Remarks on ‘A stochastic model for the primary HIV infection
Charles J. Mode · AIDS · 2003
In this interesting paper, Ribeiro and Bonhoeffer correctly state in the section entitled ‘The stochastic model’ that to study the generation of diversity of the viral quasi-species a stochastic version of the deterministic model is required. Unfortunately, the section title, ‘The stochastic model', is a misnomer, because more than one stochastic process may be associated with a system of ordinary differential equations. However, upon reading their mathematical description of the stochastic model in this section, one gets the impression that the authors are thinking about a stochastic process with integer valued random functions such that, during any small time interval of length Δt, a random function x may increase or decrease by 1 as suggested by the symbolic transitions x ← x + 1 or x ← x − 1 and others set down in this section. In the literature on applied probability and biomathematics, a widely used method for associating a stochastic process with a system of differential equations is to construct a Markov jump process in continuous time such that the infinitesimal generator of the process is determined by rate functions in the differential equations. Within this Markovian framework, the stochastic process described by the authors is actually incorrect, because the well-known result that the holding, sojourn, times in each state of the process follow an exponential distribution with a parameter determined by the current state of the process, which the authors ignore. No attempt will be made here to reproduce the notation of the authors. Rather, the basic ideas will be illustrated in terms of a general notation. Let G represent the state space of the model with elements denoted by i,j, k …. In the model considered by the authors, the symbols i,j, k … would represent vectors of integers, i.e. whole number counts of cells of various types. Let the matrix θ = (θi,j) be the infinitesimal generator of the process such that θii = 0 for all i ε G but θi,j ≥ 0 if i ≠ j. Then, is the parameter in the exponential distribution, governing the time of transitions out of state i G during any time interval. The parameter θi is equivalent to the symbol Φ defined by the authors. Suppose, for example, that at time t the process is in state i ε G. Then, it can be shown that at time t + h,h > 0, the probability that the process is still in state i is given by exp[−θih]. Furthermore, it can be shown that if the process is in state i ε G at time t > 0, then the conditional probability of a jump to the state j ε G, j ≠ i, during the time interval [t,t + h) is given by the formula where o(h)/h ← 0 as h ↓ 0. As is well known, the matrix of transition probabilities for the discrete time Markov chain embedded in the Markov jump process in continuous time is given by the formula P =(pij = θi,j/θi). Observe that in this chain only the probabilities of jumps among the states are considered and the waiting times among jumps are ignored. In terms of the notation given above, it appears that the authors computed realizations of a stochastic process such that, during any time interval of length Δt, the probability of the transition i ← j was assigned the value (θi,j/θi)Δt. When Δt = 1, the realizations of the process considered by the authors would, therefore, correspond to those of the discrete time Markov chain embedded in the Markov jump process rather than the jump process that evolves in continuous time. To be sure, if θi is large, then the probability of no jump, exp[−θih], during a time interval [t,t + h) would be small so that the conditional probability of the transition i ← j in the above formula would be approximately θi,j/θi. Nevertheless, in a population of cells infected by a new mutation in the virus, the parameter θi may not be small so that the results of Monte Carlo simulation experiments reported by the authors are of questionable validity. In closing, it may be mentioned that the problem of embedding a system of differential equations in a stochastic process has been discussed and illustrated in detail in the book by Mode and Sleeman [1]. Briefly, the approach developed in this book starts with a discrete time approximation to a Markov jump process in continuous time, which provides efficient algorithms for computing Monte Carlo realizations of the process. Then, by taking conditional expectations given in the past and using an estimation procedure, a system of difference equations may be derived in a systematic way. By letting the length of a time interval h ↓ 0 in the system of difference equations, a system of differential equations may embedded in the Markov jump process in an unambiguous manner. Although the important problem of including mutations in HIV-1 in the formulation was not considered in the book but was deferred to future research, the principles set forth therein could be applied to the problem considered by the authors to yield unambiguous results.