the necessary conditions to get the optimal solution as follows: Let (x, u) is an optimal solution of an optimal control problem Then there exists a non trivial vector function λ = (λ, λ,, λ n ) satisfying the following equalities dx = H(t,x,u,λ) λ, = H(t,x,u,λ) u, () dλ = H(t,x,u,λ) x By using the Pontryagin s Maximum Principle Pontryagin et al (96) and the theorem (4) we are going to state and prove the below theorem Theorem 4 The optimal controls η, ν given by minimizes J over the region Ω η = max{, min( η, )} and ν = max{, min( ν, )} where { } η = ( λ C λ Ĩ S )( ψ )α 3 Ñ + ( λ λ 3 )(σ σ )τĩ, { } ν = ( λ C λ Ĩ S )( ψ )α 4 Ñ Proof Using the optimality conditions H η = and H ν = Malaysian Journal of Mathematical Sciences 339

18 Athithan, S and Ghosh, M we get { } η = ( λ C λ Ĩ S )( ψ )α 3 Ñ + ( λ λ 3 )(σ σ )τĩ (= η), { } ν = ( λ C λ Ĩ S )( ψ )α (= ν) 4 Ñ These controls are bounded with upper and lower bounds as and respectively ie η = if η < and η = if η > and ν = if ν < and ν = if ν >, otherwise η = η and ν = ν Hence for this controls (η ) and (ν ) we get the optimum value of the functional J given by equation (7) Hence the theorem 5 umerical Simulation of the Optimal Control Model To exhibit the effect of optimal control, we performed the numerical simulation in this section The control profile is applied for the period of days As discussed in Athithan and Ghosh (5), here too we assume that the weight associated with detection in HIV class (C 3 ) is greater than or equal to the weight associated with detection in AIDS class (C 4 ) The optimality system in Section 4 is solved by iterative method (see Jung et al (), Lenhart and Workman (7), etc) At first we solve the state equations by the forward Euler method for the time interval, ] starting with an initial guess for the adjoint variables Then we solve the adjoint variables in the same time interval by using the obtained solutions of the state variables and the transversality conditions backward in time The following set of parameters is considered to simulate the optimal control model: Λ =, µ =, α = 5, α = 3, τ = 8, σ =, σ = 5, µ =, µ = 5, ψ = 4, ψ = 9 The weight constants C and C are taken as and C 3 and C 4 are varied From Figs 9 and, it is easy to observe that the controls take their highest value in starting period and after some period of time (nearly at the end of optimal strategic time period) the control parameter values are toned down slowly and finally comes to the level Fig 9 shows that the HIV detection rate should be maintained according to the cost applied on detecting HIV and AIDS patients over the period of control Figs -3 are showing the effects 34 Malaysian Journal of Mathematical Sciences

19 Impact of Case Detection and Treatment on HIV/AIDS of the costs C 3 and C 4 on effective control profiles of η and ν Figs 4-6 are showing the plots of adjoint variables It is evident from the figures 7-9 that the optimal control strategy gives the better result than the fixed control effort It is observed that in the case of optimal control, number of susceptible individuals starts increasing and the number of HIV and AIDS infectives start decreasing These facts are evidently showing that the optimal control program is to be conducted to detect the maximum number of infectives and to decrease the infection prevalence of HIV and AIDS infected population 9 8 (C 3 =5,, C 4 =,35) (C 3 =35, C 4 =5) 7 Control Profile η (C 3 =, C 4 =5) Figure 9: The control profile of η of the intervention strategies for different values of C 3 and C (C 3 =35, C 4 =5) (C =5, C =) 3 4 (C 3 =, C 4 =5) (C =, C =35) 3 4 Control Profile ν Figure : The control profile of ν of the intervention strategies for different values of C 3 and C 4 Malaysian Journal of Mathematical Sciences 34

20 Athithan, S and Ghosh, M 9 8 Control profiles η and ν Figure : The control profiles of η and ν of the intervention strategies for the values C 3 = and C 4 = Control profiles η and ν Figure : The control profiles of η and ν of the intervention strategies for the values C 3 = and C 4 = Control profiles η and ν Figure 3: The control profiles of η and ν of the intervention strategies for the values C 3 = 35 and C 4 = 5 34 Malaysian Journal of Mathematical Sciences

21 Impact of Case Detection and Treatment on HIV/AIDS Adjoint Variable λ λ Adjoint Variable λ Adjoint Variable λ 3 λ 6 λ Figure 4: Simulation results showing the effect of control measures on adjoint variables (shadow price) for C 3 = and C 4 = 5 Adjoint Variable λ Adjoint Variable λ λ λ 5 5 Adjoint Variable λ 3 λ Figure 5: Simulation results showing the effect of control measures on adjoint variables (shadow price) for C 3 = and C 4 = 35 Adjoint Variable λ Adjoint Variable λ λ λ 5 5 Adjoint Variable λ 3 5 λ Figure 6: Simulation results showing the effect of control measures on adjoint variables (shadow price) for C 3 = 35 and C 4 = 5 Malaysian Journal of Mathematical Sciences 343

22 Athithan, S and Ghosh, M S 5 (Optimal Control) 5 η=ν=7 η=ν= η=ν= Time (in days) Figure 7: Variation of S with time for different levels of fixed controls and optimal control I 5 5 η=ν=7 η=ν=4 η=ν= 5 (Optimal Control) Time (in days) Figure 8: Variation of I with time for different levels of fixed controls and optimal control 8 I 6 4 η=ν= η=ν=7 η=ν=4 (Optimal Control) Time (in days) Figure 9: Variation of I with time for different levels of fixed controls and optimal control 344 Malaysian Journal of Mathematical Sciences

23 Impact of Case Detection and Treatment on HIV/AIDS 6 Conclusion Here an HIV/AIDS model with standard incidence is proposed and analyzed to exhibit the effects of case detection and treatment The expression for the basic reproduction number R is obtained and its relationship with other parameters are explored graphically Equilibria of the model are found and their local and global stabilities are discussed in detail It is observed that two nonnegative equilibria are possible for the system (): the disease-free (E ) which always exists and the endemic equilibrium (E ) which exists only for R > It has been proved that the disease-free equilibrium is globally stable when R < The endemic equilibrium is locally asymptotically stable under some restriction on parameters Further, the proposed basic model is converted to an optimal control problem to know the dynamics of the disease when the control parameters become time dependent The control profile for the detection parameters are obtained and the effect of optimal control on infectives are demonstrated using numerical simulation The results of our simulation confirm that the optimal control strategies is better than fixed control as it helps in decreasing the number of infectives significantly in a specified interval of time Acknowledgement The authors thank the handling editor and anonymous referees for their valuable comments and suggestions which led to an improvement of our original manuscript References Athithan, S and Ghosh, M (5) Stability analysis and optimal control of hiv/aids model with case detection and treatment J Adv Res Dyn Control Syst, 7():66 84 Bartl, M, Li, P, and Schuster, S () Modelling the optimal timing in metabolic pathway activation- use of pontryagin s maximum principle and role of the golden section Biosystems, ():67 77 Cai, L, Guo, S, and Wang, S (4) Analysis of an extended hiv/aids epidemic model with treatment Applied Mathematics and Computation, 36:6 67 Malaysian Journal of Mathematical Sciences 345

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