HIV/AIDS Treatment Model with the Incorporation of Diffusion Equations

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1 Applied Mathematical Sciences, Vol. 12, 2018, no. 12, HIKARI Ltd HIV/AIDS Treatment Model with the Incorporation of Diffusion Equations K. F. Tireito, G. O. Lawi and C. O. Okaka Department of Mathematics, Masinde Muliro Uniersity of Science and Technology, Kenya Corresponding author Copyright c 2018 K. F. Tireito, G. O. Lawi and C. O. Okaka. This article is distributed under the Creatie Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, proided the original work is properly cited. Abstract The spread of the HIV/AIDS is still a major concern due to the mobility and trael patterns of high risk populations, thus the study through the use of diffusion equations inestigated the consequences of spatial distribution of the high risk indiiduals. Introduction of the diffusion equations shows that an introduction of infectie indiiduals results in a zone of transition from one equilibrium point to another. This results in formation of a traeling wae solution across the disease free equilibrium and the endemic equilibrium. Keywords: HIV/AIDS, Diffusion Equations, Traelling wae 1 Introduction The transmission of HIV/AIDS is associated with the spatial distribution of the high risk indiiduals [10]. For example, the spread HIV/AIDS is partially attributed to migration from rural to urban areas and its concomitant return migration [1]. The mobility of indiiduals can increase the rate of interaction and expose indiiduals to sexual partners coming from areas of higher prealence rates [7]. The moement of high risk indiiduals due to geographic diffusion influence transmission of HIV/AIDS. There are studies inestigating statistical associations between migration and reported risky behaiours and obsering that inherent characteristics of mobility leads to higher leels of

2 604 K. F. Tireito, G. O. Lawi and C. O. Okaka risky sexual behaiour (and hence HIV incidence). The insights on mobility are rarely inoked to explain the relationship between HIV/AIDS transmission and mobility of high risk indiiduals. 2 Model Formulation The large mobility of people within from one region to other influences the transmission dynamics of infectious diseases and hence, a spatial diffusion model is deeloped to capture the mobility of indiiduals. The model is summarized as; S(x, t) I(x, t) T (x, t) A(x, t) = Λ µs(t) I(t)S(t) (t) = I(t)S(t) (t) + D 2 S x 2, (ν + θ + µ)i(t) + D 2 I x 2, = δa(t) + θi(t) (γ + µ)t (t) + D 2 T x 2, = γt (t) + νi(t) (d + µ + δ)a(t) + D 2 A x 2 (1) where D is the diffusion coefficient with homogeneous eumann boundary conditions and initial conditions S = I = T = A = 0 on (0, + ) (2) S(x, 0) = S 0 (x) 0, I(x, 0) = I 0 (x) 0, T (x, 0) = T 0 (x) 0, A(x, 0) = A 0 (x) x. (3) The region is assumed to be the whole space (, + ). The fluxes of the susceptible class and infected classes are related to their concentration gradient and moe from regions of higher concentration to regions of low concentration. Since the model describes a human population, the population should remain bounded and nonnegatie. 3 Reproduction number The basic reproduction number is the aerage number of secondary infections due to a single infectious indiidual introduced in a fully susceptible population during the entire period of infectiity. From the model (1) and at diffusion free state (D = 0 ), taking the deriaties with respect to the state ariables for

3 HIV/AIDS treatment model 605 the transmission and transition terms, the matrices F and V can be found by ( ) Fi (E 0 ) F = x j ( ) Vi (E 0 ) V = (4) x j The F i are the new infections (Transmission) while the V i are the transfers of infection (Transition) from one compartment to another. Thus, I(t)S(t) (t) F : = 0 0 (ν + θ + µi(t)) V : = δa(t) θi(t) (γ + µ)t (t). (5) γt (t) νi(t) + (d + µ + δ)a(t) Hence, the transmission matrix F and the Transition matrix V are; F : = 0 0 (ν + θ + µ) V : = θ γ + µ δ (6) ν γ d + µ + δ The next generation matrix is gien by F V 1. The inerse of the matrix V exists and is gien by; 1 ν+θ+µ V 1 (d+µ+δ)θ δν d+µ+δ δ := δγ+(γ+µ)(d+µ+δ) δγ+(γ+µ)(d+µ+δ) (7) (ν+θ+µ)[δγ+(γ+µ)(d+µ+δ)] ν(γ+µ)+θγ (ν+θ+µ)[δγ+(γ+µ)(d+µ+δ)] and the next generation matrix F V 1 is F V 1 : = 0 0 = γ δγ+(γ+µ)(d+µ+δ) γ+µ δγ+(γ+µ)(d+µ+δ) 1 ν+θ+µ (d+µ+δ)θ δν (ν+θ+µ)[δγ+(γ+µ)(d+µ+δ)] ν(γ+µ)+θγ (ν+θ+µ)[δγ+(γ+µ)(d+µ+δ)] d+µ+δ δγ+(γ+µ)(d+µ+δ) γ δγ+(γ+µ)(d+µ+δ) δ δγ+(γ+µ)(d+µ+δ) γ+µ δγ+(γ+µ)(d+µ+δ) ν+θ+µ 0 (8) 0

4 606 K. F. Tireito, G. O. Lawi and C. O. Okaka The reproduction number R 0 is gien by R 0 = ρ(f V 1 ) as described in [8], where ρ(f V 1 ) is the spectral radius of the matrix (F V 1 ). Hence, R 0 = ν + θ + µ. (9) 4 Local Stability of the Disease Free Equilibria We linearize the dynamical system (1) around the arbitrary spatially homogeneous fixed point E f (S, I, T, A) for small space and time. Consider the following system µ βic λ D S βic S (ν + θ + µ) λ D 0 θ (γ + µ) λ D δ 0 ν γ (d + µ + δ) λ D The characteristic equation resulting from system (10) is gien by det (J DI λi) = 0 where I is the identity matrix, D =is the diffusion terms, and J is the Jacobian matrix of system (1) without diffusion terms. At Disease Free Equilibrium (DFE), I = T = A = 0 and = S = Λ. Hence, system (10) becomes; µ µ λ D 0 (ν + θ + µ) λ D 0 θ (γ + µ) λ D δ 0 ν γ (d + µ + δ) λ D (11) To inestigate the stability of the DFE, we compute compute the eigenalues of Equation (12) µ λ D 0 (ν + θ + µ) λ D 0 θ (γ + µ) λ D δ 0 ν γ (d + µ + δ) λ D On ealuating Equation (12), we obtain λ 1 = µ D, λ 2 = (θ + µ + ν)[r 0 1] D = 0 (10) (12)

5 HIV/AIDS treatment model 607 To compute the remaining eigenalues, we sole the reduced matrix (γ + µ) λ D δ γ (d + µ + δ) λ D = 0 (13) For stability of (13), we must hae the trace (γ + 2µ + d + δ) D < 0 and determinant (γ + µ + D)(d + µ + δ + D) δγ > 0. Hence, Equation (1) will be stable wheneer R 0 < 1 and (γ+µ+d)(d+µ+δ+d) δγ > 1. 5 Global Stability of the Disease Free Equilibria In this section the Global Stability of the DFE is examined. Theorem 1. The Disease Free Equilibrium is globally asymptotically stable when R 0 > 1. Proof. To proof the Global Stability of the DFE, we shall use the comparison theorem by Lakshmikanthan [9]. At the steady state S = I T A and using the comparison theorem we hae; I(t) T (t) = (F V ) F I(t) T (t) (14) A(t) A(t) Such that (I(t), T (t), A(t)) 0 as t hence; I(t) T (t) = (F V ) (15) A(t) where F V = µ λ D 0 (ν + θ + µ) λ D 0 θ (γ + µ) λ D δ 0 ν γ (d + µ + δ) λ D The eigenalues of (16) are µ D, θ + µ + ν)[r 0 1] D. To compute the remaining eigenalues, we sole the reduced matrix (γ + µ) λ D δ γ (d + µ + δ) λ D = 0 (17) (16)

6 608 K. F. Tireito, G. O. Lawi and C. O. Okaka For stability of (17), we must hae the trace (γ + 2µ + d + δ) D < 0 and determinant (γ + µ + D)(d + µ + δ + D) δγ > 0. Hence, Equation (1) will be stable wheneer R 0 < 1 and (γ+µ+d)(d+µ+δ+d) > 1. It follows that δγ the linearized differential equation is asymptotically stable wheneer R 0 < 1. Using Comparison theorem by Lakshmikanthan and substituting I = T = A = 0 into Equation (1), S(t) S 0 as t. Hence the DFE is globally asymptotically stable wheneer R 0 < 1. 6 Local Stability of the Endemic Equilibria To inestigate the stability at endemic equilibria E (S, I, T, A ) we use the concept of Turing stability. Linearizing the model at E (S, I, T, A ) gies; where A 2 = µ I I Y (t, x) = (D 2 + A 2 ) Y (t, x) (18) D 2 = D 0 0 D D δ 0 D S S (ν + θ + µ) 0 θ (γ + µ) δ 0 ν γ (d + µ + δ) For the linearized system, the corresponding characteristic polynomial can be expressed as follows A 2 D 2 I λi = 0 (19) Substituting the two matrices A 2 and D 2 into (19), we obtain; µ I D λ S I S (ν + θ + µ) D λ 0 θ (γ + µ) D λ δ = 0 0 ν γ (d + µ + δ) D λ (20) where S = I = ( ( ) ν + θ + µ Λ ) (ν + θ + µ) µ

7 HIV/AIDS treatment model 609 To sole Equation (20), we sole the following reduced matrices Q 2 = (γ + µ + D) λ δ γ (d + µ + δ) D λ = 0 (21) Q 1 = µ I D λ S (ν + θ + µ) D λ = 0 (22) I On soling Equation (21) we obtain S P (λ) = λ 2 + (d + 2µ + 2D + d + δ)λ + (γ + µ + D)(d + µ + δ + D) δγ = 0(23) On soling Equation (23) we hae; λ = (d + 2µ + 2D + d + δ) ± ((d + 2µ + 2D + d + δ)) 2 4[(γ + µ + D)(d + µ + δ + D) δγ] Equation (24) will always be negatie wheneer 2 ((d + 2µ + 2D + d + δ)) 2 4[(γ + µ + D)(d + µ + δ + D) δγ] < (d + 2µ + 2D + d + δ)(25) On replacing I and S in Equation (22) we obtain Q 1 = µr 0 D λ (ν + θ + µ) µ(r 0 1) D λ = 0 (26) Thus, the characteristic equation for Equation (26) is; P (λ) = λ 2 + (µr 0 + 2D)λ + D(µR 0 + D) + µ(r 0 1)(ν + θ + µ) = 0 (27) The roots of Equation (27) are λ = (µr 0 + 2D) ± (µr 0 + 2D) 2 4[D(µR 0 + D) + µ(r 0 1)(ν + θ + µ)] (28) 2 The roots of Equation (27) will always be negatie wheneer R 0 > 1 and (µr 0 + 2D) 2 4[D(µR 0 + D) + µ(r 0 1)(ν + θ + µ)] < (µr 0 + 2D) Thus the endemic equilibrium is locally asymptotically stable wheneer R 0 > 1. (24)

8 610 K. F. Tireito, G. O. Lawi and C. O. Okaka 7 Global Stability of the Endemic Equilibria Let u(x, t) = (S(x, t), I(x, t), T (x, t), A(x, t)) be any solution of model (1) and W = L(u(t, x))dx (29) be a Lyapuno functional for model (1). Calculating the time deriatie of W along the positie solution of model (1), we get; dw = L(u).(D u + f(u))dx dt = L(u).f(u)dx + L(u).D udx Hence dw dt L(u).f(u)dx + 4 D i i=1 L(u) u i u i dx (30) the second term of equation (31) is simplified using Green s formula and we thus obtain; L u i dx = L(u) ( ) u i L u i u i n dσ u i. dx (31) u i Since u n but = 0 on, then L u i dx = u i u i. ( L u i ( ) L u i. dx u i ) dx = A Lyapuno functional of model (1) at E is gien as L 2 dx (32) L L(u) = S S ln(s) + I I ln(i) + T T ln(t ) + A A ln(t ) ( ) ( βµci S L(u).f(u) = + µs 2 S ) ( Λ S S + γt 1 A T S A T ( + δa 1 T ) A + θi (1 T ) I T A T I Thus dw dt ) + νi (1 A A [ ( ) ( βµci S = + µs 2 S ) ( Λ S S + γt 1 A ) ) T + νi (1 A I S A T A I ( + δa 1 T ) A + θi (1 T ) I ] L 2 dx dx (34) T A T I L ) I I (33)

9 HIV/AIDS treatment model 611 The inequality dw = 0 holds iff (S, I, T, A) takes the equilibrium alues dt (S, I, T, A ). Since all solutions of Model (1) are positie, the largest inariant subset of the set { dw = 0}. Thus, the equilibrium point E, is globally dt attractie. 8 Existence of the traelling wae solution In this section, we focus on determining traeling wae fronts connecting the disease-free steady state and endemic steady state. To reduce the amount of parameters considered, we introduce the following nondimensional quantities. S = µ Λ S, Ī = µ Λ I, T = µ Λ T, Ā = µ Λ A t = µt, C = C Λ, γ = γ µ, d = d µ, δ = δ µ, ν = ν µ, θ = θ µ, = µ Λ, β = Λ µ β, x = µx(35) With the scalings defined in (35), Equation (1) becomes, on remoing bars for simplicity; S(x, t) I(x, t) T (x, t) A(x, t) = 1 S(t) I(t)S(t) (t) = I(t)S(t) (t) + D 2 S x 2, (ν + θ + 1)I(t) + D 2 I x 2, = δa(t) + θi(t) (γ + 1)T (t) + D 2 T x 2, = γt (t) + νi(t) (1 + d + δ)a(t) + D 2 A x 2 (36) We consider the existence of the traelling wae solutions of model (36) satisfying the following asymptotic boundary conditions. lim (S(s), I(s), T (s), A(s)) = (1, 0, 0, 0) t lim (S(s), I(s), T (s), A(s)) = t + (S, I, T, A ) (37) which accounts for transition from DFE E 0 to endemic state E. The steady states in dimensionalized model (36) are extinction states (0, 0, 0, 0), Disease free state (1, 0, 0, 0) and endemic state (S, I, T, A ) defined by; S = (ν + θ + µ) I = ( ) (θ + ν + µ) 1

10 612 K. F. Tireito, G. O. Lawi and C. O. Okaka A = T = ( ) ( ) γθ + ν(γ + 1) (γ + 1)(d + δ + 1) γδ (θ + ν + µ) 1 { δ ( γθ + ν(γ + 1) ) ( )} γ + 1 (γ + 1)(d + δ + 1) γδ (ν + θ + 1) 1 + θ ( ) γ + 1 ν + θ (38) A traelling wae solution of model (36) is a solution ( S, Ĩ, T, Ã) with the special form S(x, t) = S(x + t), Ĩ(x, t) = I(x + t), T (x, t) = T (x + t) and Ã(x, t) = A(x + t) where S, Ĩ, T, Ã C2 (R, R 3 ) and > 0 is a constant accounting for the wae speed. Substituting ( S, Ĩ, T, Ã) into Equation (36), we hae Ṡ = 1 { 1 µs(t) I(t)S(t) } + D 2 S, (t) x 2 I = 1 T = 1 A = 1 { } I(t)S(t) (ν + θ + 1)I(t) + D 2 I, (t) x { } 2 δa(t) + θi(t) (γ + 1)T (t) + D 2 T, x { 2 } γt (t) + νi(t) (1 + d + δ)a(t) + D 2 A x 2 (39) Setting D = 0 and linearizing model (39) at DFE equilibrium, we obtain; (ν + θ + 1) θ 0 (γ+1) δ 0 ν The characteristic equation of model (40) is (λ + 1 ) ( λ + (ν + θ + 1) ) { ( γ + 1 λ 2 + γ (d+δ+1) = 0 (40) + d + δ + 1 ) (γ + 1)(d + δ + 1) λ + γδ 2 2 The roots of equation (41) are λ = 1 and λ = (ν + θ + 1) and the other remaining roots are determined by the following equation λ 2 + ( γ d + δ + 1 (γ + 1)(d + δ + 1) )λ + γδ 2 = 0 2 } = 0 (41)

11 HIV/AIDS treatment model 613 which has the solution ( γ + 1 λ 1,2 = 2 If we let = ( (γ+1) + d + δ + 1 ) ± ((γ + 1) + + (d+δ+1) ) 2 4[ (γ+1)(d+δ+1) γδ ] then; 2 2 (d + δ + 1) ) 2 [ (γ + 1)(d + δ + 1) 4 γδ ] 2 2 (42) For 0 ; All the eigenalues of the Jacobian ealuated at E 0 are strictly real and hence a smooth traelling wae profile corresponding to Equation (1). For < 0 ; There exists a non-monotonic traelling wae solution connecting E 0 and E. Hence existence of traelling wae solution that is periodic and connects the disease free equilibrium and endemic equilibrium. 9 umerical Simulation of the Model Simulation of the density of infections in space and time, results showed that the traelling wae solutions conerges to a unique state as shown in Figure 1. Infecties Time t Distance x Figure 1: Infectie indiiduals in time and space. From Figure 1, the infectie indiiduals will diffuse to regions where there is no infection till an equilibrium is achieed. The numerical result shows that the moement of infecties makes the disease to preail in the population.

12 614 K. F. Tireito, G. O. Lawi and C. O. Okaka The asymptotic spreading of the infecties shows that there exists a connection between unstable steady state and stable steady state. This is depicted in Figure 2. From Figure 2, if few infecties are introduced into a completely 1.2 E 0 1 E * 0.8 (x,t) t Figure 2: Wae profile connecting DFE and EE. susceptible population, then there will a moing transition zone of the infected indiiduals. This shows that there exists a traelling wae profile connecting the disease free equilibrium and endemic equilibrium. 10 Conclusion The introduction of few infected cases at one end of equilibrium results in a wae of propagation of infected indiiduals. Therefore, a transition region from one equilibrium point to another is plausible. This shows that there exits a traeling wae profiles between the disease free equilibrium and the endemic equilibrium on introduction of infected indiiduals. References [1] A. Morozo, S. Ruan and B. Li, Patterns of patchy spread in multi-speaces reaction diffusion models, Ecological Complexity, 5 (2008), [2] J. Jiang, Dynamics of a reaction diffusion system of autocatalytic reaction, Discrete and Continuous Dynamical Systems, 21 (2008), no. 1,

13 HIV/AIDS treatment model 615 [3] J. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, USA, [4] J. Murray, Mathematical Biology, Springer Verlag, Berlin, [5] J. Van-den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math Biosci., 180 (2002), [6] S. Otto and T. Day, A Biologist s Guide to Mathematical Modeling in Ecology and Eolution, Princeton, J: Princeton Uniersity Press, [7] T. Hollingsworth, R. Anderson and C. Fraser, HIV-1 transmission by stage of infection, The Journal of Infectious Diseases, 198 (2008), [8] U. Dieckmann, J. Metz, M. Sabelis and K. Sigmund, Adaptie Dynamics of Infectious Diseases: In Pursuit of Virulence Management, ew York, Cambridge Uniersity Press, [9] V. Laksmikanthan and L. Xinzhi, On Asymptotic Stability for on- Autonomous Differential Systems, Academic Press, ew York, [10] W. Huang, M. Han and K. Liu, Dynamics of an SIS reaction-diffusion epidemic model for disease transmission, Mathematical Bioscience and Engineering, 7 (2010), no. 1, Receied: May 10, 2018; Published: May 31, 2018