Modelling Vaccine Effects on HIV-1 Viral Dynamics
JOHN GUARDIOLA and ANTONIA VECCHIO*
Istituto di Genetica e Biofisica ‘A. Buzzati Traverso’
and
Istituto per le Applicazioni del Calcolo‘Mauro Picone’*
Consiglio Nazionale delle Ricerche
via P. Castellino, 111, I-80131 Naples
ITALY
Abstract: - Mathematical models of viral dynamics are useful for describing viral progression upon infection and viral decline after drug treatment. HIV-1 models including intracellular delays and pharmacological delays appear to be a more accurate representation of the biology of the virus and to provide better estimates of the values of the kinetic parameters of infection. In the present paper, we extend integro-differential equation models, previously developed for simulating the effect of anti-retroviral inhibitors on virus progression, to the prediction of the impact of a hypothetical vaccine on HIV-1 propagation. The vaccine simulated here, either as preventive or therapeutic, is envisaged as stimulating an anti-HIV specific cytolytic T cell response of variable strength, able to kill chronically or latently infected cells. An “immunological” delay, accounting for the time required for the immune system to respond to the challenge, is also incorporated into the model.
Key-words:- Mathematical models, integro-differential equations, HIV-1 vaccine, virus progression, time delays, CTL, eradication
1
1INTRODUCTION
Emergent diseases caused by viruses constitute a relevant treat to human health due to the presently limited panoply of drugs available and to the lack of tested efficacious vaccines.
Mathematical models of viral dynamics have proved useful in describing viral progression upon infection and viral decline after drug treatment. In the case of the human immunodeficiency virus-1 (HIV-1), such models were constructed to provide the means to predict important kinetic constants of virus replication and decay in vivo, in the presence and in the absence of anti-retroviral inhibitors[8,2]. Similarly, kinetic parameters of hepatitis C virus (HCV) [6] or hepatitis B virus (HBV) [3] infections obtained by the solution of various models were found to be in good agreement with those estimated by direct determination of plasma virus levels following perturbation of the chronically infected (quasi-) steady state.
Treatment of these viruses with anti-viral drugs or combinations of them in highly active anti-retroviral therapy (HAART) causes rapid reduction in plasma virus load, viral decline occuring in several phases determined by the virus properties and by the pharmacodynamical characteristics of the drug used [3]. Current models, incorporating intracellular time-delays for virus replication in the infected cells and “pharmacological” delays related to drug efficacy, satisfactorily explain some of these phases [2,3], even though they may require ad hoc assumptions to facilitate a mathematical analysis and to enable to derive simple solutions [3]. Both intracellular delays and pharmacological delays were found useful to refine the predictions of a model and to improve the determination of some kinetic parameters of the infection.
Time-delays are introduced into these models to account for the intracellular phase of the virus life-cycle, whereby it is assumed that virus production may lag by a delay behind the infection of the cell. This implies that recruitment of virus-producing cells at time t is not given by the density of newly infected cells, but by the density of cells that were newly infected at time t-. Such a delay can be considered fixed [2,3] or distributed according to a probability density function [5].
Analogously, pharmacological delays have been incorporated into models describing the effect of drugs onto the virus replication to reflect the lag in the pharmacological effect due to the time required for drug absorption, distribution and penetration into the target organ or cell type. The assumption taken for sake of simplification is that drug effectiveness undergoes a step-wise increase from 0 to 100% after some fixed pharmacological delay which may be characteristic of different drugs. Model formulations may depend on the mechanism of action of the drug considered. For example, models in which substances inhibiting the production of virus by the infected cells, such as reverse transcriptase inhibitors [2,3] differ from models in which the action of the drug results in the production of non-infectious virus particles (see, for example, ref. 7). However, basic models of viral dynamics generally contain three unknown functions: the population of cells susceptible to infection, or target cells, S(t), the population of infected cells that produce virus, I(t) and the virus present in the plasma, V(t). A constant influx and a constant relativedeath rate is generally assumed for target S(t) cells, although this may be an oversimplification. Furthermore, to facilitate the mathematical analysis and to enable to derive simple analytical solutions when modelling the effect of inhibitory drugs, the population of uninfected cells, infected virus-producing cells and free virus are considered at a steady-state level [3].
HIV constitutes a challenge for mathematical modelling due to the complexity of the viral life cycle, in particular considering the interaction of the virus with the immune system [4], the emergence of drug resistant mutants [8], the importance of virus compartmentalization in different organs [6] or the occurrence of different phases of virus progression on significantly different time-scales [3]. Most models, however, albeit useful for parameter estimation and for supporting patient monitoring, may not accurately reflect such a complexity. In the present paper, adopting available models as the starting point, we explore the effect of a hypothetical anti-HIV vaccine. Anti-HIV vaccines can be basically envisaged in two forms: a vaccine capable of inducing neutralizing anti-HIV antibodies, acting to reduce virus load and a vaccine inducing anti-HIV specific cytolytic T cells capable of killing HIV-infected cells, controlling the number of HIV-producing cells (see ref. 1 and ref.s therein). Studies of the immune response in clinically asymptomatic seropositive patients supports in fact the contention that humoral and cellular immunity may be useful for controlling HIV infection. An ideal anti-HIV vaccine should thus contain both activities. Presently, we modify existing models featuring an intracellular delay by incorporating the effect of killer vaccines based on cytolytic T cells and having different efficacies. A “vaccine” or “immunological” delay determined by the time required for the vaccine to elicit a protective response from the immune system is also taken into consideration.
2 Problem Formulation
Several mathematical models have been proposed to describe viral dynamics of HIV-1 infection [2,3,5,7]. All models derive from a simplified phenomenological model of virus infection such as that reported in Fig. 1. Susceptible cells (S) are provided at a constant rate and removed by clearance at a constant relative rate . Infected cells (I) are derived upon infection of plasma viruses (V) of uninfected S cells at a rate of infection k and are in turn removed by clearance at a rate . Infected cells produce viral particles at a rate p, thus filling the viral plasma compartment from which viruses are removed at a clearance rate c. The effect of inhibitors used in highly active anti-retroviral therapy is exerted at different levels, depending on the mode of action of the drug considered. For example, reverse transcriptase (RT) inhibitors act at the level of infection of S cells by the virus, thus essentially reducing the value of k, whereas protease inhibitors interfere with the ability of viruses produced by I cells to cause de novo infections, thus lowering the value of p. In principle, for 100% effective drugs, k and p must reach a 0 value.
Fig. 1Phenomenological model of HIV-1 virus infection
A classical mathematical formulation of the basic model, using differential equations, is that reported by Herz et al. [3]:
where k and p are explicitly considered (in their case, but not in other models) variable with time t to account for the effect of drugs added at later times. In the variant used by Perelson et al. [7], the model incorporates the fact that in the presence of protease inhibitors plasma viruses belong to two separate pools, the infectious virus particles, VI, and the non-infectious virus particles VNI (Fig. 2), the variations of which are described by two distinct equations (see also below). S is instead considered constant. A parameter (where = 1 for a drug 100% effective) is introduced to account for the action of the protease inhibitor.
Fig. 2Phenomenological model of the effect of protease inhibitors on HIV-1 virus infection
2.1Intracellular delays
Models of HIV-1 infection based onintracellular delays are more accurately descriptive of the phenomenological aspects of virus life cycle and predictive of the values of kinetic parameters than models without delays. The intracellular delay refers to the lag which observed between the time at which the virus infects the cell and the time at which the infected cell becomes productive. A fixed delay is considered both in the model proposed by Grossman et al. [2] and in that proposed by Herz et al. [3]. In the former case, after primary infection the I cell is considered to progress through various stages until it reaches a stage of maturation in which it produces virus. The transition rate coefficient from one stage to the next, where cell death and virus release occur at the last stage, is considered constant and the delay is the time required to complete this multistep process. In the other case, it is taken that virus-producing cells at time t are those infected at time t – , where is the delay. Furthermore, a constant death rate different from the rate of virus producing cells, is assumed for cells infected, but not yet producing virus and a probability of surviving from time t – to time t is incorporated into the equation [3]. Mittler et al. [5] have modified the model proposed by Perelson et al. [7] to estimate the role of a variable intracellular delay. Such a distributed delay is accounted for by a density function , assuming that viruses which infect cells at time t were produced by cells infected t-x time units before. In this model a pharmacological delay is also introduced using the Heavyside function h(t), where h(t) = 0 when t < p and h(t) = 1 when t ≥ p and p is the pharmacological delay. The equations of this model (derived from those of Mittler et al., 1998 using the notations of Fig. 1 and 2) are the following:
The exponential factor e-mx in the first equation of problem (1) accounts for the loss of infected cells between the time of the initial infection and the release of the first virions [5]. The solution is then obtained by converting the integro-differential equations (1) into anequivalent set of ordinary differential equations [5] and by solving it numerically.
2.2 Modelling vaccine effects on viral progression
Anti-retroviral RT inhibitors affect de novo infection of target cells by plasma viruses, but do not block production of virus particles by previouslly infected cells. Nonetheless, HAART yields an esponential fall of virus titers in treated patients followed by a slower decline in residual viremia. The biphasic behaviour of this kinetics is related to the finding that most infected cells are short lived, a fact, which in combination with the inhibition of de novo infection due to the drug, explains the initial very rapid reduction of viral burden. The subsequent lower rate of decline observed in Phase II of the treatment and after HAART corresponds to longer half-lives of cells chronically and latently infected before the start of the therapy and reflects the ability of HIV-1 to attack multiple cellular compartments during its life cycle [2]. Thus, although HAART is very efficient in reducing the level of viremia during Phase I of the treatment, viruses cannot be eradicated, leading to the probability of rebound and leaving room for the accumulation of resistant mutants. As mentioned, models in which protease inhibitors are considered imply production of VI and VNI particles in various ratios, depending on the effectiveness of the drug used. De novo infection is not inhibited and in the expected decline is strongly dependent on the estimate of virus turnover rates. Furthermore, as compared to inhibitors having a different mode of action, the initial decline of plasma virus is slower because infectious viruses that were produced before the start of the therapy continue to infect cells.
Eradication of virus infection by HAART is thus hardly expected, either in real-life situation or in ideal model conditions, thus justifying the need for an efficacious HIV-1 vaccine able to remove all kinds of virus producing cells. Such a vaccine should in principle induce a cytolytic T cell (CTL) response targeted against cells infected by HIV-1 and expressing virus-specific antigens. The feasibility of this approach is supported by the evidence that anti-HIV CTL are recruited in the course of HIV-1 infection and that, in animal systems, depletion of CD8+ T cells causes a dramatic rise in plasma viremia (ref. 10 and ref.s therein). We have developed and analyzed a set of models, incorporating distributed intracellular delays, immunological delays and distributed survival probability, aimed at predicting the possibility of virus eradication by hypothetical preventive, as well as therapeutic anti-HIV-1 vaccines for which different levels of efficacy are supposed. In the present paper, we discuss a preliminary model obtained by the adaptation of the model proposed by Mittler et al. [5] and thus based on a system of Volterra integro-differential equations. Elsewhere, we shall present more refined models based ondelay integro-differential equations approaches and on visual modelling.
3Problem Solution
Our mathematical model describing the effect of a hypothetical vaccine on HIV-1 progression is based on the phenomenological model of infection summarized in Fig. 1, except that no drug inhibition is considered, but killing by cytolytic T cells of I cells is assumed. The mathematical models also refers to the in vivo kinetic pattern of progression of HIV-1 infection schematized in Fig. 3.
Fig. 3Schematic time-course of HIV-1 infection in long-term progressor patients
As it now well known, the initial phase of HIV-1 infection is characterized by a very rapid initial increase of plasma virus titer, followed by a clinically asymptomatic lag period which may last several years before another burst of viremia sets in because of a change in the properties and replication rate of the virus. The model, therefore, aims at reproducing the long term kinetics of infection and the resulting multiphasic behaviour. Furthermore, the hypothetical vaccine that we simulate can be either a preventive vaccine, the inoculation of which took place before infection or a therapeutic vaccine, inoculated immediately after the infection, during the lag period or just at the onset of the final phase of the infection. The following integro-differential equations are thus proposed:
Where
The parameters n and b describe the shape and the scale of the density function f, respectively; is the viremia resurgence delay and is the immunological delay, i.e. the time required for the immunological system to develop a reaction in response to vaccination.Note that in this model the delay, in analogy with Mittler et al. [5], is described by the probability density function f(x) assuming that viruses infecting susceptible cells were produced by cells infected x time units before. The function p(t) representing the rate of infection is defined as a step function with a discontinuity point in where is the time at which the asymptomatic lag ends and a drastic resurgence of viremia is observed. The function , representing vaccine efficacy, is also a step function which assumes null value in absence of vaccine ( ), i.e. before the time required for the immunological sysytem to develop a reaction in response to vaccination. For a fully effective vaccine we have . When , the vaccine can be considered a preventive vaccine. Assuming different values of allows instead to describe a therapeutic vaccine gaining efficacy at different time-points during the process. It must be kept in mind, however, that this constitutes an oversimplification as priming of cytolytic CD8(+) T cells by a vaccine depends on the help provided by CD4(+) T cells, which, in this case, are also the target of HIV-1: the ability of the immune system to respond to the antigenic challenge may be thus impaired in parallel with the decrease of CD4(+) cell counts typically observed in HIV-1 infected patients.This models also differs from the model of Mittler et al. [5] for the time scale considered (which includes all phases of a long term progression process) and because S is variable.
Of course, the analytical (true) solution of (2) cannot be found and a numerical (approximate) solution must be computed in order to know the behaviour of the unknown functions I(t),S(t),V(t). Before proceeding to this, we note that (as shown in [5]), thanks to the form of f(x), given in (3), the system of three integro-differential equations (2) can be transformed into an equivalent system of n+3 ordinary differential equations, n appearing in (3). What is the advantage of this transformation? Even if a variety of methods there exists for computing the numerical solution of a system of integro-differential equations (see for example [9] and the refs therein), user friendly numerical codes for solving ordinary differential equations are much more developed than those for solving integro-differential equations. Therefore, if n is not too large, it may be convenient, instead of solving a system of 3 integro-differential equations, to solve an higher dimension system of ordinary differential equations. The transformed problem is the following system of nonlinear first order ordinary differential equations
where are auxiliary functions with no biological meaning introduced by the transformation [5] and the meaning of the other symbols are already specified in (2). Problem (4) now appears as a classical Cauchy problem of the type