a new model describing the development of memory ...

CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 21, Number 2, Summer 2013

A NEW MODEL DESCRIBING THE DEVELOPMENT OF MEMORY/LATENCY DURING HIV INFECTION FEDERICO FRASCOLI, YAN WANG, BENI SAHAI AND JANE M. HEFFERNAN

ABSTRACT. An obstacle to HIV virus eradication in a patient is the persistence of the latently infected cell reservoir. Latently infected cells are long-lived and, upon reactivation, will produce virus in-host. Mathematical models have been used to describe the effects of the latently infected cell pool. However, the different avenues that produce a latently infected cell have, for the most part, been ignored. We have developed a mathematical model that delineates the CD4+ T-cell population by immune system status (na¨ıve, activated, memory) and by infection status (uninfected, infected), which captures the different methods for making an infected memory CD4+ T-cell. The model shows a number of interesting characteristics within realistic parameter space including multiple infected equilibria, Hopf bifurcations that induce periodic oscillations at low viral load, and a backward bifurcation that enables infection to persist even when the basic reproductive ratio is less than one. In addition, the model can be used to demonstrate viral blips using intermittent activation.

1 Introduction In a typical immune response, contact with antigen causes na¨ıve T-cells to proliferate and differentiate into effector cells, after which an effector T-cell may then die, or become a memory T-cell [32]. In a typical infection, after the pathogen is destroyed, most effector T-cells are eliminated, but some cells survive to form long-lived memory T-cells. The role of memory T-cells is to recognize specific antigens upon new exposures to a pathogen (or those related to it) and to mount a quick and strong immune response. In the case of HIV, however, where AMS subject classification: Primary 62P10, 92-08; Secondary 92C37, 92C99. Keywords: memory CD4+ T-cells, latently infected cells, HIV, dynamical system, backward bifurcation, Hopf bifurcation. c Copyright Applied Mathematics Institute, University of Alberta.

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CD4+ T-cells are the main target for infection by virus, memory CD4+ T-cells can harbour latent virus. Thus, when such cells come into contact with antigen, they can be activated and produce new HIV virions. During states of low-level infection, activation of latently infected T-cells may cause viral blips, whereby large amounts of virus can be produced, forming a spike in the viral load. The failure to eradicate HIV infection during long-term antiretroviral therapy demonstrates such an effect from infected memory CD4+ T-cells. The inability of the immune system to recognize cells harbouring latent virus presents a great challenge to finding a cure for HIV. Understanding the effects of latently infected CD4+ T-cells in HIV has been a major focus of the health and natural sciences. There has been extensive modelling work done to characterize viral load and pathogen– immune system interactions in HIV and the latently infected cell reservoir (see, for example, [3, 10, 11, 18, 29, 30, 31], and [33] for a review). Most mathematical models predict that viral rebound in HIV infection is very likely due to the continuous replenishment of the latent reservoir [33]. For the vast majority of these models, however (with the exception of [11]), the various biological processes that produce memory CD4+ T-cells have been ignored; i.e., activation, proliferation and differentiation. Thus, the different avenues for producing latently infected cells, specifically infection of na¨ıve CD4+ T-cells that are activated and then proceed to the memory cell class, infection of activated CD4+ T-cells, which then proceed to the memory class, or infection of memory CD4+ T-cells themselves have not been captured. In this study, we present a mathematical model that explicitly includes these different stages. The goal of the current study is to determine whether a mathematical model including activation and proliferation can capture HIV infection characteristics that are known to arise from the latently infected reservoir. For instance, it has been observed that the latently infected reservoir persists in patients with low viral loads [2, 4, 35] and that spikes in viremia (or viral blips) at low viral loads are generally not indicative of change in patient disease status [16]; i.e., the viral load and the latently infected cell pool should both be relatively stable. It has been suggested that the latent reservoir can be replenished through intermittent episodes of activation whereby virus is released, producing new infected cells that can revert to the memory state [31]. A second goal of our study is to determine whether a model including activation and proliferation can provide new insights into issues surrounding the persistence of HIV and viral blips at low viral load. The present model was motivated by Dr. Beni Sahai, and the acceler-

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ated write-up of this work is due to his unfortunate passing in 2012. In the paragraphs that follow, we introduce the mathematical model. We then present the model steady states and numerical bifurcation analysis. We also conduct a sensitivity analysis employing Latin Hypercube Sampling and we demonstrate that viral blips can be induced using current modelling techniques from the literature. The study ends with a discussion of the results and a direction for future work. 2 Methods The model consists of eight ordinary differential equations describing cellular and viral dynamics. The model includes uninfected na¨ıve, activated and memory CD4+ T-cells (xn , xa and xm ), infected na¨ıve, activated and memory CD4+ T-cells (yn , ya and ym ), and infectious and non-infectious virus (v and w). The model equations are

(2.1)

dxn dt dxa dt dxm dt dyn dt dya dt

= λ − dxn xn − βxn v − γxn (v + δ), = γxn (v + δ) + cxm v + ρxa

v − dxa xa − pxa − βxa v, v+C

= pxa − cxm v − dxm xm − βxm v, = βxn v − dyn yn − γyn v, = γyn v + βxa v + ρya

v v+C

+ cym v − dya ya − pya − ξya f (v, xa , ya ), dym = pya + βxm v − cym v − dym ym , dt dv = kqya − uv − sβv(xn + xa + xm ), dt dw = k(1 − q)ya − uw. dt Uninfected CD4+ T-cells Uninfected CD4+ T-cells are delineated by immune status (na¨ıve, activated or memory). Uninfected na¨ıve CD4+ T-cells (xn ): Uninfected na¨ıve CD4+ Tcells are assumed to be generated at a constant rate λ and die at a per capita death rate dxn . Uninfected CD4+ T-cells can be infected

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by infectious virus (v) at rate β and can be activated with rate γ to produce infected activated CD4+ T-cells (xa ). For simplicity, activation is assumed to be proportional to infectious virus (v) so that antigenpresenting cells (APCs) and other components of the immune system involved in activation do not need to be included in the model. It is assumed that there exists a small number of uninfected activated and memory T-cells (xa and xm ) when the virus enters the blood, from the early stages of infection. Thus, for simplicity, we have included a term δ which is very small to seed the initial populations of activated and memory CD4+ T-cells. This term becomes negligible as the viral load increases. Note that a population of activated uninfected CD4+ T-cells is required for this model to have a basic reproductive ratio R0 > 0 (see Section 3.1). Uninfected activated CD4+ T-cells (xa ): Uninfected activated CD4+ T-cells xa are produced by the activation of either uninfected na¨ıve or memory CD4+ T-cells (γ and c). Uninfected activated CD4+ T-cells proliferate at a rate ρ. It is assumed that the proliferation varies with the infectious virus load (v) using the function v/(v + C) where C is a half-saturation constant. Here, C is used to represent a level of immunesystem response that determines the viral load required to achieve the maximal proliferation rate ρ. When the infectious viral load (v) is sufficiently large v/(v + C) ≈ 1. This reflects the fact that proliferation of the activated CD4+ T-cell pool is bounded by underlying biological processes of the immune system. Uninfected activated CD4+ T-cells leave the system through death at a constant per capita death rate dxa , by becoming a memory cell at rate p, or infection by infectious virus (v) with rate β. Uninfected memory CD4+ T-cells (xm ): Uninfected memory CD4+ T-cells xm enter the system from the uninfected activated CD4+ T-cell population with rate p. Uninfected memory CD4+ T-cells can leave the system through death with constant per capita death rate dxm , becoming an uninfected activated CD4+ T-cell with rate c that is assumed to be proportional to the infectious virus population v, or by infection by infectious virus v with rate β. Infected CD4+ T-cells Infected CD4+ T-cells are also delineated by immune status (na¨ıve, activated or memory), and have similar dynamics to the uninfected CD4+ T-cell population. Infected na¨ıve CD4+ T-cells (yn ): Infected na¨ıve CD4+ T-cells yn are produced by the infection of uninfected na¨ıve CD4+ T-cells (xn ) with

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rate β. Infected na¨ıve CD4+ T-cells die with rate dyn and are activated with rate γ to produce infected activated CD4+ T-cells. Again, activation is assumed to be proportional to the infectious virus population v. Infected activated CD4+ T-cells (ya ): Infected activated CD4+ Tcells ya are produced by the infection of uninfected activated CD4+ Tcells (xa ) with rate β, or by the activation of infected na¨ıve or memory CD4+ T-cells (yn and ym ) with rates γ and c. Again, activation is assumed to be proportional to the infectious viral load v. Similar to uninfected activated CD4+ T-cells xa , infected activated CD4+ T-cells can proliferate at a rate ρ. It is also assumed that the proliferation rate varies with the infectious virus load v using the function v/(v + C) where C is a half-saturation constant. Infected activated CD4+ T-cells die with per capita death rate dya , can become infected memory CD4+ T-cells ym with rate p, and can be killed by the immune system with rate ξ. The CD8 T-cell population is not explicitly modelled in system (2.1). However, this population can be assumed to be proportional to the population of antigen-presenting cells (that we can represent by virus), or it could be proportional to the number of activated CD4+ T-cells xa and ya . In the following, we will assume f (v, xa , ya ) = v, but we also present a limited exploration of another form of this function, f (v, xa , ya ) = xa + ya . Infected memory CD4+ T-cells (ym ): Infected memory CD4+ Tcells ym are produced by the infection of uninfected memory CD4+ Tcells (xm ) by infectious virus v with rate β, or by the transition of infected activated CD4+ T-cells ya into the memory pools with rate p. Infected activated CD4+ T-cells die with per capita death rate dym and can be activated with rate c proportional to the infectious virus population v. Virus The virus populations are delineated by the ability to infect CD4+ T-cells. Infectious virus (v): Infectious virus v is produced by infected activated CD4+ T-cells ya with rate k where a fraction q is assumed to be infectious. Infectious virus is cleared with rate u. Infectious virus can also be lost due to infection of uninfected CD4+ T-cells xn + xa + xm with rate β. Here, s = 0 or 1, allowing us to compare system (2.1) with and without the viral loss term due to absorption into uninfected cells. In the vast majority of mathematical models of HIV infection, the viral loss term is ignored (s = 0) since it is generally assumed that u >> βxv.

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However, it has been found that the virus loss term can play a role in infection dynamics, T-cell count and viral load [7, 25, 26, 39]. Thus, this term may be important in understanding the effects of the latently infected memory CD4+ T-cell pool. Non-infectious virus (w): Non-infectious virus w is produced by infected activated CD4+ T-cells ya with rate k where a fraction 1 − q are non-infectious, and these cells are cleared with rate u. Note that explicitly modelling the non-infectious virus population separate to the infectious virus population v enables the study of the virus loss term sβv(xn + xa + xm ) in the infectious virus equation. 2.1 Parameter estimation We chose most of the parameter values from previous studies. Parameters and their values are listed in Table 1. TABLE 1: Parameter values Parameter xn

xa

xm

yn ya

ym

v w λ

Definition Uninfected na¨ıve CD4+ T-cells Uninfected activated CD4+ T-cells Uninfected memory CD4+ T-cells Infected na¨ıve CD4+ T-cells Infected activated CD4+ T-cells Infected memory CD4+ Tcells Infectious virus Non-infectious virus Birth rate

dxn

Death rate of xn

dxa

Death rate of xa

Value

Range

104 cells mL−1 day−1 0.01 day−1

103 –105

Reference

[3, 28, 29, 30, 31, 33, 39, 40]

0.1 day−1 TABLE 1 continued on next page.

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Parameter dxm

dyn dya dym u γ

c

β

δ ρ C p ξ k q

s

TABLE 1, continued from previous page. Definition Value Range Death rate of 0.001 day−1 xm Death rate of yn Death rate of ya Death rate of ym Clearance rate of v and w Activation of na¨ıve T-cells xn and yn Activation rate of memory Tcells xm and ym Infection rate of uninfected T-cells xn , xa and xm Assumed model parameter Proliferation of xa and ya Half-saturation constant Conversion rate to memory cells Killing rate of ya Bud rate of new virions Fraction of new virions that are infectious Used to study the inclusion of the virus loss term

189

Reference [3, 28, 29, 30, 31, 33, 39, 40]

same as dxn 1 day−1

[21, 27, 39, 40]

same as dxm 23 day−1

3–36

0.01 day−1

mL

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0.0001 day−1

mL

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0.00004 day−1

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0.5 virions mL−1 0.1428 day−1 1000 virions mL−1 0.001dxa /(1− 0.001) day−1 0.001 mL day−1 2000 virions cell−1 day−1 0.05

[21, 27, 39, 40] [10, 11, 28, 29, 30, 31] [10, 11, 28, 29, 30, 31]

[12, 28] 1000–10,000 [10, 11, 29, 30, 31]

50–5000

[5, 9, 17] [37]

0 or 1

The total CD4+ T-cell population is assumed to be approximately 0.5–1.6 ×106 cell/mL at the beginning of infection [15, 19, 34]. The death rate of na¨ıve CD4+ T-cells is assumed to be ∼ 0.01/day [3, 28,

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29, 30, 31, 33, 39, 40] and λ was chosen to be ∼ 103 –105 cells/(mLday) so that the initial CD4+ T-cell population target range 0.6-1.4 ×106 cell/mL could be achieved. We assume that the death rates of na¨ıve CD4+ T-cells, whether infected or not, are the same. This is based on the assumptions that (1) the immune system does not kill na¨ıve infected CD4+ T-cells, and (2) since these cells are na¨ıve, they are not producing virus, so na¨ıve infected CD4+ T-cells do not have an accelerated death rate. For similar reasons, we assume that the death rates of uninfected and infected memory CD4+ T-cells are the same. The death rate of memory CD4+ T-cells is assumed to be ∼0.001/day [3, 28, 29, 30, 31, 33, 39, 40]. Based on current estimates [21, 27] we assume a death rate of ∼1/day for infected activated CD4+ T-cells. We assume that uninfected activated CD4+ T-cells have a death rate of ∼0.1/day since these cells should have an accelerated death rate over the uninfected na¨ıve and memory CD4+ T-cell populations, but have a lower death rate than infected activated CD4+ T-cells. The virus clearance rate is assumed to be ∼23/day [21, 27]. The activation rates for na¨ıve and memory CD4+ T-cells γ and c, and the term δ were chosen so that ∼200 infected memory T-cells are present in the infected equilibrium [29, 30, 31], R0 > 1, and δ should be sufficiently small so that it has little to no effect on activation of the na¨ıve CD4+ T-cell population at the infected equilibrium. The parameter δ was chosen to be 0.5 virions/mL. The activation rates of na¨ıve and memory CD4+ T-cells c were chosen to lie in the range 0.0001–0.3 mL/day [10, 11, 28, 29, 30, 31]. A very small fraction of activated CD4+ T-cells will become memory CD4+ T-cells. It is assumed that ∼1/1000 uninfected activated CD4+ T-cells will become memory CD4+ T-cells [10, 11, 29, 30, 31]. This fraction is used to determine the value of p when the death rate of dxa is known. The value of p from this calculation is also used in the equation for ya . Thus, the fraction of infected CD4+ T-cells that become infected memory cells is less than 1/1000 since infected activated cells have a shorter lifetime than their uninfected counterparts (dya < dxa ). Activated CD4+ T-cells can proliferate. It is assumed that proliferation can only occur when virus is present in the system and that a constant proliferation rate of ρ is achieved with moderate and high levels of infection. CD4+ T-cells can proliferate approximately every 6 hours at equilibrium with stimulation of antigen, but this can vary with the level of antigen in the system [12]. The proliferation rate ρ has been chosen so that proliferation occurs within the appropriate biological range and is of similar magnitude to the death rate dxa [12, 28].

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The half-saturation constant C determines the viral load v needed to reach the maximum proliferation rate ρ. The parameter C can be used to study proliferation as an immune sytem response. For example, a small C can reflect a responsive immune system where proliferation of activated CD4+ T-cells reaches the maximum rate ρ at lower viral loads. A larger C reflects a less effective immune response since the maximum proliferation rate will be reached for higher viral load values. We have chosen to explore a range of C where C = 1000 virions/mL represents a more responsive immune system and C = 10, 000 represents a less responsive one. Infected activated cells can be killed by the immune system. Estimates for the killing rate ξ will vary depending on the functionf (v, xa , ya) that is chosen. We set the killing rate at ξ at 0.001 mL/day when the killing function f (v, xa , ya ) = v. A single activated infected CD4+ T-cell can produce between 50 and a few thousand virions in its lifetime [5, 9, 17]. We have chosen the production rate k to be 2000 virions/cell per day. We assume that a small fraction q of the virions budded are infectious. Reported ratios of infectious virus particles to total viral load range from ∼1/8 to 1/60,000 [37]. We assume that 5% of the virus particles budded are infectious (q = 0.05). 3 Results 3.1 Basic reproductive ratio R0 The basic reproductive ratio R0 (also known as the basic reproduction number) is defined as the number of secondary infections produced by one infectious individual introduced into a totally susceptible population. There are many methods that can be used to calculate R0 [6]. Using the survival function method, R0 for system (2.1) with f (v, xa , ya ) = v is (3.1)

R0 =

βxa0 kq . dya + p u + sβ(xn0 + xa0 + xm0 )

Calculation of R0 using the stability conditions given by the eigenvalues of system (2.1) gives the same result. Here, the first term gives the number of infectious virions produced during the average lifetime of an infected activated CD4+ T-cell. If the virus loss term s = 1 is included in system (2.1), the second term gives the probability that a newly generated infectious virion infects an uninfected activated CD4+ T-cell. However, if s = 0, then this term can be interpreted to give the

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total number of newly infected cells produced by an infectious virion in its average lifetime. The s = 1 case gives the correct description of the biological processes underlying HIV infection. 3.2 Equilibria 3.2.1 Uninfected Equilibrium E0 We find that there exists an uninfected equilibrium E0 given by (xn0 , xa0 , xm0 , 0, 0, 0, 0, 0) where (3.2)

xn0 =

(3.3)

xa0 =

(3.4)

xm0 =

λ , dxn + γδ (dxa

γδλ , + p)(dxn + γδ)

pγδλ . dxm (dxa + p)(dxn + γδ)

Theorem 3.1. E0 is locally asymptotically stable when R0 < 1. Proof. The Jacobian of system (2.1) evaluated at the uninfected equilibrium E0 is  −dxn − γδ  γδ   0   0  (3.5) J0 =  0   0   0 0

0 −dxa − p p 0 0 0 0 0

0 0 −dxm 0 0 0 0 0

0 0 0 0 0 −dym 0 0

0 0 0 −dyn 0 0 0 0

0 0 0 0 −dya − p p kq (1 − q)k

−βxn0 −
γxn γxn0 + Cρ − β xa0 + cxm0 −cxm0 − βxm0 βxn0 βxa0 βxm0 −u − sβX0 0

 0 0   0   0   0   0   0  −u

where X0 = xn0 + xa0 + xm0 . From the eigenvalues of J0 , it is evident that the stability of E0 is

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determined by the inequality (3.6)

1 (−dya − p − u − sβX0 ) 2 1 ± (−dya − p)2 − 2(−u − sβX0 )(−dya − p) 2 1/2 + (−u − sβX0 )2 + 4(βxa0 )(kq) 0, then x ¯n > 0 and y¯n > 0. Steady states x¯a and x ¯m will be positive if (3.9)

v+p− dxa + β¯

ρ¯ v cp¯ v > 0. − v + β¯ v v¯ + C dxm + c¯

The signs of y¯a , y¯m and w ¯ are more difficult to ascertain; however, if Eq. (3.9) is satisfied and (3.10)

v− dya + p + ξ¯

cp¯ v ρ¯ v > 0, − v v¯ + C dym + c¯

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all populations will have at least one positive solution. Rearranging Eq. (3.8), one can find a polynomial of degree 12 in v¯. Factoring, we obtain a polynomial of degree 11 (not shown). Determining roots, or the number of feasible real roots, of this polynomial is not analytically tractable. We will now turn to numerical simulations to further explore the model. 3.3 Numerical simulations Figure 1 demonstrates that the uninfected equilibrium is stable when R0 < 1, and unstable when R0 > 1. The simulations begin at the uninfected equilibrium, and one infectious virion is added to the system at day 10. The viral load is cleared if R0 < 1, and it progresses to infection if R0 > 1. This also shows that the infected equilibrium exists if R0 > 1. Figure 2 shows that the infected equilibrium is stable when R0 > 1, but that it can also exist and be stable when R0 < 1. The simulation begins at the infected equilibrium and k is reduced from 2000 to 500 virions/(cell-day) to obtain a value R0 = 0.75 < 1. From Figures 1 and 2 it is evident that the uninfected and infected equilibria are both stable when R0 = 0.75. This points to the existence of a backward bifurcation. Figure 3 shows that a backward bifurcation exists. Here we vary the bud rate k and plot the viral load v at the stable infected equilibrium (bold), the uninfected equilibrium (x-axis), and another infected steady state that is unstable (gray) and exists for some values of R0 less than unity. This is shown for both cases with and without viral loss due to infection (s = 0 (left) or 1 (right)). Note the change in stability of the uninfected equilibrium on the x-axis. This point coincides with the value R0 = 1. Figure 3 shows that the uninfected equilibrium is stable when R0 < 1 (bold on x-axis) and the infected equilibrium is stable when R0 > 1 (bold). In a region where R0 < 1, however, the infected equilibrium remains stable (bold) and a second unstable infected equilibrium (grey) also exists. Analysis performed using XPPAUT [42] confirms the existence of a backward bifurcation. This analysis has also shown that the occurrence of a backward bifurcation is consistent over all parameter sets tested, taken from the ranges listed in Table 1. The existence of a backward bifurcation has direct implications on disease outcome, as it indicates that, even if drug therapy or the immune system is successful in reducing R0 < 1, infection may not be eradicated. Figure 4 shows a bifurcation diagram for system (2.1) using the parameter values listed in Table 1 with s = 1. We plot the viral load v versus the bud rate k. This figure shows a branch point on the x-axis (point BP) at which the uninfected equilibrium changes stability. This

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point corresponds to the point R0 = 1. Figure 4 also shows a saddle node (SN) and a Hopf bifurcation (HB). The saddle node and Hopf bifurcations exist for both cases s = 0 and 1. The presence of the saddle node confirms the existence of a backward bifurcation. Interestingly, the Hopf bifurcation induces periodic orbits at low viral loads. For the case shown in Figure 4 a stable branch for limit cycles (solid circles) exists for low viral loads 5 < v < 160 virions/mL and all other limit cycles emerging from the Hopf bifurcation are unstable (open circles). Figure 5 shows an example of the oscillatory behaviour of system (2.1) using the parameter set listed in Table 1 with s = 1 and k = 86 virions/(cell-day). These oscillations at low viral load can provide a basis for viral blips to occur. A study of system (2.1) in a stochastic formulation that better

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reflects the probabilistic nature of HIV infection is needed to explore the effects of the Hopf bifurcation further. We use the half-saturation constant C to give a measurement of immune system responsiveness in system (2.1). Figure 4 shows the bifurcation diagram for system (2.1) when f (v, xa , ya ) = v and C = 1000 virions/mL. When the bifurcation analysis is performed with C = 10, 000 virions/mL, the extension of the stable branch for limit cycles changes. When C = 10, 000 virions/mL, the stable branch exists for low viral loads 10 < v < 100 virions/mL, but when C = 1000 virions/mL the stable branch has a range of 5 < v < 160 virions/mL (Figure 4). For larger C we also find a difference in the occurrence of the HB and SN points,

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FIGURE 3: Backward bifurcation. Uninfected and infected equilibrium values of the infectious virus population v are shown for system (2.1) with f (v, xa , ya ) = v when k is varied and s = 0 (left) and s = 1 (right). Bold indicates a stable equilibrium point. Parameter values are listed in Table 1, with the exception of k. The change in stability on the x-axis, corresponding to the uninfected equilibrium, denotes the point where R0 = 1. Note that the value 10−10 corresponds to a viral load of zero.

but this is very small. These results suggest that oscillations at low viral load may have a smaller magnitude if the proliferation rate is reduced at lower levels of viremia. However, these results are very similar to that found when C = 1000 virions/mL. Therefore, it is unlikely that any difference in oscillations in viral load observed between HIV patients could be attributed to different values of the half-saturation constant C. 3.4 Sensitivity analysis We conducted a sensitivity analysis on system (2.1) using Latin Hypercube Sampling (LHS) and partial rank correlation coefficients (PRCC). LHS is a stratified sampling technique that creates sets of parameters by sampling for each parameter without replacement according to a predefined probability distribution (see ref-

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FIGURE 4: One parameter bifurcation diagram. The uninfected and infected equilibrium values of the infectious virus population v are shown for system (2.1) with f (v, xa , ya ) = v when k is varied and s = 1. A branch is shown on the x-axis when R0 = 1. A saddle node (SN) and a Hopf bifurcation (HB) are also shown. Solid lines denote stable equilibria, whereas dashed lines depict unstable ones. Maxima and minima of limit cycles emanating from HB are indicated by filled (stable) and open (unstable) circles. Note that the periodic branch changes its stability at a saddle node point (SNLC), indicated for both maxima and minima of the oscillation. Parameter values are listed in Table 1, with the exception of k.

erences [1, 20] for method description). Following this procedure, for each parameter, a PRCC value is calculated that provides a measure of the relationship between model inputs and outputs. PRCC values range between −1 and 1, with the sign determining whether an increase in a parameter value will increase (+) or decrease (−) the specified model output. PRCC values that are statistically significant satisfy |P RCC| > 0.5. For each parameter, we assumed a uniform distribution across a range (1) ±20%, (2) ±50%, and (3) from the ranges listed in

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50 45 40 35 30 v 25 20 15 10 5 0 100

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FIGURE 5: Oscillations induced by a Hopf bifurcation at low viral load. A snapshot of the period solution induced by a Hopf bifurcation is shown. Parameter values are listed in Table 1 except that k = 86 virions/(cell-day).

Table 1. The system was solved using both 500 and 1000 parameter sets 2–6 times for the different ranges. The output of the sensitivity analysis was used to study the effects of the model parameters on the stable infected equilibrium and the basic reproductive ratio R0 , and to determine whether multiple infected equilibria can exist in realistic parameter space. Figure 6 shows the PRCC values for the infected equilibrium (total uninfected cells, infected cell, virus) and the basic reproductive ratio R0 , determined from one run of the LHS sensitivity analysis for s = 0 (left column) and for s = 1 (right column). Here, f (v, xa , ya ) = v and the parameter values are varied in the range ±50% of the values listed in Table 1. We also hold the value of q = 0.05 constant so as not to obscure any results pertaining to the bud rate k. Figure 6 demonstrates that R0 is sensitive to changes in the constant δ, activation rate γ,

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bud rate k, infection rate β, clearance rate u, death rate dya , death rate dxa and birth rate λ. This figure also illustrates that the infected equilibrium is sensitive to variations in the killing rate ξ, bud rate k, virus clearance rate u, and incoming rate of uninfected na¨ıve CD4+ Tcells λ. For some components of the infected equilibrium, changes in the infection rate β, proliferation rate ρ, and death rate dxa are also significant. Results from the sensitivity analysis using the range ±20% of the values listed in Table 1 and the parameter ranges listed in Table 1 have very similar results with the magnitude of the PRCCs changing very slightly from one LHS run to another. Consistent across all of the LHS and PRCC results is the sensitivity of the infected equilibrium to changes in λ, k, ξ and u. Of these parameters, an increase in the killing rate ξ, an increase in the virus clearance rate u, and a decrease in the virus production rate k are associated with an increase in total CD4+ T-cell count (X + Y = xn + xa + xm + yn + ya + ym ) and a decrease in the total viral load (v + w). This result suggests that intervention strategies that will most affect HIV disease outcome must activate the immune system to induce higher killing and clearance rates and reduce the number of virions produced by activated infected CD4+ T-cells. It is evident that multiple infected equilibria can occur for system (2.1). A sensitivity analysis can determine if multiple equilibria can exist in realistic parameter space that also result in CD4+ T-cell count and viral load levels that are representative of HIV and AIDS measurements. A CD4+ T-cell count of < 2×105 CD4+ T-cells/mL provides a diagnosis of AIDS [8, 38] and CD4+ T-cell counts lying in the range 2 × 105 to 106 cells/mL can be measured in HIV patients [8, 22, 23]. A viral load of < 107 /mL [8, 24] is also required so that viral load levels are representative of measurements from the clinical literature [8, 22, 23]. We find that for each run of the LHS, at least 20 parameter sets generate multiple infected equilibria regardless of whether s = 0 or 1. Of these parameter sets, approximately one set generates more than two infected equilibria. The majority of the parameter sets generated by the LHS present viral load and CD4+ T-cell counts that are not representative of HIV or AIDS disease, with very high viral load measurements combined with very small CD4+ T-cell counts. Of the sets generating multiple infected equilibria, at least one parameter set generates a CD4+ T-cell count > 2 × 105 cells/mL with viral < 106 cells/mL and at least one more generates a CD4+ T-cell count between 105 and 2 × 105 cells/mL with a viral load < 106 virions/mL. Therefore, we can conclude that multiple infected equilibria can exist in realistic parameter space that generate infected equilibria representative of HIV and AIDS

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viral load and CD4+ T-cell counts. Note that, due to the complexity of system (2.1) and computer system requirements, our sensitivity analysis was limited to using 500 or 1000 parameter sets per LHS simulation run. Thus, we are unable to determine the probability that multiple equilibria exist in realistic parameter space with CD4+ T-cell count and viral load values reflecting HIV or AIDS measurements. Future studies will include optimization of the LHS routine so that a larger number of parameter sets can be studied. 3.5 An ‘on-off ’ switch for activation It has been suggested that increases in viremia and viral blips in patients with low viral loads could be a result of activation of latently infected cells to the productively infected cell pool [2, 31]. Recently there has been a focus on the study of the viral protein trans-activator of transcription (Tat) on the activation of latently infected cells since it has been observed that Tat can act as x

x

ξ

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ξ C p δ c ρ γ k β u dym dya d yn dxm dxa dxn

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FIGURE 6: Partial Rank Correlation Coefficients. PRCC values are shown for the infected equilibrium X = sn + xa + xm (top row), Y = yn + ya + ym (second row). 500 simulations are used. Parameters were allowed to vary ±50% of their values as listed in Table 1.

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v

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ξ C p δ c ρ γ k β u dym dya dyn dxm dxa dxn

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FIGURE 6: (continued) Partial Rank Correlation Coefficients. PRCC values are shown for the viral load V = v + w (top row), as well as the basic reproductive ratio R0 (bottom row) for s = 0 (left) and 1 (right). 500 simulations are used. Parameters were allowed to vary ±50% of their values as listed in Table 1.

a stochastic switch to stimulate viral transcription [36, 41]. We now modify our model to include an ‘on-off’ switch representing higher and lower levels of activation. This process is used widely in the modelling literature in studies of the latently infected cell pool [11, 29, 30, 31]. It has been observed that viral blips are generally not indicative of change in patient disease status and that most blips are simply variations around a mean viral load [16]. A further observation has been that the latent reservoir persists in patients with low viral loads [2, 4, 35]. Intermittent episodes of activation can release virus that provides a level of virus replication that replenishes the latent reservoir [31]. Using the ‘on-off’ switch, we test if system (2.1) can generate a stable infected memory cell pool, a persistent low level viral load and intermittent blips in viremia. We have chosen to use a parameter set from the sensitivity analysis

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FIGURE 7: An ‘on-off’ switch. The infected equilibrium is ploted for four instances of the ‘on-off’ activation switch. The time interval between times of higher activation levels is assumed to obey a normal distribution N(50,10), and the time of increased activation is assumed to obey a uniform distribution U(4,6) [29, 30, 31]. The activation rates γ and c are assumed to fluctuate from 0.1 to 0.3 mL/day. Figure 7 shows one simulation using four instances of the ‘on-off’ switch. The simulation begins at the stable infected equilibrium, induced where drug therapy is present. Drug efficacy is modelled by reducing the infection rate by 80% in the first six equations of system (2.1) (reverse transcriptase inhibitor) and by reducing the fraction of infectious virions budded q by 30% in the last two equations (protease inhibitor). Parameter values are: λ = 2.65 × 104 cells mL−1 day−1 , dxn = dyn = 0.01 day−1 , dxa = 0.1 day−1 , dxm = dym = 0.001 day−1 , dya = 1 day−1 , u = 35 day−1 , β = 2.05 × 10−5 mL day−1 , δ = 0.5 virions/mL, ρ = 0.09 day−1 , C = 1000 virions/mL, p = 1.44 × 10−4 day−1 , ξ = 0.0038 mL day−1 , k = 1200 virions cell−1 day−1 , q = 0.05 and s = 1. Drug efficacy is modelled by reducing the infection rate by 80% in the first six equations of system (2.1) (reverse transcriptase inhibitor) and by reducing the fraction of infectious virions budded q by 30% in the last two equations (protease inhibitor).

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above that presents viral load and CD4+ T-cell count measurements reflective of HIV diagnosis , but close to the threshold where an AIDS diagnosis will occur (i.e., when the CD4+ T-cell count lies between 2– 3 ×105 CD4+ T-cells/mL). We assume that the time interval between times of higher activation levels obeys a normal distribution N (50,10), and that the time of increased activation obeys a uniform distribution U (4,6) [29, 30, 31]. The activation rates γ and c are assumed to fluctuate from 0.1 to 0.3 mL/day. Figure 7 shows one simulation using four instances of the ‘on-off’ switch. Here, we begin at the stable infected equilibrium induced where drug therapy is present. Drug efficacy is modelled by reducing the infection rate by 80% in the first six equations of system (2.1) (reverse transcriptase inhibitor) and by reducing the fraction of infectious virions budded q by 30% in the last two equations (protease inhibitor). Note that the viral loss term in the infectious virus equation is not modified, since virus loss will still occur (i.e., reverse transcription will not occur, but ‘loss’ of the virus through translocation into the cell will). Figure 7 demonstrates that system (2.1) with variations in activation rates g and c can generate a stable infected memory cell pool ym , a persistent low level viral load v + w and intermittent spikes in viremia. 3.6 A modified killing term Previously, we found the existence of a backward bifurcation and a Hopf bifurcation in system (2.1) when the killing term of infected cells includes a function f (v, xa , ya ) = v. When system (2.1) includes the modified killing function f (v, xa , ya ) = xa + ya we also find a saddle node and Hopf bifurcation similar to our previous analysis. However, for the modified killing function we also find a second Hopf bifurcation, and when s = 1 a second saddle node exists for small viral load values. Thus, bistability can occur if s = 1. The second infected equilibrium at low viral load will further complicate any chances of HIV infection eradication. A detailed bifurcation analysis studying the existence of all of the Hopf and saddle node bifurcations, and periodic solutions in an in-depth sensitivity analysis in realistic parameter space is left to a future study. 4 Discussion Latently infected cells that harbour virus unrecognizable by the immune system can lay dormant for very long periods, and, upon activation, can allow the persistence of HIV even in patients with very low viral loads. Mathematical modelling studies have improved the understanding of the effects of the latently infected cell pool; however, the vast majority of these models have ignored the biological

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processes whereby this cell pool can be developed; i.e., infection of na¨ıve CD4+ T-cells that are activated and then proceed to the memory cell class, infection of activated CD4+ T-cells that then proceed to the memory class, or infection of memory CD4+ T-cells. In the current work, we have developed a mathematical model that explicitly includes all of these possibilities. We found that system (2.1) has an uninfected equilibrium and at least one infected steady state. We also calculated the basic reproductive ratio R0 and showed that the uninfected equilibrium is locally asymptotically stable when R0 < 1. Sensitivity analysis demonstrated that the parameters that greatly affect the infected equilibrium are the killing rate of activated infected cells ξ, production rate of virus particles k, virus clearance rate u and birth rate of uninfected na¨ıve CD4+ T-cells λ. Of these parameters, an increase in the killing rate ξ, an increase in the virus clearance rate u and a decrease in the virus production rate k are associated with an increase in CD4+ T-cell count and a decrease in viral load. Current drug therapy regimens focus on decreasing the transmission rate β (which was not shown to be significant for changes in the infected equilibrium for all model populations), and decreasing the production of infectious virus q, but these have not been effective in clearing HIV infection. There is currently an effort to develop HIV vaccines specifically directed at inducing CD8+ T-cell responses, which will increase the killing rate of the infected activated CD4+ T-cell population (see [14] for a review). HIV vaccines inducing antibody activities are also a current focus [13]. Our sensitivity analysis results are in agreement with these directions of research. Our results also suggest that the development of a drug therapy or vaccine that reduces the virus production rate k could also be considered. It is unknown, however, if there is a biological mechanism or therapeutic option that could be developed to achieve this. Viral blips, where viral loads increase to very high levels and then decrease over a short period of time, have been observed in patients on effective drug therapies. It has been suggested, however, that these blips are not indicative of any change in disease status [2, 4, 16, 35]. Following current modelling studies of the latently infected cells pools [29, 30, 31], we demonstrated that an ‘on-off’ swtich that varies activation rates of na¨ıve and memory CD4+ T-cells can generate a stable infected memory cell pool ym , a persistent low level viral load v + w and intermittent spikes in viremia. Mathematical models are built on assumptions and trade-offs between representing the underlying area of application and the complexity of the

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model. As a result, in studies of HIV infection in-host, assumptions are usually made to represent aspects of the immune system using parameters or surrogate variables. We compared our model with and without the viral loss term in the infectious virus equations (a term that is ignored in the vast majority of modelling studies of HIV infection in-host), and with different killing functions where the CD8 T-cell population that performs the killing of activated infected CD4+ T-cells is represented by virus v or activated CD4+ T-cell xa + ya populations. We found that a backward bifurcation exists when s = 0 or 1 for both killing functions studied, irrespective of the value of s and the killing function. This result demonstrates that, even if R0 is reduced below unity, HIV infection can persist. The numerical bifurcation analysis also uncovered a Hopf bifurcation for both killing functions irrespective of the value of s. Interestingly, the Hopf bifurcation induces periodic oscillations at low viral loads and can provide a basis for viral blips that are observed in HIV patients. A direction for future work is to develop a stochastic model of system (2.1) so that the Hopf bifurcation combined with stochastic fluctuations that can produce viral blips that are consistent with that observed in HIV patients can be studied. When the modified killing function f (v, xa , ya ) = xa + ya is incorporated into system (2.1), another Hopf bifurcation exists in conjunction with a second saddle node when s = 1. This is an interesting result as it shows that bistability can occur with one stable infected equilibrium at a low viral load, making it very difficult for HIV infection to be eradicated. Recently, Zhang et al. [43, 44] showed that a mathematical model with two equations describing the uninfected and infected CD4+ T-cell populations can exhibit a Hopf bifurcation that can be used to represent viral blips. Futhermore, they showed that two Hopf bifurcations can result in a 3-dimensional system describing uninfected and infected CD4+ T-cells and activated CD8+ T-cells (CTL). An in-depth comparison of these models with system (2.1) is a direction for future work. In the current model, we compared two killing functions that use different model populations to represent the action of CD8+ effector cells. Many of the other terms in the model use the virus v as a representative of antigen-presenting cells. These terms can be modified to use the infected acivated CD4+ T-cell population ya instead, which is a type of APC. Future work will include a sensitivity analysis whereby different surrogate model variables of the APC population will be compared. The goal of this study was to determine whether a mathematical

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model including activation and proliferation can capture known characteristics of the latently infected reservoir in HIV infection, and determine whether this model can provide new insights into issues surrounding the existence and persistence of HIV and viral blips at low viral load. We have demonstrated that system (2.1) can generate a stable infected memory cell pool ym , a persistent low level viral load v + w and intermittent spikes in viremia. We have also found, however, that system (2.1) can incorporate one or two Hopf bifurcations at low viral loads that can induce viral blips, and a backward bifurcation that demonstrates that HIV eradication in-host may not be achievable. Model results have also shown that, depending on the killing function assumed and if s = 1, bistability can result with one stable infected equilibrium existing at low viral load. Thus, more effective drug therapies or HIV vaccines are required to overcome this barrier to HIV infection eradication. Acknowledgments The authors would like to acknowledge funding from the Natural Sciences and Engineering Council of Canada (NSERC) and the National Natural Science Foundation of China (No. 11301543).

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