Mathematical Models for HIV infection in vivo - A Review

Transcription

1 ing of ODE DDE SDE s Mathematical s for HIV infection in vivo - A Department of Mathematics and Statistics Indian Institute of Technology, Kanpur Kanpur, , India peeyush@iitk.ac.in January 20, 2010 International Conference on Mathematical ing and Non-linear Equations, BNMIT, Bangalore

2 ing of ODE DDE SDE s,, Basic and Analysis, of Mathematical s, Ordinary Differential Equation models, Delay Differential Equation models, Stochastic differential Equation models, models, and s.

3 HIV/AIDS ing of ODE DDE SDE s Over the past two decades, hundreds of thousands of journal articles pertaining to HIV/AIDS have been published. Among these are hundreds of mathematical and statistical models of various aspects of the disease. Broadly speaking, we can classify the various mathematical models of HIV dynamics into one of two categories. Epidemiological models, which attempt to assess the spread of infection through a population using quantitative and theoretical means, and Immunological models, which examine the within-host cellular responses to an invader.

4 ing of ODE DDE SDE s The utility of mathematical models to the HIV-immune system interaction is twofold. (i) A good model can predict with a great deal of accuracy what will transpire in the very long term. (ii) To help treatment with drug therapy at different regimes.

5 Mathematical ing ing of ODE DDE SDE s

6 HIV in vivo ing of AIDS (Acquired Immune Deficiency Syndrome) ODE DDE SDE s

7 HIV in vivo ing of AIDS (Acquired Immune Deficiency Syndrome) HIV (Human Immunodeficiency Virus) ODE DDE SDE s

8 HIV in vivo ing of ODE DDE SDE s AIDS (Acquired Immune Deficiency Syndrome) HIV (Human Immunodeficiency Virus) occurs via transfer of body fluid

9 HIV in vivo ing of ODE DDE SDE s AIDS (Acquired Immune Deficiency Syndrome) HIV (Human Immunodeficiency Virus) occurs via transfer of body fluid Once inside body it attacks cells of immune system

10 Immune System ing of ODE DDE SDE s The immune system is a collection of cells (white blood cells) and organs that work together synergistically. T cells are subset of WBCs: CD 4 + T cell ( helper T cells, give signals when invaders enter.) CD 8 + T cell ( killer T cells, produce antibodies to kill invader.) HIV targets CD 4 + T cells Destruction and damage of CD 4 + T cells causes their depletion. Strong signal is not sent and hence CD 8 + T cell ( killer T cells ) remain unaware. This causes immunodeficiency (Stage of AIDS).

11 The process of infection ing of Fusion - Attachment of HIV with T cell and transfusion ODE DDE SDE s

12 The process of infection ing of Fusion - Attachment of HIV with T cell and transfusion Reverse Transcription - Conversion of RNA to DNA ODE DDE SDE s

13 The process of infection ing of ODE DDE SDE s Fusion - Attachment of HIV with T cell and transfusion Reverse Transcription - Conversion of RNA to DNA Integration - Integration of viral DNA to T cell DNA

14 The process of infection ing of ODE DDE SDE s Fusion - Attachment of HIV with T cell and transfusion Reverse Transcription - Conversion of RNA to DNA Integration - Integration of viral DNA to T cell DNA Cleaving - cleaving of different proteins

15 The process of infection ing of ODE DDE SDE s Fusion - Attachment of HIV with T cell and transfusion Reverse Transcription - Conversion of RNA to DNA Integration - Integration of viral DNA to T cell DNA Cleaving - cleaving of different proteins Budding - gathering near cell membrane and budding out

16 The virus ing of ODE DDE SDE s

17 Life cycle of virus in CD4+ T cells ing of ODE DDE SDE s

18 Schematic Diagram ing of ODE DDE SDE s Schematic Diagram for Virus

19 The Basic ing of Important Populations ODE DDE SDE s

20 The Basic ing of Important Populations Virus Population (V (t)), CD 4 + T cell Population, ODE DDE SDE s

21 The Basic ing of Important Populations Virus Population (V (t)), CD 4 + T cell Population, Uninfected CD 4 + T cell Population (T (t)), Infected CD 4 + T cell Population (T (t)), ODE DDE SDE s

22 The Basic ing of ODE DDE SDE s Important Populations Virus Population (V (t)), CD 4 + T cell Population, Uninfected CD 4 + T cell Population (T (t)), Infected CD 4 + T cell Population (T (t)), Assumptions

23 The Basic ing of ODE DDE SDE s Important Populations Virus Population (V (t)), CD 4 + T cell Population, Uninfected CD 4 + T cell Population (T (t)), Infected CD 4 + T cell Population (T (t)), Assumptions occurs through T cells only,

24 The Basic ing of ODE DDE SDE s Important Populations Virus Population (V (t)), CD 4 + T cell Population, Uninfected CD 4 + T cell Population (T (t)), Infected CD 4 + T cell Population (T (t)), Assumptions occurs through T cells only, Populations are homogeneously mixed,

25 The Basic ing of ODE DDE SDE s Important Populations Virus Population (V (t)), CD 4 + T cell Population, Uninfected CD 4 + T cell Population (T (t)), Infected CD 4 + T cell Population (T (t)), Assumptions occurs through T cells only, Populations are homogeneously mixed, Interaction is mass action type,

26 The Basic ing of ODE DDE SDE s Important Populations Virus Population (V (t)), CD 4 + T cell Population, Uninfected CD 4 + T cell Population (T (t)), Infected CD 4 + T cell Population (T (t)), Assumptions occurs through T cells only, Populations are homogeneously mixed, Interaction is mass action type, Cells once infected start producing virus - latency period ignored,

27 The Basic ing of ODE DDE SDE s Important Populations Virus Population (V (t)), CD 4 + T cell Population, Uninfected CD 4 + T cell Population (T (t)), Infected CD 4 + T cell Population (T (t)), Assumptions occurs through T cells only, Populations are homogeneously mixed, Interaction is mass action type, Cells once infected start producing virus - latency period ignored, Infected cells produce same number of virus particles.

28 ing of ODE DDE SDE s We generally seek steady-state (equilibrium) solutions. s contain at least a healthy steady state and an endemically infected steady state. The healthy steady state represents clearance of infection. The goal of treatment is to reduce infection levels OR at the very least to ensure that solutions remain close to the infected steady state. When this steady state becomes unstable, infection levels may increase unboundedly, and this represents progression to AIDS. For this reason stability analysis is performed.

29 The Basic ing of The models dealing with the primary infection of HIV has mainly considered the interaction of CD 4 + T cells and HIV. Most of the models are of the following form: ODE DDE SDE s dt dt dv = Λ kvt µt, = kvt δt, (1) = NδT cv. Nowak and Bangham (1996), Science.

30 The Basic ing of ODE DDE SDE s The models dealing with the primary infection of HIV has mainly considered the interaction of CD 4 + T cells and HIV. Most of the models are of the following form: dt dt dv = Λ kvt µt, = kvt δt, (1) = NδT cv. Λ: inflow rate of uninfected T cells, µ: natural death rate, k: interaction infection rate of T cells with virus, δ : death rate of infected cells which also includes the possibility of death by bursting of cells hence δ µ, c: clearance rate of virus and N: burst size. All parameters are non-negative. Nowak and Bangham (1996), Science.

31 Steady States ing of ODE DDE SDE s The model system (1), has following two steady states: ( ) 1 Λ the infection free steady state E 1 = µ, 0, 0 and 2 the infected ( steady state E 2 = Λ µr 0, Λ δ (1 1R0 ), NΛ c (1 1R0 )). Existence of E 2 is guaranteed whenever R 0 = NΛk µc > 1.

32 Steady States ing of ODE DDE SDE s The model system (1), has following two steady states: ( ) 1 Λ the infection free steady state E 1 = µ, 0, 0 and 2 the infected ( steady state E 2 = Λ µr 0, Λ δ (1 1R0 ), NΛ c (1 1R0 )). Existence of E 2 is guaranteed whenever R 0 = NΛk µc > 1. R 0 is the basic reproductive ratio, which is defined as the number of newly infected cells that arise from any one infected cell when almost all cells are uninfected.

33 Stability Analysis ing of ODE DDE SDE s For local stability analysis the system (1) is linearized around a steady state, and Jacobian matrix evaluated at steady state is obtained as follows: M 1 = µ kv ss 0 kt ss kv ss δ kt ss. 0 Nδ c Remark If all the eigen values of Jacobian matrix evaluated at a given steady state are with negative real part, then the steady state is said to be locally asymptotically stable.

34 Stability Results ing of All the eigen values of M 1 evaluated at E 1 are with negative real part when R 0 < 1. Hence the infection free steady state E 1 is locally asymptotically stable whenever R 0 < 1 and it becomes unstable when R 0 > 1. ODE DDE SDE s

35 Stability Results ing of ODE DDE SDE s All the eigen values of M 1 evaluated at E 1 are with negative real part when R 0 < 1. Hence the infection free steady state E 1 is locally asymptotically stable whenever R 0 < 1 and it becomes unstable when R 0 > 1. Further, using Routh-Hurwitz criterion we note that all the roots of characteristic equation of M 1 evaluated at E 2 have negative real parts whenever R 0 > 1. Hence E 2 is locally asymptotically stable whenever it exists.

36 Conclusion ing of Thus if R 0 > 1 the infection persists and if R 0 < 1 it is cleared. ODE DDE SDE s

37 Conclusion ing of ODE DDE SDE s Thus if R 0 > 1 the infection persists and if R 0 < 1 it is cleared. This model gives the basic spike like viral population graph during primary infection and then the viral population settles to a low state, which is in accordance with experimental observations.

38 Conclusion ing of ODE DDE SDE s Thus if R 0 > 1 the infection persists and if R 0 < 1 it is cleared. This model gives the basic spike like viral population graph during primary infection and then the viral population settles to a low state, which is in accordance with experimental observations. An important insight obtained from these models is that HIV does not remain latent during the asymptomatic phase of the infection, but it continuously replicates with high turn-over rate. This enables the virus to evolve at a fast rate.

39 Numerical Simulation ing of ODE DDE SDE s We used ODE45 in MATLAB R to solve the system, for following set of parameters: Λ = 10 mm 3 perday, k = mm 3 perday, c = 3.4 perday, δ = 0.16 perday, µ = 0.01 perday and N = 1000.

40 Numerical Simulation ing of ODE DDE SDE s We used ODE45 in MATLAB R to solve the system, for following set of parameters: Λ = 10 mm 3 perday, k = mm 3 perday, c = 3.4 perday, δ = 0.16 perday, µ = 0.01 perday and N = Here R 0 = NΛk µc = 7.06 Hence system stabilizes to the infected equilibrium E 2 = (291.6, 27.2, ).

41 Numerical results (for T) ing of ODE DDE SDE s

42 Numerical results (for T ) ing of ODE DDE SDE s

43 Numerical results (for V) ing of ODE DDE SDE s

44 Progression of HIV in a typical patient ing of ODE DDE SDE s

45 Further modification ing of ODE DDE SDE s Proliferation of CD 4 + T cells may be considered. Interaction may be taken different from mass action type (TV ). Another population of Latently infected cells may be considered. The natural immune response may be considered. Various delays may be considered. Stochasticity of parameters can be taken into account. Age structured s may be involved. models along with above techniques used.

46 Modification of Basic ing of ODE DDE SDE s Observation: It is observed that when infection with virus takes place inside T cell, the viral RNA may not be completely reverse transcribed. Since the un-integrated virus present in the cell is labile and degrades with time, a fraction of infected T cells revert back to uninfected class. Rong et al. (2007), Journal of Theoretical Biology.

47 Modification of Basic ing of ODE DDE SDE s Observation: It is observed that when infection with virus takes place inside T cell, the viral RNA may not be completely reverse transcribed. Since the un-integrated virus present in the cell is labile and degrades with time, a fraction of infected T cells revert back to uninfected class. Rong et al. (2007) modified the basic model by including a class of infected cells that are not yet producing virus. Rong et al. (2007), Journal of Theoretical Biology.

48 Rong et al. (2007) ing of Rong et al. [Rong et al.(2007)rong, Gilchrist, Feng, and Perelson] considered following ODE model: ODE DDE SDE s dt dt 1 dt dv = Λ kvt µt + bt 1, = kvt (b + φ + δ 1 )T 1, = φt 1 δt, = NδT cv, Cells in the eclipse phase revert to the uninfected T class at a constant rate b. In addition, they may alternatively progress to the productively infected class T at the rate φ, or die at the rate δ 1.

49 Srivastava & (2010) ing of Srivastava & [Srivastava and (2010)] first considered a simplified model similar to the basic model, to account for the observation that a fraction of infected cells revert back to uninfected class as the reverse transcription may not be completed in the eclipse phase for all infected cells. ODE DDE SDE s Srivastava and (2010), Nonlinear Analysis: RWA.

50 Srivastava & (2010) ing of ODE DDE SDE s Srivastava & [Srivastava and (2010)] first considered a simplified model similar to the basic model, to account for the observation that a fraction of infected cells revert back to uninfected class as the reverse transcription may not be completed in the eclipse phase for all infected cells. Hence following ODE model is proposed: dt dt dv = Λ kvt µt + bt, = kvt (b + δ)t, = NδT cv, with T (0) = T 0 > 0, T (0) = 0 and V (0) = V 0 > 0. Srivastava and (2010), Nonlinear Analysis: RWA.

51 ing of ODE DDE SDE s Srivastava & [Srivastava and (2010)] showed that 1 The infection free steady state is globally asymptotically stable if R 0 = NδkΛ cµ(b+δ) 1 and is unstable if R 0 > 1. 2 The infected steady state is locally stable, if R 0 > 1. 3 The infected steady state is globally stable, when R 0 > 1 and µ < δ. 4 They also studied effect of delay in activation of T cells and established stability of infected steady state.

52 Wang and Song (2007) ing of ODE DDE SDE s The interaction may be less than linear in V due to saturation at high virus concentration and may be greater than linear if infectious fraction was very small or if multiple exposures were necessary for infection. Hence Wang and Song [Wang and Song(2007)] took interaction term as kt q V. ( dt = Λ µt (t) + rt (t) 1 T (t) + T ) (t) kt q (t)v (t), T Max dt = kt q (t)v (t) δt (t) dv = NδT (t) cv (t).

53 Wang and Song (2007) ing of ODE DDE SDE s The interaction may be less than linear in V due to saturation at high virus concentration and may be greater than linear if infectious fraction was very small or if multiple exposures were necessary for infection. Hence Wang and Song [Wang and Song(2007)] took interaction term as kt q V. ( dt = Λ µt (t) + rt (t) 1 T (t) + T ) (t) kt q (t)v (t), T Max dt = kt q (t)v (t) δt (t) dv = NδT (t) cv (t). free steady state is globally asymptotically stable if it is the only non-negative steady of the system. Infected steady state is globally stable under certain restrictions on parameters (Li and Muldowney [Li and Muldowney(1996)] method of geometric approach to global stability).

54 The Delay Differential Equation s: ing of The fact that the biological systems do not react immediately implies that there is an inherent delay in the system. ODE DDE SDE s

55 The Delay Differential Equation s: ing of The fact that the biological systems do not react immediately implies that there is an inherent delay in the system. In particular for HIV infection, there are cascade of events which take place inside CD 4 + T cells during infection. ODE DDE SDE s

56 The Delay Differential Equation s: ing of ODE DDE SDE s The fact that the biological systems do not react immediately implies that there is an inherent delay in the system. In particular for HIV infection, there are cascade of events which take place inside CD 4 + T cells during infection. Uninfected Latently Infected Productively Infected Production of Virus

57 The Delay Differential Equation s: ing of ODE DDE SDE s The fact that the biological systems do not react immediately implies that there is an inherent delay in the system. In particular for HIV infection, there are cascade of events which take place inside CD 4 + T cells during infection. Uninfected Latently Infected Productively Infected Production of Virus Therefore, there is a time lag for infected cells to become actively infected and start producing virus.

58 The Delay Differential Equation s: ing of ODE DDE SDE s The fact that the biological systems do not react immediately implies that there is an inherent delay in the system. In particular for HIV infection, there are cascade of events which take place inside CD 4 + T cells during infection. Uninfected Latently Infected Productively Infected Production of Virus Therefore, there is a time lag for infected cells to become actively infected and start producing virus. This delay has been accounted in some models using Delay Differential Equations.

59 by Perelson et. al., MBSc, 1993 ing of ODE DDE SDE s Perelson et al. [Perelson et al.(1993)perelson, Kirschner, and Boer] considered the model with latently infected class. dt dt 1 dt dv = Λ + rt ( 1 T ) tot kvt µt, T max = kvt µ 1 T 1 αt 1, = αt 1 δt, where T tot = T + T 1 + T. = NδT cv kvt.

60 Culshaw and Ruan (2000) ing of ODE DDE SDE s Culshaw and Ruan [Culshaw and Ruan(2000)] to account for the latently infected cells by using discrete delay: dt dt dv ( = Λ + rt (t) 1 T (t) + T ) (t) µt (t) kv (t)t (t), T Max = k V (t τ)t (t τ) µ I T (t) = Nµ b T (t) kv (t)t (t) cv (t). loss of virus due to fusion during infection µ I, µ b : blanket and lytic death rate for infected T cells respectively, k : the rate at which infected cells become actively infected τ: discrete time delay.

61 ing of ODE DDE SDE s The system has following two equilibrium points: A. The infection free equilibrium E 1 = (T 0, 0, 0), where T 0 = r µ+[(r µ)2 +4rsTmax] 1 1/2 2rTmax 1 B. The infected equilibrium E 2 = (T, T, V ), where T = It is found that E 2 exists only if c Nk k, T = k T V δ, V = δ[(s+(r µ)t )Tmax rt 2 ] T [k rt +kδt max ] N > N crit = c + kt 0 k T 0 i.e. R 0 = Nk T 0 c + kt 0 > 1

62 Stability results for τ = 0 ing of Following results hold for τ = 0: ODE DDE SDE s If R 0 1, then the infection free equilibrium E 1 is locally asymptotically stable. If R 0 > 1, then the infection free equilibrium E 1 becomes unstable and the infected equilibrium E 2 is locally asymptotically stable. Stability of the model was studied and under certain conditions, for a critical value of τ model may show Hopf bifurcation and delay induced oscillations.

63 Jiang et al. (2008) ing of ODE DDE SDE s Jiang et al. [Jiang et al.(2008)jiang, Zhou, Shi, and Song] considered that a fraction ρt of infected cell population will revert to uninfected class due to therapy. They considered the following equivalent model: dt dt dv ( = Λ + rt (t) 1 T ) + µt (t) kv (t)t (t) + ρt (t), T Max = k V (t τ)t (t τ) µ I T (t) ρt (t) = Nµ b T (t) cv (t). where T + = T (t) + T (t).

64 Jiang et al. (2008) ing of ODE DDE SDE s Jiang et al. [Jiang et al.(2008)jiang, Zhou, Shi, and Song] considered that a fraction ρt of infected cell population will revert to uninfected class due to therapy. They considered the following equivalent model: dt dt dv ( = Λ + rt (t) 1 T ) + µt (t) kv (t)t (t) + ρt (t), T Max = k V (t τ)t (t τ) µ I T (t) ρt (t) = Nµ b T (t) cv (t). where T + = T (t) + T (t). The local stability was studied and it was observed that for a critical value of delay parameter τ the system will undergo Hopf bifurcation. They also found the direction and stability of bifurcating periodic orbits.

65 Li and Ma (2007) ing of Li and Ma [Li and Ma(2007)] considered Holling-Type II interaction in model (1) for the interaction between T cells and virus population and proposed following delay differential equation model: ODE DDE SDE s dt dt dv kv (t)t (t) = Λ µt (t) 1 + V (t), kv (t τ)t (t τ) = δt (t) 1 + V (t τ) = NδT (t) cv (t).

66 Li and Ma (2007) ing of ODE DDE SDE s Li and Ma [Li and Ma(2007)] considered Holling-Type II interaction in model (1) for the interaction between T cells and virus population and proposed following delay differential equation model: dt dt dv kv (t)t (t) = Λ µt (t) 1 + V (t), kv (t τ)t (t τ) = δt (t) 1 + V (t τ) = NδT (t) cv (t). They established that delay has no effect on local asymptotic stability of infected equilibrium, whenever it exists, also discussed global asymptotic stability for viral free equilibrium of the delay model.

67 Cai and Li (2009) ing of ODE DDE SDE s Cai and Li [Cai and Li(2009)] proposed a model considering the proliferation of infected T cells also, which is given as follows: dt dt dv ( = Λ + rt (t) 1 T (t) + T ) (t) µt (t) kv (t)t (t), T Max ( = kv (t τ)t (t τ) + rt (t) 1 T (t) + T ) (t) δt (t) T Max = NδT (t) cv (t).

68 Cai and Li (2009) ing of ODE DDE SDE s Cai and Li [Cai and Li(2009)] proposed a model considering the proliferation of infected T cells also, which is given as follows: dt dt dv ( = Λ + rt (t) 1 T (t) + T ) (t) µt (t) kv (t)t (t), T Max ( = kv (t τ)t (t τ) + rt (t) 1 T (t) + T ) (t) δt (t) T Max = NδT (t) cv (t). They established the sufficiency criteria for delay independent stability of infected equilibrium. Then obtained critical value of delay for which Hopf bifurcation may occur.

69 Stochastic Differential equation s: ing of ODE DDE SDE s Real life is more stochastic rather than deterministic, particularly when modeling biological phenomenon. Stochastic differential equation models have played a relevant role in many application areas including biology, epidemiology and population dynamics, mostly because they can provide an additional degree of realism to physical systems as compared to their deterministic counterparts. This has also been used for HIV dynamics.

70 Stochastic Differential equation s: ing of ODE DDE SDE s Real life is more stochastic rather than deterministic, particularly when modeling biological phenomenon. Stochastic differential equation models have played a relevant role in many application areas including biology, epidemiology and population dynamics, mostly because they can provide an additional degree of realism to physical systems as compared to their deterministic counterparts. This has also been used for HIV dynamics. (This is because the different cells and infective virus particles reacting in same environment can often give different results. These models produce more useful output than deterministic models as by running a stochastic model many times we can build a distribution of predicted outcomes, while in deterministic case we have only single predicted value.)

71 Tuckwell and Le Corfec (1998) ing of ODE DDE SDE s Tuckwell and Le Corfec [Tuckwell and Corfec(1998)] considered the stochastic extension of Phillips model by introducing additional noise term. Their model, which is a multi-dimensional diffusion process, includes activated uninfected CD4 + T cells (T), latently (L) and actively infected (I) CD4 + T cells and virus population (V). Stochastic effects are assumed to arise in the process of infection of CD4 + T cells and transitions may occur from uninfected to latently or actively infected cells by chance mechanisms.

72 ing of ODE DDE SDE s They considered following model dt = (Λ k 1 TV µt ) k 1 TV dw, dl = (k 1 ptv (µ + α)l) + k 1 ptv dw, di = (k 1 (1 p)tv + αl ai ) + k 1 (1 p)tv dw, dv = (ci γv k 2 TV ) k 2 TV dw, where W is standard Weiner process (i.e. mean 0 and variance t at time t).

73 ing of ODE DDE SDE s They considered following model dt = (Λ k 1 TV µt ) k 1 TV dw, dl = (k 1 ptv (µ + α)l) + k 1 ptv dw, di = (k 1 (1 p)tv + αl ai ) + k 1 (1 p)tv dw, dv = (ci γv k 2 TV ) k 2 TV dw, where W is standard Weiner process (i.e. mean 0 and variance t at time t). The results were drawn by solving the system numerically. They estimated for maximum viral load and its time of occurrence. In addition the effect of perturbation on some parameters of the model were also studied.

74 Dalal et. al., JMAA, 2008 ing of ODE DDE SDE s Recently Dalal et al. [Dalal et al.(2008)dalal, Greenhalgh, and Mao] proposed and analysed an ODE model with HAART. To study the effect of environmental stochasticity they introduced randomness into the model by considering stochastically perturbed death rates for populations. dt = (Λ µt (1 η 1 )ktv ) σ 1 TdB 1, dt = ((1 η 1 )ktv δt ) σ 1 T db 1, dv = ((1 η 2 )NδT (1 η 1 )ktv cv ) σ 2 VdB 2, where η 1 and η 2 are drug efficacy of RT & Protease Inhibitor respectively.

75 Dalal et. al., JMAA, 2008 ing of ODE DDE SDE s Recently Dalal et al. [Dalal et al.(2008)dalal, Greenhalgh, and Mao] proposed and analysed an ODE model with HAART. To study the effect of environmental stochasticity they introduced randomness into the model by considering stochastically perturbed death rates for populations. dt = (Λ µt (1 η 1 )ktv ) σ 1 TdB 1, dt = ((1 η 1 )ktv δt ) σ 1 T db 1, dv = ((1 η 2 )NδT (1 η 1 )ktv cv ) σ 2 VdB 2, where η 1 and η 2 are drug efficacy of RT & Protease Inhibitor respectively. For the above model they showed analytically that it possess non-negative solutions and studied the asymptotic behavior of the model. They proved that the number of infected cells and virus particles tended asymptotically to zero exponentially almost surely.

76 s: ing of ODE DDE SDE s In these models life-cycle of individual cells has been neglected. Since in HIV infection the age since infection of CD 4 + T cells is one important factor for the progression of infection, the attempts have been made to model the primary infection by means of age structured models. Structured models bridge the gap between the individual and the population level.

77 Nelson et. al. (2004) ing of Nelson et al. [Nelson et al.(2004)nelson, Gilchrist, Coombs, Hyman, and Perelson] proposed the model considering the dynamics of uninfected CD 4 + T cells T (t), infected CD 4 + T cells structured by age, a, of their infection, T (a, t), and virus V (t). The equations defining the model are as follows: ODE DDE SDE s T (t) t dt (t) + T (t) da a dv (t) = s dt (t) kv (t)t (t), = δ(a)t (a, t) = 0 p(a)t (a, t)da cv (t).

78 ing of ODE DDE SDE s The virion production rate p(a) and the death rate δ(a) of infected cell of age a T (a, t) are the function of age since infection a. c is clearance rate of virus. It is assumed that da = 1. Also the initial condition for the model are given as T (0, t) = kv (t)t (t) and the function p(a) is taken first delayed exponential and then a Hill type function. p(a) = { p (1 e θ(a a1) ), a a 1 ; 0, else, where θ determines how quickly p(a) reaches its saturation level p. a 1 is the age at whic RT is completed. Existence of equilibrium is established and stability analysis is performed.

79 Drug ing of ODE DDE SDE s Anti retroviral drugs e.g., Fusion inhibitors (FI), Reverse Transcriptase inhibitors (RTI) and Protease inhibitors (PI) have been developed so as to attack on different phases of viral life cycle during infection. FI inhibits the fusion of viral particles in the host cell, which causes low infection rate. RTI inhibits the reverse transcription of viral RNA into DNA, which decreases the number of productively infected cells. PI inhibits the cleaving of polyprotein encompassing reverse transcriptase, protease, integrase, etc. This causes the production of non-infectious viral particles. Usually a combination of these drugs is used. A huge number of articles have been published related to drug therapy.

80 Target for ing of ODE DDE SDE s

81 Drug therapy:rt inhibitor ing of ODE DDE SDE s The following model is considered by Perelson and Nowak [Perelson and Nelson(1999)] to include RTI drug therapy. dt dt dv = Λ + rt ( 1 T ) (1 η RT )kvt µt, T max = (1 η RT )kvt δt, = NδT cv. Here η RT represents efficacy of RT-inhibitor.

82 Multiple drug therapy: RT+ Protease inhibitor ing of ODE DDE SDE s They modified it in presence of Protease inhibitor as follows: dt dt dv I dv NI = s + rt ( 1 T ) (1 η RT )kv I T µt T max = (1 η RT )kv I T δt, = (1 η PI )NδT cv I, = η PI NδT cv NI, Here η PI represents efficacy of protease-inhibitor.

83 Srivastava, Banerjee,, JBS (2009) ing of ODE DDE SDE s Srivastava et al. [Srivastava et al.(2009)srivastava, Banerjee, and ] considered following model with RT-Inhibitor: dt dt 1 dt dv = s kvt µt + (b + ηα)t 1, = kvt (b + µ 1 + α)t 1, = (1 η)αt 1 δt, = NδT cv, with T (0) = T 0, T 1 (0) = 0, T (0) = 0, V (0) = V 0. Where T 1 is pre-rt infected class and T is post-rt infected class.

84 Results ing of ODE DDE SDE s The above system has the following two steady states: (1) E 1 = ( T = s µ, 0, 0, 0 ) (2) E 2 = ( T, T, I, V ). The components of E 2 are given as T = (b+µ1+α)c Nαk(1 η), T 1 = s µt µ, T 1+α(1 η) = α(1 η) δ Feasible existence of E 2 is ensured whenever T > T. T 1 and V = Nδ c T.

85 Contd... ing of ODE DDE SDE s From the existence condition T > T for E 2, we may find the critical value for the drug efficacy: η crit = 1 µc(µ + α + b). Nαks Whenever η < η crit both E 1 and E 2 coexist and when η > η crit, the infection is cleared and only uninfected steady state E 1 will exist.

86 Numerical Simulation ing of ODE DDE SDE s Graph of η crit for various magnitudes of α and N, within the range 0.2 α 0.8 and 500 N 2500.

87 I ing of ODE DDE SDE s L. Cai and X. Li. Stability and Hopf bifurcation in a delayed model for HIV infection of CD4 + cells. Chaos, Solitons & Fractals, 42:1 11, W.A. Coppel. Stability and Asymptotic Behavior of Differential Equations. Health, Boston, R.V. Culshaw. of HIV models: the role of the natural immune response and implications for treatment. J. Biol. Sys, 12: , 2004.

88 II ing of ODE DDE SDE s R.V. Culshaw and S. Ruan. A delay differential equation madel of HIV infection of CD4+ T cells. Mathematical Biosciences, 165:27 39, N. Dalal, D. Greenhalgh, and X. Mao. A stochastic model for internal HIV dynamics. J. Math. Anal. Appl., 341: , X. Jiang, X. Zhou, X. Shi, and X. Song. Analysis of stability and Hopf bifurcation for a delay-differential equation model of HIV infection of CD4 + T cells. Chaos, Solitons & Fractals, 38:447460, 2008.

89 III ing of ODE DDE SDE s D. Li and W. Ma. Asymptotic properties of a HIV-1 infection model with time delay. Journal of Mathematical Analysis and Applications, 335: , M.Y. Li and J.S. Muldowney. A geometric approach to the global-stability problems. SIAM J. Math. Anal., 27: , J.S. Muldowney. Compound matrices and ordinary differential equations. Rocky Mount. J. Math., 20: , 1990.

90 IV ing of ODE DDE SDE s P.W. Nelson, M. Gilchrist, D. Coombs, J. Hyman, and A.S. Perelson. An Age-structured model of HIV infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells. Mathematical Biosciences and Engineering, 1:267 88, M.A. Nowak and R.M. May. Virus Dynamics A.S. Perelson and P.W. Nelson. Mathematical analysis of HIV-1 dynamics in vivo. SIAM, 41:3 44, 1999.

91 V ing of ODE DDE SDE s A.S. Perelson, D.E. Kirschner, and R.De Boer. Dynamics of HIV infection of CD4 + T cells. Mathematical Biosciences, 114:81 125, L. Rong, M.A. Gilchrist, Z. Feng, and A.S. Perelson. ing within host HIV-1 dynamics and the evolution of drug resistance: Trade offs between virul enzyme function and drug susceptibility. J. Theor. Biol., 247: , P. K. Srivastava and. ing the dynamics of HIV and CD4 + T cells during primary infection. Nonlinear Analysis: Real World Applications, 11: , 2010.

92 VI ing of ODE DDE SDE s P. K. Srivastava, Malay Banerjee, and. ing the drug therapy for HIV infection. Journal of Biological Systems, 17 (2): , H. C. Tuckwell and E. Le Corfec. A stochastic model for early HIV-1 population dynamics. Journal of Theoretical Biology, 195: , X. Wang and X. Song. Global stability and periodic solution of a model for HIV infection of CD4 + T cells. Applied Mathematics and Computation, 189: , 2007.

93 ing of ODE DDE SDE s THANK YOU

94 ing of ODE DDE SDE s

95 Li & Muldowney Criteria for Global stability ing of ODE DDE SDE s Now we discuss the global stability of E 2 for model (1)-(2) using the approach developed by Li and Muldowney [Li and Muldowney(1996)], which we briefly summarize here. Let the map x f (x) from an open subset D R n to R n be such that the solution x(t) to the differential equation x = f (x) (2) is uniquely determined by its initial value x(0) = x 0. We denote this solution by x(t, x 0 ).

96 Li & Muldowney Criteria for Global stability ing of ODE DDE SDE s Now we discuss the global stability of E 2 for model (1)-(2) using the approach developed by Li and Muldowney [Li and Muldowney(1996)], which we briefly summarize here. Let the map x f (x) from an open subset D R n to R n be such that the solution x(t) to the differential equation x = f (x) (2) is uniquely determined by its initial value x(0) = x 0. We denote this solution by x(t, x 0 ). Further, we assume that (H 1 ) D is simply connected, (H 2 ) x is the only equilibrium point of (2) in D, and (H 3 ) there is a compact absorbing set E D.

97 contd... ing of ODE DDE SDE s A set E is called absorbing in D for system (2) if x(t, E 1 ) E for each compact set E 1 E for sufficiently large t. For a square matrix B, the Lozinskiĭ measure [Coppel(1965)] with respect to induced matrix norm. is defined as I + hb 1 µ(b) = lim h 0 h For x D, consider a map x Q(x) where Q(x) is an ( ( n 2) n ) 2 matrix valued C 1 function and Q 1 (x) exists; define B = Q f Q 1 + QJ [2] Q 1. The matrix Q f is obtained by replacing each entry q ij of Q by its derivative in the direction of f, and J [2] is second additive compound matrix [Muldowney(1990)] of the Jacobian matrix J of system (2). For Lozinskiĭ measure µ on R (n 2) ( n 2), define a quantity q2 as

98 contd... ing of ODE DDE SDE s A set E is called absorbing in D for system (2) if x(t, E 1 ) E for each compact set E 1 E for sufficiently large t. For a square matrix B, the Lozinskiĭ measure [Coppel(1965)] with respect to induced matrix norm. is defined as I + hb 1 µ(b) = lim h 0 h For x D, consider a map x Q(x) where Q(x) is an ( ( n 2) n ) 2 matrix valued C 1 function and Q 1 (x) exists; define B = Q f Q 1 + QJ [2] Q 1. The matrix Q f is obtained by replacing each entry q ij of Q by its derivative in the direction of f, and J [2] is second additive compound matrix [Muldowney(1990)] of the Jacobian matrix J of system (2). For Lozinskiĭ measure µ on R (n 2) ( n 2), define a quantity q2 as q 2 = lim sup t sup x 0 E 1 t t 0 µ(b(x(s, x 0 )))ds (3)

99 contd... ing of ODE DDE SDE s The following result is established in Li and Muldowney [Li and Muldowney(1996)]. For the system (2), assume that assumptions (H 1 ), (H 2 ), and (H 3 ) hold. Then the unique equilibrium x is globally asymptotically stable in D if there exist a function Q(x) and a Lozinskiĭ measure µ such that q 2 < 0.