The effect of random virus failure following cell entry on infection outcome and the success of antiviral therapy

A virus infection can be initiated with very few or even a single infectious virion, and as such can become extinct, i.e. stochastically fail to take hold or spread significantly. There are many ways that a fully competent infectious virion, having successfully entered a cell, can fail to cause a productive infection, i.e. one that yields infectious virus progeny. Though many stochastic models (SMs) have been developed and used to estimate a virus infection’s establishment probability, these typically neglect infection failure post virus entry. The SM presented herein introduces parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1]$$\end{document}γ∈(0,1] which corresponds to the probability that a virion’s entry into a cell will result in a productive cell infection. We derive an expression for the likelihood of infection establishment in this new SM, and find that prophylactic therapy with an antiviral reducing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma$$\end{document}γ is at least as good or better at decreasing the establishment probability, compared to antivirals reducing the rates of virus production or virus entry into cells, irrespective of the SM parameters. We investigate the difference in the fraction of cells consumed by so-called extinct versus established virus infections, and find that this distinction becomes biologically meaningless as the probability of establishment approaches zero. We explain why the release of virions continuously over an infectious cell’s lifespan, rather than as a single burst at the end of the cell’s lifespan, does not result in an increased risk of infection extinction. We show, instead, that the number of virus released, not the timing of the release, affects infection establishment and associated critical antiviral efficacy.

Important differences with our more complex SM (Eqn. (1)) are that, for the simpler SM, the number of each transition occurring over a time step is Poisson distributed, there is a single compartment for the eclipse and infectious phase, and one successful virion is assumed to cause the infection of one cell.

S2 Inoculum size
In the main text, the initial number of virions V 0 = 1 IV and we explored varying the efficacy ε. Now, we fix ε = 0.8 and we explore varying V 0 . Fig S2(A) and Fig S2(B) shows the establishment probability or the median fraction of cells consumed by established infections, respectively, as a function of V 0 for antivirals with efficacy ε = 0.8 reducing β, p or γ. As V 0 increases, the establishment probability tends to 100% and differences in the establishment probability for the 3 antiviral modes of action disappear. For    Table S1) where our additional infection parameters were set to γ = 1 cell/IV and s = 1 mL.

S3 Parameter set with a higher burst size
Here, we investigate the other parameter set explored in Czuppon et al. [3]. The difference between the two parameter sets is a 10-fold decrease of the number of cells N cells , a 10-fold increase of the virus production rate p (hence, of the average burst size B = pτ I ) and a corresponding ∼10-fold decrease of βN cells (see Table  S1). For the parameter set with the higher burst size, an antiviral reducing γ or β have a similar effect on the establishment probability, better than that for an antiviral reducing p. Due to the higher burst size, the establishment probability is approximately given by the probability that the initial infectious virion causes a productive cell infection, P V → Establishment = P V → I − 1/B ≈ P V → I for n I = 1 (see Methods for more details). The probability that the initial infectious virion will cause a productive cell infection is given by the ratio between the rate of successful cell infection per infectious virion and the rate of virion loss (γβN cells /s)/(c + βN cells /s). Since the rate of virion entry into cells is much lower than the rate of virion loss of infectivity (βN cells /s ≪ c) the rate of virion loss is mostly governed by the rate of virion loss of infectivity ((c + βN cells /s) ≈ c). Therefore, the establishment probability is approximately given by (γβN cells /s)/c which is affected by γ or β the same but not affected by p.
For the parameter set with the higher burst size, an antiviral reducing either β, γ or p results in the same median fraction of cells consumed by established infections at equal efficacy. This is because, for burst size (B = pτ I ) sufficiently large that γpτ I − 1 ≈ γpτ I , we have T * /N cells = [c/(βN cells /s)]/[γpτ I − 1] ≈ [c/(βN cells /s)]/[γpτ I ]. As such, an antiviral acting on any of β, p or γ at the same efficacy will reduce the median fraction of cells infected by established infections (a monotonically decreasing fraction of T * /N cells ) by the same amount.

S4 Post-exposure antiviral therapy
In the main text, we have considered pre-exposure antiviral therapy for an infection initiated with a number of infectious virions. Now, as Czuppon et al. [3] have done, let us also explore antiviral therapy for an infection initiated with only one infectious cell. This may be representative of post-exposure antiviral therapy as it is possible that by the time an antiviral has been given after exposure, the virus has had time to cause some infectious cells.
With n I = 1, the establishment probability given that there is initially one infectious cell (see Methods for derivation) is given by  Like Czuppon et al. [3], with initially one infectious cell, we find that an antiviral reducing p is better than an antiviral reducing β to reduce the establishment probability. Reducing p affects only the denominator in the expression for the establishment probability given that there is initially one infectious cell (Eqn. (S10)). Whereas, reducing β affects both the numerator and denominator (Eqn. (S10)). Unlike Czuppon et al. [3], with initially one infectious cell, we find that an antiviral reducing p or γ have the same effect on the establishment probability (Eqn. (S10)).
In addition, there is no noticeable difference in the median fraction of cells consumed by established infections given that there is initially one infectious virion or one infectious cell. The initial number of infectious virions or cells is small and hence has a negligible effect on the median fraction of cells consumed by established infections (see Eqn. (42)).

S5 Infection risk reduction
Conway et al. [2] showed that, under pre-exposure HIV antiviral therapy, reverse transcriptase inhibitors (RTIs) reducing the cell infection rate β are more effective than protease inhibitors reducing the virus production rate p, at reducing the risk of infection.
In Conway et al. [2], the probability of having n infectious virions in the exposure inoculum is given by the binomial probability mass function, N0 n Q n c (1 − Q c ) N0−n , where N 0 is the total number of virions in the exposure inoculum and Q c represents the fraction of virions that are infectious. The risk of infection, i.e. the likelihood of infection establishment given the exposure inoculum, is then given by where P V → Extinction is the extinction probability given an infection initiated with only one infectious virion (denoted by q therein). For n I = 1, as in Conway et al. [2], our expression for P V → Extinction reduces to In this case, the difference between our expression for the extinction probability and theirs, is the inclusion of parameter γ. RTIs prevent the transcription of viral DNA from viral RNA, a necessary replication step post virus entry. One could then argue that an antiviral reducing γ better captures the mode of action of a RTI. With the addition of parameter γ in P V → Extinction , we can then test, under pre-exposure HIV antiviral therapy, if a RTI represented as reducing γ is more effective than one reducing β at reducing the risk of infection.
In Conway et al. [2], all parameters are fixed except N 0 which is assumed to be uniformly distributed over the interval [0, N max ], where N max represents the maximum inoculum size, and βN cells over a log-normal distribution f (βN cells ). The parameters of the log-normal distribution f (βN cells ) are estimated from R 0 data from Ribeiro et al. [4] using the expression βN cells = (cR 0 /τ I )/(p−1/τ I ). The maximum inoculum size N max is then determined such that, without antivirals, the risk of infection averaged over N 0 and βN cells , is ∼0.3%. For sake of simplicity, herein, we fixed βN cells = (cR 0 /τ I )/(p − 1/τ I ), where R 0 is equal to the median value of 8.04 in Ribeiro et al. [4], and N 0 such that, without antivirals, the risk of infection ∼0.3%. Other infection parameters were taken from Conway et al. [2], i.e. n I = 1, τ I = 24 h, c = 23/24 h −1 , Q c = 10 −3 , and our additional infection parameters were set to γ = 1 cell/IV and s = 1 mL. Fig S5(A-C) shows the infection risk reduction as a function of antiviral efficacy for pre-exposure antivirals reducing β or γ, for different values of the infectious virion production rate p explored in Conway et al. [2]. Fig S5(D-F) shows the infection risk reduction in (A-C) for an antiviral reducing γ minus that for an antiviral reducing β. For some of the parameters sets explored, under pre-exposure antiviral therapy, a RTI represented as reducing γ is shown to be better than one reducing β, at reducing the risk of infection. where either n I = 1 (dark colours) or n I = 60 (pale colours). Unless otherwise specified, the parameters were the same as in Fig 4.