Acta Biotheor DOI 10.1007/s10441-015-9263-y REGULAR ARTICLE
Dynamics of a Class of HIV Infection Models with Cure of Infected Cells in Eclipse Stage Mehdi Maziane1 • El Mehdi Lotfi1 • Khalid Hattaf1,2 • Noura Yousfi1
Received: 1 October 2014 / Accepted: 9 June 2015 Springer Science+Business Media Dordrecht 2015
Abstract In this paper, we propose two HIV infection models with specific nonlinear incidence rate by including a class of infected cells in the eclipse phase. The first model is described by ordinary differential equations (ODEs) and generalizes a set of previously existing models and their results. The second model extends our ODE model by taking into account the diffusion of virus. Furthermore, the global stability of both models is investigated by constructing suitable Lyapunov functionals. Finally, we check our theoretical results with numerical simulations. Keywords
HIV Nonlinear incidence rate Diffusion Global stability
1 Introduction The human immunodeficiency virus (HIV) is a lentivirus that causes acquired immunodeficiency syndrome (AIDS) by infecting CD4? T cells. Recently, many mathematical models have been presented and developed for to better understand the dynamics of HIV infection. Rong et al. (2007) extended the basic model presented in Nowak and Bangham (1996), Nowak and May (2000) by including a
& Khalid Hattaf [email protected] 1
Department of Mathematics and Computer Science, Faculty of Sciences Ben M’sik, Hassan II University, P.O. Box 7955, Sidi Othman, Casablanca, Morocco
2
Centre Re´gional des Me´tiers de l’Education et de la Formation (CRMEF), 20340 Derb Ghalef, Casablanca, Morocco
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class of infected cells that are not yet producing virus, i.e., cells in the eclipse phase. They proposed the following nonlinear differential equation model dT ¼ k lT TðtÞ bTðtÞVðtÞ þ qEðtÞ; dt dE ¼ bTðtÞVðtÞ ðlE þ q þ cÞEðtÞ; dt dI ¼ cEðtÞ lI IðtÞ; dt dV ¼ kIðtÞ lV VðtÞ; dt
ð1Þ
where T(t), E(t), I(t) and V(t) denote the densities of uninfected CD4? T cells, infected cells in the eclipse stage (unproductive cells), productive infected cells and free virus particles at time t, respectively. The constant k is the recruitment rate of the uninfected cells. The constants lT , lE , lI and lV represent the death rates of uninfected cells, unproductive cells, productive cells and virus, respectively. The constant q is the rate at which the unproductive infected cells may revert to the uninfected cells. The term bTðtÞVðtÞ describes the incidence of HIV infection of health CD4? T cells, where b is the infection rate. The constant c is the rate at which infected cells in the eclipse stage become productive infected cells and the constant k is the rate of production of virions by infected cells. The local stability of system (1) has been analyzed in Rong et al. (2007) and its global stability has been established by Buonomo and Vargas-De-Leo´n (2012). Hu et al. (2014) replaced the bilinear incidence rate bTðtÞVðtÞ by a saturated infection rate and they obtained the following model dT bTðtÞVðtÞ ¼ k lT TðtÞ þ qEðtÞ; dt 1 þ aVðtÞ dE bTðtÞVðtÞ ¼ ðlE þ q þ cÞEðtÞ; dt 1 þ aVðtÞ dI ¼ cEðtÞ lI IðtÞ; dt dV ¼ kIðtÞ lV VðtÞ: dt
ð2Þ
The global stability of the disease-free equilibrium and the infected equilibrium of system (2) has been studied in Hu et al. (2014) by using the Lyapunov function and LaSalle’s invariance principle, respectively. The first aim of this paper is to generalize the models cited above by proposing the following model
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dT ¼ k lT TðtÞ f ðTðtÞ; VðtÞÞVðtÞ þ qEðtÞ; dt dE ¼ f ðTðtÞ; VðtÞÞVðtÞ ðlE þ q þ cÞEðtÞ; dt dI ¼ cEðtÞ lI IðtÞ; dt dV ¼ kIðtÞ lV VðtÞ; dt
ð3Þ
where f ðT; VÞ ¼ 1þa1 TþabT2 Vþa3 TV is the incidence function proposed by Hattaf et al. [see Section 5, in Hattaf et al. (2013)] and a1 ; a2 ; a3 0 are constants. It is very important to note if a1 ¼ a2 ¼ a3 ¼ 0, model (3) is reduced to the model (1) with bilinear incidence rate which was introduced by Rong et al. (2007). If a1 ¼ a3 ¼ 0, model (3) is reduced to the model (2) with saturation response which was discussed by Hu et al. (2014). Furthermore, when a3 ¼ 0, the Hattaf’s response is simplified to Beddington–DeAnglis functional response which was introduced by Beddington (1975) and DeAngelis et al. (1975) and was used in Huang et al. (2009, 2011). Also, when a3 ¼ a1 a2 , the Hattaf’s response is simplified to the Crowley–Martin functional response which was introduced by Crowley and Martin (1989) and was used by Zhou and Cui (2011). Recently, the more generalized incidence function f(T, V) is used in Adnani et al. (2013), Lotfi et al. (2014), Hattaf and Yousfi (2014). On the other hand, the system (3) ignores the mobility of cells and viruses. For this reason, we take into account the spatial effect by proposing the following model oT ot oE ot oI ot oV ot
¼ k lT Tðx; tÞ f ðTðx; tÞ; Vðx; tÞÞVðx; tÞ þ qEðx; tÞ; ¼ f ðTðx; tÞ; Vðx; tÞÞVðx; tÞ ðlE þ q þ cÞEðx; tÞ; ð4Þ ¼ cEðx; tÞ lI Iðx; tÞ; ¼ dV DVðx; tÞ þ kIðx; tÞ lV Vðx; tÞ;
where T(x, t), E(x, t), I(x, t) and V(x, t) represent the densities of uninfected CD4? T cells, unproductive cells, productive infected cells, and free virus particles at location x and time t, respectively. The positive constant dV is the diffusion coefficient of virus. Further, all the parameters used in system (3) and system (4) have the same biological meanings as in model (1). In this paper, we consider system (4) with homogeneous Neumann boundary conditions oV ¼ 0; om
on
ð5Þ
oX ð0; þ1Þ;
and initial conditions Tðx; 0Þ ¼ /1 ðxÞ 0; Iðx; 0Þ ¼ /3 ðxÞ 0;
Eðx; 0Þ ¼ /2 ðxÞ 0; Vðx; 0Þ ¼ /4 ðxÞ 0;
x 2 X;
ð6Þ
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where D ¼
Pn
o2 i¼1 ox2i
is the Laplacian operator, X is a bounded domain in IRn with
smooth boundary oX, and oV om denotes the outward normal derivative on oX. The rest of paper is organized as follows. The mathematical analysis of the ODE model (3) and the diffused model (4) is presented in Sects. 2 and 3, respectively. In Sect. 4, we present parameters estimation for both models associated with HIV infection. Numerical simulations of our theoretical results are given in Sect. 5. Finally, the conclusion of our paper is the Sect. 6.
2 Analysis of the Generalized ODE Model In this section, we analyse the ODE model (3). First, we determine the equilibria and the basic reproduction number of system (3). After, we discuss the global stability of these equilibria. 2.1 Equilibria and Basic Reproduction Number It is easy to see that system (3) has a disease-free equilibrium Qf
k lT
; 0; 0; 0 .
Therefore, the basic reproduction number of (3) is given by R0 ¼
kbkc : lI lV ðka1 þ lT Þðq þ lE þ cÞ
ð7Þ
Biologically, R0 denotes the average number of secondary infections produced by one productive infected cell during the period of infection when all cells are uninfected, and the disease-free equilibrium Qf represents the extinction of the free virus. Theorem 2.1 (i)
If R0 1,then the system (3) has a unique disease-free equilibrium of the k form Qf l ; 0; 0; 0 . T
(ii) If R0 [ 1, the disease-free equilibrium is still present and the system (3) has a unique chronic infection equilibrium of the form Q ðT1 ; E1 ; I1 ; V1 Þ with T1 2 0; lk , E1 [ 0, I1 [ 0 and V1 [ 0. T
Proof Obviously, Qf is the unique steady state of system (3) when R0 1. To find the other equilibria, we resolve the following system
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k lT TðtÞ f ðT; VÞVðtÞ þ qEðtÞ ¼ 0;
ð8Þ
f ðT; VÞVðtÞ ðlE þ q þ cÞEðtÞ ¼ 0;
ð9Þ
cEðtÞ lI IðtÞ ¼ 0;
ð10Þ
Dynamics of a Class of HIV Infection Models with Cure of…
kIðtÞ lV VðtÞ ¼ 0:
ð11Þ
cðklT TÞ kcðklT TÞ TT From (8)–(11), we get E ¼ kl lE þc , I ¼ lI ðlE þcÞ , V ¼ lI lV ðlE þcÞ and kcðk lT T1 Þ ðq þ lE þ cÞlV lI : f T; ¼ lI lV ðlE þ cÞ kc TT We have E ¼ kl l þc 0, which implies that T E
equilibrium point when T [
k lT .
Then, there is no positive
k lT .
h i Now we consider the following function defined on the interval 0; lk by T kcðk lT T1 Þ ðq þ lE þ cÞlV lI : gðTÞ ¼ f T; lI lV ðlE þ cÞ kc
þcÞlV lI þcÞlV lI Clearly, gð0Þ ¼ ðqþlE kc \0, gðlk Þ ¼ ðqþlE kc ðR0 1Þ and T
g0 ðTÞ ¼
of kclT of [ 0: oT lI lV ðlE þ cÞ oV
Hence, if R0 [ 1, the system admits a unique endemic equilibrium Q ðT1 ; E1 ; I1 ; V1 Þ, with T1 2 ð0; lk Þ, E1 [ 0, I1 [ 0 and V1 [ 0. This completes the T proof. h 2.2 Global Stability of Equilibria Now, we establish the global stability of our system (3). First we have Theorem 2.2 R0 1.
The disease-free equilibrium Qf is globally asymptotically stable if
Proof To study the global stability of Qf for (3), we propose the following Lyapunov functional Z T f ðT0 ; 0Þ dS W0 ðT; E; I; VÞ ¼ T T0 T0 f ðS; 0Þ q þ ðT T0 þ EÞ2 2ð1 þ a1 T0 ÞðlT þ lE þ cÞT0 q þ lE þ c l ðq þ lE þ cÞ IþEþ I V; þ c kc where T0 ¼ lk . It is not difficult to show that the functional W0 is non-negative. T The time derivative of W0 along the solution of system (3) is given by
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f ðT0 ; 0Þ _ q _ Tþ 1 ðT T0 þ EÞðT_ þ EÞ f ðT; 0Þ ð1 þ a1 T0 ÞðlT þ lE þ cÞT0 q þ lE þ c _ _ lI ðq þ lE þ cÞ _ IþEþ V; þ c kc
dW0 ¼ dt
Using k ¼ lT T0 , we get dW0 f ðT0 ; 0Þ f ðT0 ; 0Þf ðT; VÞ f ðT0 ; 0Þ V þq 1 ¼ 1 l ðT0 TÞ þ E f ðT; 0Þ T f ðT; 0Þ f ðT; 0Þ dt qlT ðT T0 Þ2 qðlE þ cÞE2 ð1 þ a1 T0 ÞðlT þ lE þ cÞT0 ð1 þ a1 T0 ÞðlT þ lE þ cÞT0 qE l l ðq þ lE þ cÞ V þ ðT0 TÞ I V ð1 þ a1 T0 ÞT0 kc 1 q lT ðT T0 Þ2 qðlE þ cÞE2 þ ¼ T ðlT þ lE þ cÞT0 1 þ a1 T0 ð1 þ a1 T0 ÞðlT þ lE þ cÞT0
qðT T0 Þ2 E l l ðq þ lE þ cÞ ðR0 1ÞV þ I V ð1 þ a1 T0 ÞTT0 kc ða2 þ a3 TÞV f ðT0 ; 0ÞV: 1 þ a1 T þ a2 V þ a3 TV
0 Therefore, dW dt 0 if R0 1. In addition, it is not hard to verify that the largest 0 compact invariant set in fðT; E; I; VÞj dW dt ¼ 0g is just the singleton Qf . From LaSalle (1976) invariance principle, we deduce that Qf is globally asymptotically stable. h
Finally, we establish the global stability of the chronic infection equilibrium Q when R0 [ 1. Theorem 2.3 The chronic infection equilibrium Q is globally asymptotically stable if R0 [ 1 and R0 1 þ
½lT lI lV ðlE þ cÞ þ a2 lT kkcðlE þ q þ cÞ þ qa3 kck2 : qlI lV ðlE þ q þ cÞðlT þ a1 kÞ
ð12Þ
Proof To study the global stability of Q for (3), we propose the following Lyapunov functional Z T f ðT1 ;V1 Þ dS W1 ðT;E;I;VÞ ¼T T1 T1 f ðS;V1 Þ qð1 þ a2 V1 Þ þ ðT T1 þ E E1 Þ2 2ð1 þ a1 T1 þ a2 V1 þ a3 T1 V1 ÞðlT þ lE þ cÞT1 q þ lE þ c I E l ðq þ lE þ cÞ V I1 U V1 U þ þ E1 U þ I ; c I1 E1 kc V1
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where UðxÞ ¼ x 1 lnðxÞ. Calculating the time derivative of W1 along the positive solutions of the system (3), we get dW1 f ðT1 ; V1 Þ _ qð1 þ a2 V1 ÞðT T1 þ E E1 Þ _ Tþ ¼ 1 ðT_ þ EÞ f ðT; V1 Þ ð1 þ a1 T1 þ a2 V1 þ a3 T1 V1 ÞðlT þ lE þ cÞT1 dt q þ lE þ c I1 _ E1 _ lI ðq þ lE þ cÞ V1 _ 1 1 Iþ 1 Eþ V: þ c kc I E V By applying k ¼ lT1 þ f ðT1 ; V1 ÞV1 qE1 , we obtain dW1 f ðT1 ;V1 Þ f ðT1 ;V1 Þf ðT;VÞ f ðT1 ; V1 Þ ¼ 1 V þq 1 lT ðT1 TÞ þ E f ðT;V1 Þ f ðT;V1 Þ f ðT; V1 Þ dt lT qð1 þ a2 V1 Þ ðT T1 Þ2 ð1 þ a1 T1 þ a2 V1 þ a3 T1 V1 ÞðlT þ lE þ cÞT1 qð1 þ a2 V1 ÞðlE þ cÞ ðE E1 Þ2 ð1 þ a1 T1 þ a2 V1 þ a3 T1 V1 ÞðlT þ lE þ cÞT1 qð1 þ a2 V1 ÞðE E1 ÞðT T1 Þ f ðT1 ;V1 Þ þ 1 ððlE þ cÞE1 þ qEÞ ð1 þ a1 T1 þ a2 V1 þ a3 T1 V1 ÞT1 f ðT;V1 Þ I1 E E1 V1 I f ðT; VÞV f ðT1 ;V1 ÞV1 f ðT1 ;V1ÞV f ðT1 ;V1 ÞV1 IE1 I1 V E f ðT1 ; V1 Þ f ðT1 ;V1 Þf ðT; VÞ V þ 3f ðT1 ;V1 ÞV1 ¼ 1 l ðT1 TÞ þ f ðT; V1 Þ T f ðT; V1 Þ f ðT1 ;V1 Þ lT qð1 þ a2 V1 Þ þq 1 ðT T1 Þ2 E f ðT;V1 Þ ð1 þ a1 T1 þ a2 V1 þ a3 T1 V1 ÞðlT þ lE þ cÞT1 qð1 þ a2 V1 ÞðlE þ cÞ ðE E1 Þ2 ð1 þ a1 T1 þ a2 V1 þ a3 T1 V1 ÞðlT þ lE þ cÞT1 qð1 þ a2 V1 ÞðE E1 ÞðT T1 Þ2 ðf ðT1 ;V1 ÞÞ2 V1 þ 4f ðT1 ; V1 ÞV1 ð1 þ a1 T1 þ a2 V1 þ a3 T1 V1 ÞT1 T f ðT; V1 Þ I1 E E1 V1 I : f ðT1 ;V1ÞV f ðT1 ;V1 ÞV1 f ðT; VÞV f ðT1 ;V1 ÞV1 IE1 I1 V E
Hence,
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dW1 ð1 þ a2 V1 ÞðT T1 Þ2 qlT T ¼ ðlT T1 qE1 Þ þ þ qE lT þ lE þ c dt TT1 ð1 þ a1 T1 þ a2 V1 þ a3 T1 V1 Þ qðE E1 Þ2 ð1 þ a2 V1 ÞðlE þ cÞ T1 ð1 þ a1 T1 þ a2 V1 þ a3 T1 V1 ÞðlT þ lE þ cÞ f ðT1 ; V1 Þ I1 E f ðT; VÞ VE1 V1 I f ðT; V1 Þ þ f ðT1 ; V1 ÞV1 5 f ðT; V1 Þ E1 I f ðT1 ; V1 Þ V1 E VI1 f ðT; VÞ
f ðT1 ; V1 ÞV1 ð1 þ a1 TÞða2 þ a3 TÞðV V1 Þ2 : V1 ð1 þ a1 T þ a2 V1 þ a3 TV1 Þð1 þ a1 T þ a2 V þ a3 TVÞ
Since the arithmetic mean is greater than or equal to the geometric mean, it is clear that 5
f ðT1 ; V1 Þ I1 E f ðT; VÞ VE1 V1 I f ðT; V1 Þ 0; f ðT; V1 Þ E1 I f ðT1 ; V1 Þ V1 E VI1 f ðT; VÞ
and the equality holds only for T ¼ T1 , E ¼ E1 , I ¼ I1 and V ¼ V1 . 1 Clearly, dW dt 0 if R0 [ 1 and qE1 6 lT T1 . In addition, it is not hard to show that the condition qE1 6 lT T1 is equivalent to R0 1 þ
½lT lI lV ðlE þ cÞ þ a2 lT kkcðlE þ q þ cÞ þ qa3 kck2 ; qlI lV ðlE þ q þ cÞðlT þ a1 kÞ
1 and the largest compact invariant set in fðT; E; I; VÞj dW dt ¼ 0g is just the singleton Q . By LaSalle invariance principle LaSalle (1976), we conclude that Q is globally asymptotically stable. h
Remark 2.4
We have
½lT lI lV ðlE þ cÞ þ a2 lT kkcðlE þ q þ cÞ þ qa3 kck2 ¼ 1: q!0 qlI lV ðlE þ q þ cÞðlT þ a1 kÞ lim
½lT lI lV ðlE þ cÞ þ a2 lT kkcðlE þ q þ cÞ þ qa3 kck2 ¼ 1: c!1 qlI lV ðlE þ q þ cÞðlT þ a1 kÞ lim
From Theorem 2.3, we deduce that the chronic infection equilibrium Q is globally asymptotically stable if R0 [ 1 and q ¼ 0. (ii) From Theorem 2.3, we deduce that the chronic infection equilibrium Q is globally asymptotically stable if R0 [ 1 and c sufficiently large.
(i)
3 Analysis of the Diffused Model In this section, we analyse the diffused model (4). First, we establish the global existence, positivity and boundedness of solutions.
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Proposition 3.1 For any given initial data satisfying the condition (6), there exists a unique solution of problem (4)–(6) defined on ½0; þ1Þ and this solution remains non-negative and bounded for all t [ 0. Proof The system can be written abstractly in the Banach space X ¼ CðXÞ of the form CðXÞ U 0 ðtÞ ¼ AUðtÞ þ FðUðtÞÞ; Uð0Þ ¼ U0 2 X;
t[0
ð13Þ
where U ¼ colðT; E; I; VÞ, U0 ¼ colð/1 ; /2 ; /3 ; /4 Þ, AUðtÞ ¼ colð0; 0; 0; dV MVÞ and 1 0 bTðx;tÞVðx;tÞ k lT Tðx;tÞ þ qEðx;tÞ C B 1 þ a1 Tðx;tÞ þ a2 Vðx;tÞ þ a3 Tðx;tÞVðx;tÞ C B C B bTðx;tÞVðx;tÞ B ðq þ lE þ cÞEðx;tÞ C FðUðtÞÞ ¼ B C: C B 1 þ a1 Tðx;tÞ þ a2 Vðx;tÞ þ a3 Tðx;tÞVðx;tÞ C B A @ cEðx;tÞ lI Iðx;tÞ kIðx;tÞ lV Vðx;tÞ It is clear that F is locally Lipschitz in X. From Pazy (1983), we deduce that system (4) admits a unique local solution on ½0;Tmax Þ, where Tmax is the maximal existence time for solution of system (4). In addition, system (4) can be written of the form oT ¼ F1 ðT; E; I; VÞ ot oE ¼ F2 ðT; E; I; VÞ ot oI ¼ F3 ðT; E; I; VÞ ot oV dV DVðx; tÞ ¼ F4 ðT; E; I; VÞ: ot It easy to see that the functions Fi ðT; E; I; VÞ, 1 i 4, are continuously differenbTV tiable satisfying: F1 ð0; E; I; VÞ ¼ K þ qE 0, F2 ðT; 0; I; VÞ ¼ 1þa1 Tþa 2 Vþa3 TV 0; F3 ðT; E; 0; VÞ ¼ cE 0 and F4 ðT; E; I; 0Þ ¼ kI 0, for all T; E; I; V 0. Since initial data of system (4) are nonnegative and from Smoller (1983), we deduce the positivity of the local solution. Now, we show the boundedness of solution. Let Sðx; tÞ ¼ Tðx; tÞ þ Eðx; tÞ þ Iðx; tÞ: From system (4), we get oSðx; tÞ ¼ k lT T lE E lI I ot k mSðx; tÞ;
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where m ¼ minðlT ; lE ; lI Þ. Then, Sðx; tÞ Sðx; 0Þemt þ
k ð1 emt Þ: m
Therefore, T, E and I are bounded. It remains to prove the boundedness of V. From system (4), we obtain 8 oVðx; tÞ > > dV MV ¼ kI lV V; > > < ot oVðx; tÞ > ¼ 0; > > > : on Vðx; 0Þ ¼ /4 ðxÞ k/4 k1 ¼ maxx2X /4 ðxÞ;
ð14Þ
By the comparison principle Protter and Weinberger (1967), we have Vðx; tÞ VðtÞ, ¼ /4 ðxÞelV t þ k kIkð1 elV t Þ is the solution of the problem where VðtÞ lV 8 dV > > ¼ kkIk lV V < dt > > : Vð0Þ ¼ k/4 k1 : ½0; Tmax Þ, we have that V is maxf k kIk; k/4 k1 g, 8ðx; tÞ 2 X Since VðtÞ lV bounded. From the above, we have proved that T(x, t), E(x, t), I(x, t) and V(x, t) are ½0; Tmax Þ. Therefore, it follows from the standard theory for bounded on X semilinear parabolic systems Henry (1993) that Tmax ¼ þ1. This completes the proof of the proposition. h Now, we determine the global stability for reaction–diffusion equations (4)–(6) by constructing a suitable Lyapunov functionals. These Lyapunov functionals are obtained from those for ordinary differential equations (3) by applying the method of Hattaf and Yousfi presented in Hattaf and Yousfi (2013). Therefore, we obtain the following result. Theorem 3.2
For all diffusion coefficients, we have
(i)
If R0 6 1, the disease-free equilibrium Qf of problem (4)–(6) is globally asymptotically stable. (ii) If condition (12) holds, the chronic infection equilibrium Q of problem (4)– (6) is globally asymptotically stable.
Proof Let L0 and L1 be a Lyapunov functionals obtained, respectively, from Lyapunov functional W0 and W1 by applying the method of Hattaf and Yousfi (2013). It is easy to show that W0 and W1 verified the condition (15) given in Hattaf and Yousfi (2013). Hence, it follows from [Hattaf and Yousfi (2013), Proposition 2.1] that L0 and L1 are a Lyapunov functionals for the reaction-diffusion system (4)–
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(6) at Qf when R0 6 1 and at Q when R0 [ 1 and the condition (12) is satisfied. This completes the proof. h
4 Parameters Estimation From our theoretical results, both models (3) and (4) have a disease-free equilibrium of the form Qf
k lT
; 0; 0; 0 that is an uninfected state in which no infection and no
virus are present. Hence, the quantity lk represents the total number of healthy T CD4? T cells. Based on experimental data, Bourgeois et al. (2008) estimated the average half-life of naive CD4 cells to be 50 days, which means that lT ¼ ln502 ’ 0:0139 day1 . On the other hand, the total number of CD4? T cells in the blood of a healthy individuals without HIV infection ranges between approximately 430 and 1740 cells/ll (Uppal et al. 2003). Therefore, we obtain k between 5.9770 and 24.1860 cells ll1 day1 . For the rate of virion infection of CD4? T cells, b, it was estimated by Perelson et al. (1996) to be b ¼ 2:4 105 ll virion1 day1 . Also, Stafford et al. (2000) estimated the parameter b for 10 patients to be between 1:9 104 to 4:8 103 ll virion1 day1 . In general, the death rate of unproductive infected cells, lE , is larger than the death rate of uninfected cells, lT , because the cytotoxic T lymphocytes (CTL) cells kill infected cells before they become productive. The half-life of unproductive infected cells was estimated in Nowak et al. (1997) to be between 10 and 20 days, giving lE between 0.0347 and 0.0693 day1 . The productive infected cells are also short-lived, with a half-life 2:1 0:5 days in Ho et al. (1995), 2 days in Wei et al. (1995) and 1:55 0:57 days in Perelson et al. (1996). Hence, lI is between 0.2666 and 0.7073 day1 . Clearance of free virus, lV , is the more rapid process occurring on a time scale of hours. Values of lV are reported to range between 2.06 and 3:81 day1 in Perelson et al. (1996), which means that half-life of free virus is between 0.18 and 0.34 days. The cure rate of infected cells in the eclipse stage, q, is not known precisely. It has been estimated that only a small fraction of cells in the eclipse phase will revert to the uninfected state Essunger and Perelson (1994). For this reason, Rong et al. (2007) assumed that q ¼ 0:01 day1 . The rate to become productive infected cells, c, was to be 3 103 Perelson et al. (1993), 1:2 104 Perelson (1989) and 1:1 day1 in Rong et al. (2007). Finally, we estime rate of virion production, k, by assuming that each productive infected cell produces N virions during its lifetime l1 . Then k ¼ NlI . The mumber N I
is not known precisely and its estimate varies from 103 to 104 De Boer et al. (2010), from 102 to 103 Wang et al. (1999), Bajaria et al. (2002), from 200 to 1000 Chen and Cloyd (1999), Hill et al. (2014) and from 0:92 103 to 5:3 103 Tsai et al. (1996). Hence, we can assume that N is between 102 and 104 . Therefore, we estimate the parameter k to be between 27 and 7073 virion cell1 day1 .
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M. Maziane et al. Table 1 Parameter values for the models (3) and (4) with their sources Parameters
Meaning
Value
Source
k
Production rate of uninfected cells
5.9770–24.1860 cells ll1 day1
Calculated
lT
Death rate of uninfected cells
0.0139 day-1
Bourgeois et al. (2008)
b
Viral infectivity rate
2:4 105 – 4:8 103 ll virion1 day1
Perelson et al. (1996), Stafford et al. (2000)
lE
Death rate of unproductive infected cells
0.0347–0.0693 day-1
Nowak et al. (1997)
lI
Death rate of productive infected cells
0.2666–0.7073 day-1
Ho et al. (1995), Wei et al. (1995), Perelson et al. (1996)
lV
Clearance rate of virus
2.06–3.81 day-1
Perelson et al. (1996)
-4
-1
c
Rate to become productive cells
1.2 9 10 –1.1 day
Rong et al. (2007), Perelson et al. (1993), Perelson (1989)
k
Virion production rate
27–7073 virion cell-1 day-1
Calculated
q
Cure rate of infected cells in the eclipse stage
0.01 day-1
Rong et al. (2007)
dV
Diffusion coefficient of virus
Varied
–
A complete list of parameters and their estimated values for both models is given in Table 1.
5 Numerical Simulations In this section, we present some numerical simulations in order to illustrate our theoretical results. The parameter values for these simulations are selected from biological values given in Table 1. In addition, we consider system (4) with initial conditions Tðx; 0Þ ¼ 700 cells mm3 , Eðx; 0Þ ¼ 2 cells mm3 , Iðx; 0Þ ¼ 0 cells mm3 , Vðx; 0Þ ¼ 102 virions mm3 , and under Neumann boundary conditions oV om ¼ 0, t [ 0, x ¼ 0; 1. First, we choose the following parameter values of system (4): K ¼ 10, lT ¼ 0:0139, b ¼ 2:4 105 , a1 ¼ 0:1, a2 ¼ 0:01, a3 ¼ 0:00001, q ¼ 0:01 c ¼ 1:1, lI ¼ 0:27, lE ¼ 0:0347, k ¼ 600, lV ¼ 3 and dV ¼ 0:01. By calculation, we have R0 ¼ 0:1685. Hence, system (4) has a disease-free equilibrium Qf ¼ ð719:4245; 0; 0; 0Þ. By using Theorem 3.2 (i), we see that Qf is globally asymptotically stable and the solution of (4) converges to Qf (see Fig. 1). In this case, we see the number of CD4? T cells increases and converges to the value k lT ¼ 719:4245, while the numbers of other variables converge to the value 0, which can be explained by the elimination of infected cells and virus, means that the patient will be completely cured, the virus is cleared and the infection dies out.
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Dynamics of a Class of HIV Infection Models with Cure of…
Fig. 1 Demonstration of the stability of the disease-free equilibrium Qf
Now, we choose b ¼ 0:0012 and we keep the other parameter values. By calculation, we have R0 ¼ 8:4247 and ½lT lI lV ðlE þ cÞ þ a2 lT kkcðlE þ q þ cÞ þ qa3 kck2 ¼ 113:9642: qlI lV ðlE þ q þ cÞðlT þ a1 kÞ Hence, the condition (12) is satisfied. By applying Theorem 3.2 (ii), the unique chronic infection equilibrium Q ð92:0039; 7:6859; 31:3128; 6262:56Þ is globally asymptotically stable which means that the virus persists in the host and the infection becomes chronic (see Fig. 2). With a bigger rate of infection, we see that the numbers of unproductive cells, productive infected cells and virus increase, while the number of CD4? T cells decreases and converges to the value 92.0039 which is less than 200 per mm3 . In this case, the patient enters the phase AIDS and must follow a treatment. In Fig. 3, we see that the time for arriving at V1 with large diffusion coefficient is faster than that with small diffusion coefficient. This larger coefficient diffusion signifies the faster mobility of viruses in the blood. The same result was observed in Hattaf and Yousfi (2015) for HBV infection. In Fig. 4, we see that the dynamics of HIV infection converges to steady state Q for all initial conditions. However, the condition (12) is not satisfied with R0 ¼ 2
E þcÞþa2 lT kkcðlE þqþcÞþqa3 kck 5:3425 and 1 þ ½lT lI lV ðlql ¼ 4:7466: Therefore, the condiI lV ðlE þqþcÞðlT þa1 kÞ tion (12) is not necessary.
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Fig. 2 Demonstration of the stability of the chronic infection equilibrium Q
4
4
x 10
x 10
d =0.01
2
V
V
1.8
1.6
1.6
1.4
1.4
V(x,t)
V(x,t)
1.8
d =0.05
2
1.2 1
1.2 1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
100
100 80
80
10 60
Time, t
10 60
8 6
40
4
20 0
2 0
8 6
40
Time, t
Space, x
4
20 0
2 0
Space, x
4
x 10
d =0.5
2
V
1.8
16000 14000
1.4
V(x,t)
V(x,t)
dV=5
18000
1.6
12000
1.2
10000
1 0.8
8000
0.6
6000
0.4
4000
0.2
2000
100
100
80
10 60
8
Time, t
80
10 60
6
40
4
20 0
2 0
Space, x
Time, t
Fig. 3 Distribution of virus under different diffusion coefficients
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8 6
40
4
20 0
2 0
Space, x
1000
400
Unproductive cells
Uninfected CD4+ T cells
Dynamics of a Class of HIV Infection Models with Cure of…
800 600 400 200 0
0
50
100
150
200
250
300 200 100 0
300
0
50
100
150
200
250
300
250
300
Time (days) 4
4 x 10
70 60
3
50
Virus
Productive infected cells
Time (days)
40 30 20
2 1
10 0
0
50
100
150
200
250
Time (days)
300
0
0
50
100
150
200
Time (days)
Fig. 4 Dynamics of HIV infection with k ¼ 10, lT ¼ 0:0139, b ¼ 0:0012, a1 ¼ 0:1, a2 ¼ 0:01, a3 ¼ 0:00001, q ¼ 0:01, c ¼ 0:01, lI ¼ 0:27, lE ¼ 0:0347, lV ¼ 3 and k ¼ 2000
6 Discussion and Conclusion In this work, we first have proposed an ODE model describing the interaction between CD4? T cells and HIV virus and taking into account the cure of infected cells in the eclipse stage. Our main results obtained by studying this ODE model extend the corresponding results in papers in Rong et al. (2007), Buonomo and Vargas-De-Leo´n (2012), Hu et al. (2014) and those in Cuifang et al. (2014) when we ignored the effect of CTL immune response. Second, we have extended our ODE model to a system with partial differential equations in order to take into account the mobility of the virus. In reality, the infection transmission process is unknown in detail and the bilinear incidence rate is not reasonable to describe this process during the full course of infection because the total number of healthy CD4? T cells is limited in vivo and the HIV infection will approach saturation with more and more virus produced. For these reasons, the infection transmission process in both models is modeled by a specific nonlinear incidence rate that includes the traditional bilinear incidence rate, the saturated incidence rate, the Beddington–DeAngelis functional response and the Crowley–Martin functional response. This incidence rate can be used for other viral infections such as the hepatitis B virus (HBV), hepatitis C virus (HCV) and human T cell leukemia virus (HTLV); for epidemiological diseases such as tuberculosis (TB); for vector-borne diseases such as dengue fever and malaria, and for social epidemics such as alcoholism. Due to the high similarity between computer virus and biological virus (Chen et al. 2015), our incidence rate can be used to investigate the virus spreading on the Internet. In addition, our functional response can be adapted for fisheries applications to model the predation rate of class i on class 0 for
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a fish population submitted to fishing and is structured by age class (see Touzeau and Gouze´ 1998). The global stability of both models is investigated by constructing suitable Lyapunov functionals. More precisely, the global stability of the disease-free equilibrium Qf is characterized by R0 1. When R0 [ 1, the chronic infection equilibrium Q is globally asymptotically stable provided that condition (12) is satisfied. On the other hand, the basic reproduction number is independent of the diffusion coefficient, dV . Hence, we conclude that the diffusion of virus has no effect on the convergence of solution to problem (4)–(6), but it may affect the time for arriving at the infection steady state (see Fig. 3). Based on the above, we derive a practical result for clinics that consists to compute the basic reproduction number R0 for each patient. If this number is less than or equal to unity, then in the beginning of the infection, each productive infected cell produces on average less than or equal to one newly infected cell. Hence, the infection cannot spread, and the system returns to the uninfected state. In this case, the patient does not require treatment. When the basic reproduction number is greater than unity, then initially each productive infected cell produces on average more than one newly infected cell. In this case, the virus persists in the host and the patient must follow a treatment to improve their quality of life. From the Remark 2.4, we see that with a very small value of q that corresponds to small fraction of cells in the eclipse phase will revert to the uninfected state, or with a very large value of c that corresponds to quicker reverse transcription, the condition (12) is satisfied and the global stability of the chronic infection equilibrium Q is only characterized by R0 [ 1. For numerical simulations, we deduce that the condition (12) is not necessary. Therefore, we can conjecture that Q is globally asymptotically only if R0 [ 1. It will be very interesting to prove this conjecture in the future work. During HIV viral infection, the adaptive immune system has two main responses. The first is called cellular immune response mediated by CTL cells which are responsible to kill the infected cells. The second is called humoral immune response based on the antibodies which are proteins that are produced by B cells and are specifically programmed to neutralize the viruses. Thus, modeling the adaptive immune response in HIV infection will become also our future work. Acknowledgments We would like to thank the editor and three anonymous referees for their very helpful comments and suggestions that greatly improved the quality of this work.
References Adnani J, Hattaf K, Yousfi N (2013) Stability analysis of a stochastic SIR epidemic model with specific nonlinear incidence rate. Int J Stoch Anal Article ID 431257, 4 p Bajaria SH, Webb G, Cloyd M, Kirschner D (2002) Dynamics of naive and memory CD4? T lymphocytes in HIV-1 disease progression. J Acquir Immune Defic Syndr 30:41–58 Beddington JR (1975) Mutual interference between parasites or predators and its effect on searching efficiency. J Anim Ecol 44:331–340 Bourgeois C, Hao Z, Rajewsky K, Potocnik AJ, Stockinger B (2008) Ablation of thymic export causes accelerated decay of naive CD4 T cells in the periphery because of activation by environmental antigen. Proc Natl Acad Sci 105(25):8691–8696
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Dynamics of a Class of HIV Infection Models with Cure of… Buonomo B, Vargas-De-Leo´n C (2012) Global stability for an HIV-1 infection model including an eclipse stage of infected cells. J Math Anal Appl 385:709–720 Chen L, Hattaf K, Sun J (2015) Optimal control of a delayed SLBS computer virus model. Phys A Stat Mech Appl 427:244–250 Chen JJY, Cloyd MW (1999) The potential importance of HIV-induction of lymphocyte homing to lymph nodes. Int Immunol 11(10):1591–1594 Crowley PH, Martin EK (1989) Functional responses and interference within and between year classes of a dragonfly population. J North Am Benthol Soc 8:211–221 Cuifang Lv, Huang L, Yuan Y (2014) Global stability for an HIV-1 infection model with Beddington– DeAngelis incidence rate and CTL immune response. Commun Nonlinear Sci Numer Simul 19:121–127 De Boer RJ, Ribeiro RM, Perelson AS (2010) Current estimates for HIV-1 production imply rapid viral clearance in lymphoid tissues. Plos Comput Biol 6(9):e1000906 DeAngelis DL, Goldstein RA, O’Neill RV (1975) A model for trophic interaction. Ecology 56:881–892 Essunger P, Perelson AS (1994) Modeling HIV infection of CD4? T-cell subpopulations. J Theor Biol 170(4):367–391 Hattaf K, Yousfi N, Tridane A (2013) Stability analysis of a virus dynamics model with general incidence rate and two delays. Appl Math Comput 221:514–521 Hattaf K, Yousfi N (2013) Global stability for reaction–diffusion equations in biology. Comput Math Appl 66:1488–1497 Hattaf K, Yousfi N (2014) Global dynamics of a delay reaction–diffusion model for viral infection with specific functional response. Comput Appl Math. doi:10.1007/s40314-014-0143-x Hattaf K, Yousfi N (2015) A generalized HBV model with diffusion and two delays. Comput Math Appl 69:31–40 Henry D (1993) Geometric theory of semilinear parabolic equations, vol 840 of Lecture notes in mathematics. Springer, Berlin Hill AL, Rosenbloom DIS, Fu F, Nowak MA, Siliciano RF (2014) Predicting the outcomes of treatment to eradicate the latent reservoir for HIV-1. Proc Natl Acad Sci USA 111:13475–13480 Ho DD, Neumann AU, Perelson AS, Chen W, Leonard JM, Markowitz M (1995) Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection. Nature 373:123–126 Hu Z, Pang W, Liao F, Ma W (2014) Analysis of a CD4? T cell viral infection model with a class of saturated infection rate. Discret Contin Dyn Syst Ser B 19(3):735–745 Huang G, Ma W, Takeuchi Y (2009) Global properties for virus dynamics model with Beddington– DeAngelis functional response. Appl Math Lett 22:1690–1693 Huang G, Ma W, Takeuchi Y (2011) Global analysis for delay virus dynamics model with Beddington– DeAngelis functional response. Appl Math Lett 24(7):1199–1203 LaSalle JP (1976) The stability of dynamical systems. In: Regional conference series in applied mathematics. SIAM, Philadelphia Lotfi EM, Maziane M, Hattaf K, Yousfi N (2014) Partial differential equations of an epidemic model with spatial diffusion. Int J Partial Differ Equ Article ID 186437, 6 p Nowak MA, Bonhoeffer S, Shaw GM, May R (1997) Anti-viral drug treatment: dynamics of resistance in free virus and infected cell populations. J Theor Biol 184:203–217 Nowak MA, Bangham CRM (1996) Population dynamics of immune responses to persistent viruses. Science 272:74–79 Nowak MA, May RM (2000) Virus dynamics: mathematical principles of immunology and virology. Oxford University Press, New York Pazy A (1983) Semigroups of linear operators and applications to partial differential equations, vol 44 of applied mathematical sciences. Springer, New York Perelson AS (1989) Modeling the interaction of the immune system with HIV. In: Castillo-Chavez C, (ed) Mathematical and statistical approaches to AIDS epidemiology (Lect. Notes Biomath., vol 831). Springer, New York, pp 350–370 Perelson AS, Kirschner DE, De Boer R (1993) Dynamics of HIV Infection of CD4? T cells. Math Biosci 114:81–125 Perelson AS, Neumann AU, Markowitz M, Leonard JM, Ho DD (1996) HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time. Science 271:1582–1586 Protter MH, Weinberger HF (1967) Maximum principles in differential equations. Prentice Hall, Englewood Cliffs
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M. Maziane et al. Rong L, Gilchrist MA, Feng Z, Perelson AS (2007) Modeling within-host HIV-1 dynamics and the evolution of drug resistance: trade-offs between viral enzyme function and drug susceptibility. J Theor Biol 247:804–818 Smoller J (1983) Shock waves and reaction–diffusion equations. Springer, Berlin Stafford MA, Corey L, Cao Y, Daar ES, Ho DD, Perelson AS (2000) Modeling plasma virus concentration during primary HIV infection. J Theor Biol 203(3):285–301 Touzeau S, Gouze´ J-L (1998) On the stock–recruitment relationships in fish population models. Environ Model Assess 3:87–93 Tsai WP, Conley SR, Kung HF, Garrity RR, Nara PL (1996) Preliminary in vitro growth cycle and transmission studies of HIV-1 in an autologous primary cell assay of bloodderived macrophages and peripheral blood mononuclear cells. Virology 226:205–216 Uppal SS, Verma Shashi, Dhot PS (2003) Normal values of CD4 and CD8 lymphocyte subsets in healthy Indian adults and the effects of sex, age, ethnicity, and smoking. Cytom Part B Clin Cytom 52(1):32–36 Wang L, Chen JJY, Gelman BB, Konig R, Cloyd MW (1999) A novel mechanism of CD4 lymphocyte depletion involves effects of HIV on resting lymphocytes: induction of lymph node homing and apoptosis upon secondary signaling through homing receptors. J Immunol 162:268–276 Wei X, Ghosh SK, Taylor ME et al (1995) Viral dynamics in human immunodeficiency virus type 1 infection. Nature 373:117–122 Zhou X, Cui J (2011) Global stability of the viral dynamics with Crowley–Martin functional response. Bull Korean Math Soc 48(3):555–574
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Dynamics of a Class of HIV Infection Models with Cure of Infected Cells in Eclipse Stage Mehdi Maziane1 • El Mehdi Lotfi1 • Khalid Hattaf1,2 • Noura Yousfi1
Received: 1 October 2014 / Accepted: 9 June 2015 Springer Science+Business Media Dordrecht 2015
Abstract In this paper, we propose two HIV infection models with specific nonlinear incidence rate by including a class of infected cells in the eclipse phase. The first model is described by ordinary differential equations (ODEs) and generalizes a set of previously existing models and their results. The second model extends our ODE model by taking into account the diffusion of virus. Furthermore, the global stability of both models is investigated by constructing suitable Lyapunov functionals. Finally, we check our theoretical results with numerical simulations. Keywords
HIV Nonlinear incidence rate Diffusion Global stability
1 Introduction The human immunodeficiency virus (HIV) is a lentivirus that causes acquired immunodeficiency syndrome (AIDS) by infecting CD4? T cells. Recently, many mathematical models have been presented and developed for to better understand the dynamics of HIV infection. Rong et al. (2007) extended the basic model presented in Nowak and Bangham (1996), Nowak and May (2000) by including a
& Khalid Hattaf [email protected] 1
Department of Mathematics and Computer Science, Faculty of Sciences Ben M’sik, Hassan II University, P.O. Box 7955, Sidi Othman, Casablanca, Morocco
2
Centre Re´gional des Me´tiers de l’Education et de la Formation (CRMEF), 20340 Derb Ghalef, Casablanca, Morocco
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class of infected cells that are not yet producing virus, i.e., cells in the eclipse phase. They proposed the following nonlinear differential equation model dT ¼ k lT TðtÞ bTðtÞVðtÞ þ qEðtÞ; dt dE ¼ bTðtÞVðtÞ ðlE þ q þ cÞEðtÞ; dt dI ¼ cEðtÞ lI IðtÞ; dt dV ¼ kIðtÞ lV VðtÞ; dt
ð1Þ
where T(t), E(t), I(t) and V(t) denote the densities of uninfected CD4? T cells, infected cells in the eclipse stage (unproductive cells), productive infected cells and free virus particles at time t, respectively. The constant k is the recruitment rate of the uninfected cells. The constants lT , lE , lI and lV represent the death rates of uninfected cells, unproductive cells, productive cells and virus, respectively. The constant q is the rate at which the unproductive infected cells may revert to the uninfected cells. The term bTðtÞVðtÞ describes the incidence of HIV infection of health CD4? T cells, where b is the infection rate. The constant c is the rate at which infected cells in the eclipse stage become productive infected cells and the constant k is the rate of production of virions by infected cells. The local stability of system (1) has been analyzed in Rong et al. (2007) and its global stability has been established by Buonomo and Vargas-De-Leo´n (2012). Hu et al. (2014) replaced the bilinear incidence rate bTðtÞVðtÞ by a saturated infection rate and they obtained the following model dT bTðtÞVðtÞ ¼ k lT TðtÞ þ qEðtÞ; dt 1 þ aVðtÞ dE bTðtÞVðtÞ ¼ ðlE þ q þ cÞEðtÞ; dt 1 þ aVðtÞ dI ¼ cEðtÞ lI IðtÞ; dt dV ¼ kIðtÞ lV VðtÞ: dt
ð2Þ
The global stability of the disease-free equilibrium and the infected equilibrium of system (2) has been studied in Hu et al. (2014) by using the Lyapunov function and LaSalle’s invariance principle, respectively. The first aim of this paper is to generalize the models cited above by proposing the following model
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dT ¼ k lT TðtÞ f ðTðtÞ; VðtÞÞVðtÞ þ qEðtÞ; dt dE ¼ f ðTðtÞ; VðtÞÞVðtÞ ðlE þ q þ cÞEðtÞ; dt dI ¼ cEðtÞ lI IðtÞ; dt dV ¼ kIðtÞ lV VðtÞ; dt
ð3Þ
where f ðT; VÞ ¼ 1þa1 TþabT2 Vþa3 TV is the incidence function proposed by Hattaf et al. [see Section 5, in Hattaf et al. (2013)] and a1 ; a2 ; a3 0 are constants. It is very important to note if a1 ¼ a2 ¼ a3 ¼ 0, model (3) is reduced to the model (1) with bilinear incidence rate which was introduced by Rong et al. (2007). If a1 ¼ a3 ¼ 0, model (3) is reduced to the model (2) with saturation response which was discussed by Hu et al. (2014). Furthermore, when a3 ¼ 0, the Hattaf’s response is simplified to Beddington–DeAnglis functional response which was introduced by Beddington (1975) and DeAngelis et al. (1975) and was used in Huang et al. (2009, 2011). Also, when a3 ¼ a1 a2 , the Hattaf’s response is simplified to the Crowley–Martin functional response which was introduced by Crowley and Martin (1989) and was used by Zhou and Cui (2011). Recently, the more generalized incidence function f(T, V) is used in Adnani et al. (2013), Lotfi et al. (2014), Hattaf and Yousfi (2014). On the other hand, the system (3) ignores the mobility of cells and viruses. For this reason, we take into account the spatial effect by proposing the following model oT ot oE ot oI ot oV ot
¼ k lT Tðx; tÞ f ðTðx; tÞ; Vðx; tÞÞVðx; tÞ þ qEðx; tÞ; ¼ f ðTðx; tÞ; Vðx; tÞÞVðx; tÞ ðlE þ q þ cÞEðx; tÞ; ð4Þ ¼ cEðx; tÞ lI Iðx; tÞ; ¼ dV DVðx; tÞ þ kIðx; tÞ lV Vðx; tÞ;
where T(x, t), E(x, t), I(x, t) and V(x, t) represent the densities of uninfected CD4? T cells, unproductive cells, productive infected cells, and free virus particles at location x and time t, respectively. The positive constant dV is the diffusion coefficient of virus. Further, all the parameters used in system (3) and system (4) have the same biological meanings as in model (1). In this paper, we consider system (4) with homogeneous Neumann boundary conditions oV ¼ 0; om
on
ð5Þ
oX ð0; þ1Þ;
and initial conditions Tðx; 0Þ ¼ /1 ðxÞ 0; Iðx; 0Þ ¼ /3 ðxÞ 0;
Eðx; 0Þ ¼ /2 ðxÞ 0; Vðx; 0Þ ¼ /4 ðxÞ 0;
x 2 X;
ð6Þ
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where D ¼
Pn
o2 i¼1 ox2i
is the Laplacian operator, X is a bounded domain in IRn with
smooth boundary oX, and oV om denotes the outward normal derivative on oX. The rest of paper is organized as follows. The mathematical analysis of the ODE model (3) and the diffused model (4) is presented in Sects. 2 and 3, respectively. In Sect. 4, we present parameters estimation for both models associated with HIV infection. Numerical simulations of our theoretical results are given in Sect. 5. Finally, the conclusion of our paper is the Sect. 6.
2 Analysis of the Generalized ODE Model In this section, we analyse the ODE model (3). First, we determine the equilibria and the basic reproduction number of system (3). After, we discuss the global stability of these equilibria. 2.1 Equilibria and Basic Reproduction Number It is easy to see that system (3) has a disease-free equilibrium Qf
k lT
; 0; 0; 0 .
Therefore, the basic reproduction number of (3) is given by R0 ¼
kbkc : lI lV ðka1 þ lT Þðq þ lE þ cÞ
ð7Þ
Biologically, R0 denotes the average number of secondary infections produced by one productive infected cell during the period of infection when all cells are uninfected, and the disease-free equilibrium Qf represents the extinction of the free virus. Theorem 2.1 (i)
If R0 1,then the system (3) has a unique disease-free equilibrium of the k form Qf l ; 0; 0; 0 . T
(ii) If R0 [ 1, the disease-free equilibrium is still present and the system (3) has a unique chronic infection equilibrium of the form Q ðT1 ; E1 ; I1 ; V1 Þ with T1 2 0; lk , E1 [ 0, I1 [ 0 and V1 [ 0. T
Proof Obviously, Qf is the unique steady state of system (3) when R0 1. To find the other equilibria, we resolve the following system
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k lT TðtÞ f ðT; VÞVðtÞ þ qEðtÞ ¼ 0;
ð8Þ
f ðT; VÞVðtÞ ðlE þ q þ cÞEðtÞ ¼ 0;
ð9Þ
cEðtÞ lI IðtÞ ¼ 0;
ð10Þ
Dynamics of a Class of HIV Infection Models with Cure of…
kIðtÞ lV VðtÞ ¼ 0:
ð11Þ
cðklT TÞ kcðklT TÞ TT From (8)–(11), we get E ¼ kl lE þc , I ¼ lI ðlE þcÞ , V ¼ lI lV ðlE þcÞ and kcðk lT T1 Þ ðq þ lE þ cÞlV lI : f T; ¼ lI lV ðlE þ cÞ kc TT We have E ¼ kl l þc 0, which implies that T E
equilibrium point when T [
k lT .
Then, there is no positive
k lT .
h i Now we consider the following function defined on the interval 0; lk by T kcðk lT T1 Þ ðq þ lE þ cÞlV lI : gðTÞ ¼ f T; lI lV ðlE þ cÞ kc
þcÞlV lI þcÞlV lI Clearly, gð0Þ ¼ ðqþlE kc \0, gðlk Þ ¼ ðqþlE kc ðR0 1Þ and T
g0 ðTÞ ¼
of kclT of [ 0: oT lI lV ðlE þ cÞ oV
Hence, if R0 [ 1, the system admits a unique endemic equilibrium Q ðT1 ; E1 ; I1 ; V1 Þ, with T1 2 ð0; lk Þ, E1 [ 0, I1 [ 0 and V1 [ 0. This completes the T proof. h 2.2 Global Stability of Equilibria Now, we establish the global stability of our system (3). First we have Theorem 2.2 R0 1.
The disease-free equilibrium Qf is globally asymptotically stable if
Proof To study the global stability of Qf for (3), we propose the following Lyapunov functional Z T f ðT0 ; 0Þ dS W0 ðT; E; I; VÞ ¼ T T0 T0 f ðS; 0Þ q þ ðT T0 þ EÞ2 2ð1 þ a1 T0 ÞðlT þ lE þ cÞT0 q þ lE þ c l ðq þ lE þ cÞ IþEþ I V; þ c kc where T0 ¼ lk . It is not difficult to show that the functional W0 is non-negative. T The time derivative of W0 along the solution of system (3) is given by
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f ðT0 ; 0Þ _ q _ Tþ 1 ðT T0 þ EÞðT_ þ EÞ f ðT; 0Þ ð1 þ a1 T0 ÞðlT þ lE þ cÞT0 q þ lE þ c _ _ lI ðq þ lE þ cÞ _ IþEþ V; þ c kc
dW0 ¼ dt
Using k ¼ lT T0 , we get dW0 f ðT0 ; 0Þ f ðT0 ; 0Þf ðT; VÞ f ðT0 ; 0Þ V þq 1 ¼ 1 l ðT0 TÞ þ E f ðT; 0Þ T f ðT; 0Þ f ðT; 0Þ dt qlT ðT T0 Þ2 qðlE þ cÞE2 ð1 þ a1 T0 ÞðlT þ lE þ cÞT0 ð1 þ a1 T0 ÞðlT þ lE þ cÞT0 qE l l ðq þ lE þ cÞ V þ ðT0 TÞ I V ð1 þ a1 T0 ÞT0 kc 1 q lT ðT T0 Þ2 qðlE þ cÞE2 þ ¼ T ðlT þ lE þ cÞT0 1 þ a1 T0 ð1 þ a1 T0 ÞðlT þ lE þ cÞT0
qðT T0 Þ2 E l l ðq þ lE þ cÞ ðR0 1ÞV þ I V ð1 þ a1 T0 ÞTT0 kc ða2 þ a3 TÞV f ðT0 ; 0ÞV: 1 þ a1 T þ a2 V þ a3 TV
0 Therefore, dW dt 0 if R0 1. In addition, it is not hard to verify that the largest 0 compact invariant set in fðT; E; I; VÞj dW dt ¼ 0g is just the singleton Qf . From LaSalle (1976) invariance principle, we deduce that Qf is globally asymptotically stable. h
Finally, we establish the global stability of the chronic infection equilibrium Q when R0 [ 1. Theorem 2.3 The chronic infection equilibrium Q is globally asymptotically stable if R0 [ 1 and R0 1 þ
½lT lI lV ðlE þ cÞ þ a2 lT kkcðlE þ q þ cÞ þ qa3 kck2 : qlI lV ðlE þ q þ cÞðlT þ a1 kÞ
ð12Þ
Proof To study the global stability of Q for (3), we propose the following Lyapunov functional Z T f ðT1 ;V1 Þ dS W1 ðT;E;I;VÞ ¼T T1 T1 f ðS;V1 Þ qð1 þ a2 V1 Þ þ ðT T1 þ E E1 Þ2 2ð1 þ a1 T1 þ a2 V1 þ a3 T1 V1 ÞðlT þ lE þ cÞT1 q þ lE þ c I E l ðq þ lE þ cÞ V I1 U V1 U þ þ E1 U þ I ; c I1 E1 kc V1
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where UðxÞ ¼ x 1 lnðxÞ. Calculating the time derivative of W1 along the positive solutions of the system (3), we get dW1 f ðT1 ; V1 Þ _ qð1 þ a2 V1 ÞðT T1 þ E E1 Þ _ Tþ ¼ 1 ðT_ þ EÞ f ðT; V1 Þ ð1 þ a1 T1 þ a2 V1 þ a3 T1 V1 ÞðlT þ lE þ cÞT1 dt q þ lE þ c I1 _ E1 _ lI ðq þ lE þ cÞ V1 _ 1 1 Iþ 1 Eþ V: þ c kc I E V By applying k ¼ lT1 þ f ðT1 ; V1 ÞV1 qE1 , we obtain dW1 f ðT1 ;V1 Þ f ðT1 ;V1 Þf ðT;VÞ f ðT1 ; V1 Þ ¼ 1 V þq 1 lT ðT1 TÞ þ E f ðT;V1 Þ f ðT;V1 Þ f ðT; V1 Þ dt lT qð1 þ a2 V1 Þ ðT T1 Þ2 ð1 þ a1 T1 þ a2 V1 þ a3 T1 V1 ÞðlT þ lE þ cÞT1 qð1 þ a2 V1 ÞðlE þ cÞ ðE E1 Þ2 ð1 þ a1 T1 þ a2 V1 þ a3 T1 V1 ÞðlT þ lE þ cÞT1 qð1 þ a2 V1 ÞðE E1 ÞðT T1 Þ f ðT1 ;V1 Þ þ 1 ððlE þ cÞE1 þ qEÞ ð1 þ a1 T1 þ a2 V1 þ a3 T1 V1 ÞT1 f ðT;V1 Þ I1 E E1 V1 I f ðT; VÞV f ðT1 ;V1 ÞV1 f ðT1 ;V1ÞV f ðT1 ;V1 ÞV1 IE1 I1 V E f ðT1 ; V1 Þ f ðT1 ;V1 Þf ðT; VÞ V þ 3f ðT1 ;V1 ÞV1 ¼ 1 l ðT1 TÞ þ f ðT; V1 Þ T f ðT; V1 Þ f ðT1 ;V1 Þ lT qð1 þ a2 V1 Þ þq 1 ðT T1 Þ2 E f ðT;V1 Þ ð1 þ a1 T1 þ a2 V1 þ a3 T1 V1 ÞðlT þ lE þ cÞT1 qð1 þ a2 V1 ÞðlE þ cÞ ðE E1 Þ2 ð1 þ a1 T1 þ a2 V1 þ a3 T1 V1 ÞðlT þ lE þ cÞT1 qð1 þ a2 V1 ÞðE E1 ÞðT T1 Þ2 ðf ðT1 ;V1 ÞÞ2 V1 þ 4f ðT1 ; V1 ÞV1 ð1 þ a1 T1 þ a2 V1 þ a3 T1 V1 ÞT1 T f ðT; V1 Þ I1 E E1 V1 I : f ðT1 ;V1ÞV f ðT1 ;V1 ÞV1 f ðT; VÞV f ðT1 ;V1 ÞV1 IE1 I1 V E
Hence,
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dW1 ð1 þ a2 V1 ÞðT T1 Þ2 qlT T ¼ ðlT T1 qE1 Þ þ þ qE lT þ lE þ c dt TT1 ð1 þ a1 T1 þ a2 V1 þ a3 T1 V1 Þ qðE E1 Þ2 ð1 þ a2 V1 ÞðlE þ cÞ T1 ð1 þ a1 T1 þ a2 V1 þ a3 T1 V1 ÞðlT þ lE þ cÞ f ðT1 ; V1 Þ I1 E f ðT; VÞ VE1 V1 I f ðT; V1 Þ þ f ðT1 ; V1 ÞV1 5 f ðT; V1 Þ E1 I f ðT1 ; V1 Þ V1 E VI1 f ðT; VÞ
f ðT1 ; V1 ÞV1 ð1 þ a1 TÞða2 þ a3 TÞðV V1 Þ2 : V1 ð1 þ a1 T þ a2 V1 þ a3 TV1 Þð1 þ a1 T þ a2 V þ a3 TVÞ
Since the arithmetic mean is greater than or equal to the geometric mean, it is clear that 5
f ðT1 ; V1 Þ I1 E f ðT; VÞ VE1 V1 I f ðT; V1 Þ 0; f ðT; V1 Þ E1 I f ðT1 ; V1 Þ V1 E VI1 f ðT; VÞ
and the equality holds only for T ¼ T1 , E ¼ E1 , I ¼ I1 and V ¼ V1 . 1 Clearly, dW dt 0 if R0 [ 1 and qE1 6 lT T1 . In addition, it is not hard to show that the condition qE1 6 lT T1 is equivalent to R0 1 þ
½lT lI lV ðlE þ cÞ þ a2 lT kkcðlE þ q þ cÞ þ qa3 kck2 ; qlI lV ðlE þ q þ cÞðlT þ a1 kÞ
1 and the largest compact invariant set in fðT; E; I; VÞj dW dt ¼ 0g is just the singleton Q . By LaSalle invariance principle LaSalle (1976), we conclude that Q is globally asymptotically stable. h
Remark 2.4
We have
½lT lI lV ðlE þ cÞ þ a2 lT kkcðlE þ q þ cÞ þ qa3 kck2 ¼ 1: q!0 qlI lV ðlE þ q þ cÞðlT þ a1 kÞ lim
½lT lI lV ðlE þ cÞ þ a2 lT kkcðlE þ q þ cÞ þ qa3 kck2 ¼ 1: c!1 qlI lV ðlE þ q þ cÞðlT þ a1 kÞ lim
From Theorem 2.3, we deduce that the chronic infection equilibrium Q is globally asymptotically stable if R0 [ 1 and q ¼ 0. (ii) From Theorem 2.3, we deduce that the chronic infection equilibrium Q is globally asymptotically stable if R0 [ 1 and c sufficiently large.
(i)
3 Analysis of the Diffused Model In this section, we analyse the diffused model (4). First, we establish the global existence, positivity and boundedness of solutions.
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Proposition 3.1 For any given initial data satisfying the condition (6), there exists a unique solution of problem (4)–(6) defined on ½0; þ1Þ and this solution remains non-negative and bounded for all t [ 0. Proof The system can be written abstractly in the Banach space X ¼ CðXÞ of the form CðXÞ U 0 ðtÞ ¼ AUðtÞ þ FðUðtÞÞ; Uð0Þ ¼ U0 2 X;
t[0
ð13Þ
where U ¼ colðT; E; I; VÞ, U0 ¼ colð/1 ; /2 ; /3 ; /4 Þ, AUðtÞ ¼ colð0; 0; 0; dV MVÞ and 1 0 bTðx;tÞVðx;tÞ k lT Tðx;tÞ þ qEðx;tÞ C B 1 þ a1 Tðx;tÞ þ a2 Vðx;tÞ þ a3 Tðx;tÞVðx;tÞ C B C B bTðx;tÞVðx;tÞ B ðq þ lE þ cÞEðx;tÞ C FðUðtÞÞ ¼ B C: C B 1 þ a1 Tðx;tÞ þ a2 Vðx;tÞ þ a3 Tðx;tÞVðx;tÞ C B A @ cEðx;tÞ lI Iðx;tÞ kIðx;tÞ lV Vðx;tÞ It is clear that F is locally Lipschitz in X. From Pazy (1983), we deduce that system (4) admits a unique local solution on ½0;Tmax Þ, where Tmax is the maximal existence time for solution of system (4). In addition, system (4) can be written of the form oT ¼ F1 ðT; E; I; VÞ ot oE ¼ F2 ðT; E; I; VÞ ot oI ¼ F3 ðT; E; I; VÞ ot oV dV DVðx; tÞ ¼ F4 ðT; E; I; VÞ: ot It easy to see that the functions Fi ðT; E; I; VÞ, 1 i 4, are continuously differenbTV tiable satisfying: F1 ð0; E; I; VÞ ¼ K þ qE 0, F2 ðT; 0; I; VÞ ¼ 1þa1 Tþa 2 Vþa3 TV 0; F3 ðT; E; 0; VÞ ¼ cE 0 and F4 ðT; E; I; 0Þ ¼ kI 0, for all T; E; I; V 0. Since initial data of system (4) are nonnegative and from Smoller (1983), we deduce the positivity of the local solution. Now, we show the boundedness of solution. Let Sðx; tÞ ¼ Tðx; tÞ þ Eðx; tÞ þ Iðx; tÞ: From system (4), we get oSðx; tÞ ¼ k lT T lE E lI I ot k mSðx; tÞ;
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where m ¼ minðlT ; lE ; lI Þ. Then, Sðx; tÞ Sðx; 0Þemt þ
k ð1 emt Þ: m
Therefore, T, E and I are bounded. It remains to prove the boundedness of V. From system (4), we obtain 8 oVðx; tÞ > > dV MV ¼ kI lV V; > > < ot oVðx; tÞ > ¼ 0; > > > : on Vðx; 0Þ ¼ /4 ðxÞ k/4 k1 ¼ maxx2X /4 ðxÞ;
ð14Þ
By the comparison principle Protter and Weinberger (1967), we have Vðx; tÞ VðtÞ, ¼ /4 ðxÞelV t þ k kIkð1 elV t Þ is the solution of the problem where VðtÞ lV 8 dV > > ¼ kkIk lV V < dt > > : Vð0Þ ¼ k/4 k1 : ½0; Tmax Þ, we have that V is maxf k kIk; k/4 k1 g, 8ðx; tÞ 2 X Since VðtÞ lV bounded. From the above, we have proved that T(x, t), E(x, t), I(x, t) and V(x, t) are ½0; Tmax Þ. Therefore, it follows from the standard theory for bounded on X semilinear parabolic systems Henry (1993) that Tmax ¼ þ1. This completes the proof of the proposition. h Now, we determine the global stability for reaction–diffusion equations (4)–(6) by constructing a suitable Lyapunov functionals. These Lyapunov functionals are obtained from those for ordinary differential equations (3) by applying the method of Hattaf and Yousfi presented in Hattaf and Yousfi (2013). Therefore, we obtain the following result. Theorem 3.2
For all diffusion coefficients, we have
(i)
If R0 6 1, the disease-free equilibrium Qf of problem (4)–(6) is globally asymptotically stable. (ii) If condition (12) holds, the chronic infection equilibrium Q of problem (4)– (6) is globally asymptotically stable.
Proof Let L0 and L1 be a Lyapunov functionals obtained, respectively, from Lyapunov functional W0 and W1 by applying the method of Hattaf and Yousfi (2013). It is easy to show that W0 and W1 verified the condition (15) given in Hattaf and Yousfi (2013). Hence, it follows from [Hattaf and Yousfi (2013), Proposition 2.1] that L0 and L1 are a Lyapunov functionals for the reaction-diffusion system (4)–
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(6) at Qf when R0 6 1 and at Q when R0 [ 1 and the condition (12) is satisfied. This completes the proof. h
4 Parameters Estimation From our theoretical results, both models (3) and (4) have a disease-free equilibrium of the form Qf
k lT
; 0; 0; 0 that is an uninfected state in which no infection and no
virus are present. Hence, the quantity lk represents the total number of healthy T CD4? T cells. Based on experimental data, Bourgeois et al. (2008) estimated the average half-life of naive CD4 cells to be 50 days, which means that lT ¼ ln502 ’ 0:0139 day1 . On the other hand, the total number of CD4? T cells in the blood of a healthy individuals without HIV infection ranges between approximately 430 and 1740 cells/ll (Uppal et al. 2003). Therefore, we obtain k between 5.9770 and 24.1860 cells ll1 day1 . For the rate of virion infection of CD4? T cells, b, it was estimated by Perelson et al. (1996) to be b ¼ 2:4 105 ll virion1 day1 . Also, Stafford et al. (2000) estimated the parameter b for 10 patients to be between 1:9 104 to 4:8 103 ll virion1 day1 . In general, the death rate of unproductive infected cells, lE , is larger than the death rate of uninfected cells, lT , because the cytotoxic T lymphocytes (CTL) cells kill infected cells before they become productive. The half-life of unproductive infected cells was estimated in Nowak et al. (1997) to be between 10 and 20 days, giving lE between 0.0347 and 0.0693 day1 . The productive infected cells are also short-lived, with a half-life 2:1 0:5 days in Ho et al. (1995), 2 days in Wei et al. (1995) and 1:55 0:57 days in Perelson et al. (1996). Hence, lI is between 0.2666 and 0.7073 day1 . Clearance of free virus, lV , is the more rapid process occurring on a time scale of hours. Values of lV are reported to range between 2.06 and 3:81 day1 in Perelson et al. (1996), which means that half-life of free virus is between 0.18 and 0.34 days. The cure rate of infected cells in the eclipse stage, q, is not known precisely. It has been estimated that only a small fraction of cells in the eclipse phase will revert to the uninfected state Essunger and Perelson (1994). For this reason, Rong et al. (2007) assumed that q ¼ 0:01 day1 . The rate to become productive infected cells, c, was to be 3 103 Perelson et al. (1993), 1:2 104 Perelson (1989) and 1:1 day1 in Rong et al. (2007). Finally, we estime rate of virion production, k, by assuming that each productive infected cell produces N virions during its lifetime l1 . Then k ¼ NlI . The mumber N I
is not known precisely and its estimate varies from 103 to 104 De Boer et al. (2010), from 102 to 103 Wang et al. (1999), Bajaria et al. (2002), from 200 to 1000 Chen and Cloyd (1999), Hill et al. (2014) and from 0:92 103 to 5:3 103 Tsai et al. (1996). Hence, we can assume that N is between 102 and 104 . Therefore, we estimate the parameter k to be between 27 and 7073 virion cell1 day1 .
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M. Maziane et al. Table 1 Parameter values for the models (3) and (4) with their sources Parameters
Meaning
Value
Source
k
Production rate of uninfected cells
5.9770–24.1860 cells ll1 day1
Calculated
lT
Death rate of uninfected cells
0.0139 day-1
Bourgeois et al. (2008)
b
Viral infectivity rate
2:4 105 – 4:8 103 ll virion1 day1
Perelson et al. (1996), Stafford et al. (2000)
lE
Death rate of unproductive infected cells
0.0347–0.0693 day-1
Nowak et al. (1997)
lI
Death rate of productive infected cells
0.2666–0.7073 day-1
Ho et al. (1995), Wei et al. (1995), Perelson et al. (1996)
lV
Clearance rate of virus
2.06–3.81 day-1
Perelson et al. (1996)
-4
-1
c
Rate to become productive cells
1.2 9 10 –1.1 day
Rong et al. (2007), Perelson et al. (1993), Perelson (1989)
k
Virion production rate
27–7073 virion cell-1 day-1
Calculated
q
Cure rate of infected cells in the eclipse stage
0.01 day-1
Rong et al. (2007)
dV
Diffusion coefficient of virus
Varied
–
A complete list of parameters and their estimated values for both models is given in Table 1.
5 Numerical Simulations In this section, we present some numerical simulations in order to illustrate our theoretical results. The parameter values for these simulations are selected from biological values given in Table 1. In addition, we consider system (4) with initial conditions Tðx; 0Þ ¼ 700 cells mm3 , Eðx; 0Þ ¼ 2 cells mm3 , Iðx; 0Þ ¼ 0 cells mm3 , Vðx; 0Þ ¼ 102 virions mm3 , and under Neumann boundary conditions oV om ¼ 0, t [ 0, x ¼ 0; 1. First, we choose the following parameter values of system (4): K ¼ 10, lT ¼ 0:0139, b ¼ 2:4 105 , a1 ¼ 0:1, a2 ¼ 0:01, a3 ¼ 0:00001, q ¼ 0:01 c ¼ 1:1, lI ¼ 0:27, lE ¼ 0:0347, k ¼ 600, lV ¼ 3 and dV ¼ 0:01. By calculation, we have R0 ¼ 0:1685. Hence, system (4) has a disease-free equilibrium Qf ¼ ð719:4245; 0; 0; 0Þ. By using Theorem 3.2 (i), we see that Qf is globally asymptotically stable and the solution of (4) converges to Qf (see Fig. 1). In this case, we see the number of CD4? T cells increases and converges to the value k lT ¼ 719:4245, while the numbers of other variables converge to the value 0, which can be explained by the elimination of infected cells and virus, means that the patient will be completely cured, the virus is cleared and the infection dies out.
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Fig. 1 Demonstration of the stability of the disease-free equilibrium Qf
Now, we choose b ¼ 0:0012 and we keep the other parameter values. By calculation, we have R0 ¼ 8:4247 and ½lT lI lV ðlE þ cÞ þ a2 lT kkcðlE þ q þ cÞ þ qa3 kck2 ¼ 113:9642: qlI lV ðlE þ q þ cÞðlT þ a1 kÞ Hence, the condition (12) is satisfied. By applying Theorem 3.2 (ii), the unique chronic infection equilibrium Q ð92:0039; 7:6859; 31:3128; 6262:56Þ is globally asymptotically stable which means that the virus persists in the host and the infection becomes chronic (see Fig. 2). With a bigger rate of infection, we see that the numbers of unproductive cells, productive infected cells and virus increase, while the number of CD4? T cells decreases and converges to the value 92.0039 which is less than 200 per mm3 . In this case, the patient enters the phase AIDS and must follow a treatment. In Fig. 3, we see that the time for arriving at V1 with large diffusion coefficient is faster than that with small diffusion coefficient. This larger coefficient diffusion signifies the faster mobility of viruses in the blood. The same result was observed in Hattaf and Yousfi (2015) for HBV infection. In Fig. 4, we see that the dynamics of HIV infection converges to steady state Q for all initial conditions. However, the condition (12) is not satisfied with R0 ¼ 2
E þcÞþa2 lT kkcðlE þqþcÞþqa3 kck 5:3425 and 1 þ ½lT lI lV ðlql ¼ 4:7466: Therefore, the condiI lV ðlE þqþcÞðlT þa1 kÞ tion (12) is not necessary.
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Fig. 2 Demonstration of the stability of the chronic infection equilibrium Q
4
4
x 10
x 10
d =0.01
2
V
V
1.8
1.6
1.6
1.4
1.4
V(x,t)
V(x,t)
1.8
d =0.05
2
1.2 1
1.2 1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
100
100 80
80
10 60
Time, t
10 60
8 6
40
4
20 0
2 0
8 6
40
Time, t
Space, x
4
20 0
2 0
Space, x
4
x 10
d =0.5
2
V
1.8
16000 14000
1.4
V(x,t)
V(x,t)
dV=5
18000
1.6
12000
1.2
10000
1 0.8
8000
0.6
6000
0.4
4000
0.2
2000
100
100
80
10 60
8
Time, t
80
10 60
6
40
4
20 0
2 0
Space, x
Time, t
Fig. 3 Distribution of virus under different diffusion coefficients
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8 6
40
4
20 0
2 0
Space, x
1000
400
Unproductive cells
Uninfected CD4+ T cells
Dynamics of a Class of HIV Infection Models with Cure of…
800 600 400 200 0
0
50
100
150
200
250
300 200 100 0
300
0
50
100
150
200
250
300
250
300
Time (days) 4
4 x 10
70 60
3
50
Virus
Productive infected cells
Time (days)
40 30 20
2 1
10 0
0
50
100
150
200
250
Time (days)
300
0
0
50
100
150
200
Time (days)
Fig. 4 Dynamics of HIV infection with k ¼ 10, lT ¼ 0:0139, b ¼ 0:0012, a1 ¼ 0:1, a2 ¼ 0:01, a3 ¼ 0:00001, q ¼ 0:01, c ¼ 0:01, lI ¼ 0:27, lE ¼ 0:0347, lV ¼ 3 and k ¼ 2000
6 Discussion and Conclusion In this work, we first have proposed an ODE model describing the interaction between CD4? T cells and HIV virus and taking into account the cure of infected cells in the eclipse stage. Our main results obtained by studying this ODE model extend the corresponding results in papers in Rong et al. (2007), Buonomo and Vargas-De-Leo´n (2012), Hu et al. (2014) and those in Cuifang et al. (2014) when we ignored the effect of CTL immune response. Second, we have extended our ODE model to a system with partial differential equations in order to take into account the mobility of the virus. In reality, the infection transmission process is unknown in detail and the bilinear incidence rate is not reasonable to describe this process during the full course of infection because the total number of healthy CD4? T cells is limited in vivo and the HIV infection will approach saturation with more and more virus produced. For these reasons, the infection transmission process in both models is modeled by a specific nonlinear incidence rate that includes the traditional bilinear incidence rate, the saturated incidence rate, the Beddington–DeAngelis functional response and the Crowley–Martin functional response. This incidence rate can be used for other viral infections such as the hepatitis B virus (HBV), hepatitis C virus (HCV) and human T cell leukemia virus (HTLV); for epidemiological diseases such as tuberculosis (TB); for vector-borne diseases such as dengue fever and malaria, and for social epidemics such as alcoholism. Due to the high similarity between computer virus and biological virus (Chen et al. 2015), our incidence rate can be used to investigate the virus spreading on the Internet. In addition, our functional response can be adapted for fisheries applications to model the predation rate of class i on class 0 for
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a fish population submitted to fishing and is structured by age class (see Touzeau and Gouze´ 1998). The global stability of both models is investigated by constructing suitable Lyapunov functionals. More precisely, the global stability of the disease-free equilibrium Qf is characterized by R0 1. When R0 [ 1, the chronic infection equilibrium Q is globally asymptotically stable provided that condition (12) is satisfied. On the other hand, the basic reproduction number is independent of the diffusion coefficient, dV . Hence, we conclude that the diffusion of virus has no effect on the convergence of solution to problem (4)–(6), but it may affect the time for arriving at the infection steady state (see Fig. 3). Based on the above, we derive a practical result for clinics that consists to compute the basic reproduction number R0 for each patient. If this number is less than or equal to unity, then in the beginning of the infection, each productive infected cell produces on average less than or equal to one newly infected cell. Hence, the infection cannot spread, and the system returns to the uninfected state. In this case, the patient does not require treatment. When the basic reproduction number is greater than unity, then initially each productive infected cell produces on average more than one newly infected cell. In this case, the virus persists in the host and the patient must follow a treatment to improve their quality of life. From the Remark 2.4, we see that with a very small value of q that corresponds to small fraction of cells in the eclipse phase will revert to the uninfected state, or with a very large value of c that corresponds to quicker reverse transcription, the condition (12) is satisfied and the global stability of the chronic infection equilibrium Q is only characterized by R0 [ 1. For numerical simulations, we deduce that the condition (12) is not necessary. Therefore, we can conjecture that Q is globally asymptotically only if R0 [ 1. It will be very interesting to prove this conjecture in the future work. During HIV viral infection, the adaptive immune system has two main responses. The first is called cellular immune response mediated by CTL cells which are responsible to kill the infected cells. The second is called humoral immune response based on the antibodies which are proteins that are produced by B cells and are specifically programmed to neutralize the viruses. Thus, modeling the adaptive immune response in HIV infection will become also our future work. Acknowledgments We would like to thank the editor and three anonymous referees for their very helpful comments and suggestions that greatly improved the quality of this work.
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