The reproduction number and its probability distribution for stochastic viral dynamics

Abstract

We consider stochastic models of individual infected cells. The reproduction number, R, is understood as a random variable representing the number of new cells infected by one initial infected cell in an otherwise susceptible (target cell) population. Variability in R results partly from heterogeneity in the viral burst size (the number of viral progeny generated from an infected cell during its lifetime), which depends on the distribution of cellular lifetimes and on the mechanism of virion release. We analyse viral dynamics models with an eclipse phase: the period of time after a cell is infected but before it is capable of releasing virions. The duration of the eclipse, or the subsequent infectious, phase is non-exponential, but composed of stages. We derive the probability distribution of the reproduction number for these viral dynamics models, and show it is a negative binomial distribution in the case of constant viral release from infectious cells, and under the assumption of an excess of target cells. In a deterministic model, the ultimate in-host establishment or extinction of the viral infection depends entirely on whether the mean reproduction number is greater than, or less than, one, respectively. Here, the probability of extinction is determined by the probability distribution of R, not simply its mean value. In particular, we show that in some cases the probability of infection is not an increasing function of the mean reproduction number.

Keywords: Erlang distribution; eclipse phase; infectious phase; reproduction number; stochastic model; viral burst size.

Figure 1.

Model with constant viral release rate. Upon infection by a free (i.e. extracellular) infectious virion V, a target cell, T, enters the eclipse phase E followed by the infectious phase I. The arrows show the possible transitions and their corresponding rates, or the distribution of times, in the case of non-exponentially (Erlang) distributed transition times. Free virions are cleared with rate c, and eclipse-phase cells with rate ν_E. Infectious cells are cleared by the immune system with rate ν_I, or undergo virus-induced cell death. In (2.1), the eclipse and infectious phases consist of n_E and n_I stages, respectively.

Figure 2.

Counting secondary infections using a Markov chain. At each of b steps, one for each released virion, the process either moves one state to the right, corresponding to infection of a target cell, or stays put, corresponding to loss of a virion. ξ_i = β(T₀ − i)/(β(T₀ − i) + c) is the probability of moving from state i to i + 1, where T₀ is the initial number of uninfected target cells available. The position after b steps is the number of secondary infections generated by b extracellular virions.

Figure 3.

Histograms for the probability distribution of the reproduction number, R, for different values of β and T₀. For each pair of parameter values, the distributions obtained from methods of Case 1 using (2.11) (number of target cells is constant) and Case 2 using (2.12) (number of target cells decreases as they become infected) are shown. All other parameter values are fixed to the values in table 3, with ν_I = 0 (so the reproduction number distribution is equivalent under budding or bursting assumptions).

Figure 4.

Heatmap showing the Hellinger distance between the two distributions of the reproduction number, R, calculated using Case 1 (2.11) and Case 2 (2.12), for different values of β and T₀. All other parameter values are fixed to the values in table 3, with ν_I = 0 (so the reproduction number distribution is equivalent under budding or bursting assumptions).

Figure 5.

Reproduction number probability distributions calculated using the expression in table 2 for the model of viral release by budding, for different values of the immune killing rate of infectious cells, ν_I. Other parameters are set to the values in table 3. The values of ν_I used are 0, 0.25, 0.5, 1 and 1.6, per cell per day, corresponding to values of $\overline{R}$ of 735, 623, 535, 407 and 308, respectively.

Figure 6.

Reproduction number probability distributions calculated using the expression in table 2 for the model of viral release by bursting, for different values of the immune killing rate of infectious cells, ν_I. Other parameters are set to the values in table 3. The values of ν_I used are 0, 0.25, 0.5, 1 and 1.6, per cell per day, corresponding to values of $\overline{R}$ of 735, 524, 377, 201 and 99, respectively. (a) Probability of zero secondary infections for each value of ν_I. (b) Probability distribution of R conditioned on positive values of the reproduction number.

Figure 7.

Probability of viral extinction as a function of the initial number of infected cells, for the model of viral release by bursting, for different values of the immune killing rate of infectious cells, ν_I. The probability of extinction starting with one cell is calculated by numerically finding the smallest fixed point of the p.g.f. in (2.14). To obtain the probability of extinction starting with i infected cells, one takes the probability for one cell to the power of i. Other parameters are set to the values in table 3.

Figure 8.

Reproduction number distributions calculated using the expression in table 2 for the bursting model, for different numbers of infectious phase stages, n_I. For each value of n_I, the mean of the Erlang-distributed time until cell burst is kept fixed to the value of τ_I in table 3. Other parameters are also set to the values in table 3, and the rate of immune killing of infectious cells is ν_I = 1.6 per cell per day. (a) Probability of zero secondary infections for each value of n_I. (b) Probability distribution of R conditioned on positive values of the reproduction number.

Figure 9.

Plots to show how the probability of viral extinction depends on the number of infectious phase stages, n_I, for the model of viral release by bursting. The probability of extinction is calculated by numerically finding the smallest fixed point of the p.g.f. in (2.14). (a) The probability of viral extinction starting from one infected cell as a function of θ, for different values of n_I. The value of θ obtained from the values of β, c and T₀ in table 3 is indicated by the dashed line. For each value of n_I, the mean of the Erlang-distributed time until cell burst is kept fixed to the value of τ_I in table 3. The viral production rate is also fixed to the value of p in table 3, and the rate of immune killing of infectious cells is ν_I = 1.6 per cell per day. (b) The probability of viral extinction starting from one infected cell is shown for different values of n_I, for θ = 0.59 (the value indicated by the dashed line in the left plot). Along with the number of infectious stages, the mean reproduction number increases along the x-axis.