Fig 1.
Schematic of model for viral dynamics in a patient undergoing antiviral treatment.
(A) Diagram of basic viral dynamics model incorporating the infected cell maturation process. The population of cells is comprised of healthy cells x, one or more stages of immature infected cells w, and mature (virus-producing) infected cells y. The maturation time is the time it takes for an infected cell to pass through all the maturation phases. (B) Probability distribution functions of maturation times, when the maturation process happens as a series of n consecutive steps. The maturation times are gamma-distributed with the same average of 2 days, for different numbers of maturation steps (n = 1, 2, 10, 25). (C) The viral infectivity β(t) relative to the viral infectivity in the absence of the drug β0 fluctuates in response to drug levels (blue line). In the simple on-off model (a step function), drug levels are “on” for a fraction f of the time between doses (red shading), reducing viral fitness to zero, and “off” for the rest of the interval (viral fitness returns to baseline). (D) Time course of infection levels when the maturation time is fixed to τ = 2 days (yellow line) and τ = 3 days (blue line), and the drug dosage is modeled as a periodic step function. The synchronized strain (maturation time of 2 days, yellow line) reaches higher time-averaged infection level than the unsynchronized strain. In these examples, we use drug period T = 2 days and drug efficacy f = 0.85.
Table 1.
Parameter values used in the viral dynamics and drug treatment models.
Fig 2.
Equilibrium infection level for the single-strain deterministic model, as a function of the average maturation time.
Viral dynamics were simulated under periodic antiviral therapy given by the simple on-off model with a period (T) of 2 days and varying drug efficacy (f). The infection level (heat map color) is measured as the concentration of mature infected cells (y) once a steady-state has been reached. Each calculation included only a single virus strain with average maturation time 1/m (maturation rate of nm for each stage). (A) Results with n = 1 maturation steps. (B) Results with n = 10 maturation steps. (C) Results with n = 25 maturation steps. (D) Results with fixed maturation time τ = 1/m. The white dotted lines show where the average maturation time is equal to an integer multiple of the drug period. For all shown simulations, we assume the death rate of immature cells to be zero (dw = 0). Results shown for 41 different drug efficacies between f = 0.6 and f = 1.0, for 101 different strains with average maturation times between 1 and 6 days. A version of the results with more resolution around the threshold drug efficacy (f = 0.9) is in S2 Fig and a version with dw > 0 is in S3 Fig.
Fig 3.
Competition between viral strains with different life cycle times.
Viral dynamics were simulated under periodic antiviral therapy given by the simple on-off model with a drug efficacy of 85% (f = 0.85). The infection level (y-axis) is measured as the concentration of mature infected cells (y) once a steady-state has been reached. Each simulation included a collection of viral strains with the full range of maturation times shown. Each strain is labeled by its average maturation time 1/m (maturation rate of nm for each stage). A-C Fixed maturation time, varying drug dosing period (T). (A) Drug dosing period T = 4 days. (B) Drug dosing period T = 3 days. (C) Drug dosing period T = 1 day. D-F Drug dosing period (T) of 2 days, varying distribution of maturation times. (D) Results with n = 1 maturation step. (E) Results with n = 10 maturation steps. (F) Results with fixed maturation time τ = 1/m. For all shown simulations, we assume the death rate of immature cells to be zero (dw = 0). Data shown for competitions between 57, 85, 165, 501, 501, and 108 different strains, for panels from A to F respectively, with average maturation times between 1 and 6 days.
Fig 4.
Competition between viral strains with different life cycle times when infected cells can die before maturing.
Viral dynamics were simulated under periodic antiviral therapy given by the simple on-off model with a period (T) of 2 days, a drug efficacy of 85% (f = 0.85), and an immature cell death rate between 0 and dy (mature infected cell death rate). The infection level (heat map color) is measured as the concentration of mature infected cells (y) once a steady-state has been reached. Each simulation included a collection of viral strains with the full range of maturation times shown, all with the same death rate of immature infected cells. Each strain is labeled by its average maturation time 1/m (maturation rate of nm for each stage). (A) Results with n = 1 maturation step. (B) Results with n = 10 maturation steps. Data shown for 110 different immature cell death rates between dw = 0.001 and dw = 1, for competitions between 501 different strains with average maturation times between 1 and 6 days.
Fig 5.
Stochastic competition between viral strains with different life cycle times.
Viral dynamics were simulated under periodic antiviral therapy given by the simple on-off model with a period (T) of 2 days and varying drug efficacy (f). The fixation probability (heat map color) is measured as the fraction of simulations in which a strain was the last surviving in the population and continued on to reach a steady state. (A) Results with n = 10 maturation steps. (B) Results with fixed maturation time τ = 1/m. The white dotted lines show where the average maturation time is equal to an integer multiple of the drug period. For all shown simulations, we assume the death rate of immature cells to be zero (dw = 0). 500 simulations were run for each drug efficacy level. Data shown for 21 different drug efficacies between f = 0.8 and f = 1.0, for competitions between 126 strains with average maturation times between 1 and 6 days. A version of the results for all drug efficacies is in S4 Fig, and a version with dw > 0 is in S5 Fig.
Fig 6.
Modified basic reproductive ratio , unlike the time-averaged
, accurately predicts infection outcome under a periodic drug treatment.
(A) Time course of infection levels (concentration of mature infected cells, y(t)) for an unsynchronized strain (T = 2 days and τ = 5 days). The maturation time is fixed and drug levels are modeled as a periodic step function. Unsynchronized strains are more exposed to the drug effects, as they overlap less with the off-windows of the drug treatment. (B) Synchronized strains (τ = 2) are less exposed to the drug effects, as they overlap with the off-windows of the drug treatment. The off-windows in the drug treatment are represented by blue shading. In this example, we use drug period T = 2 days and drug efficacy f = 0.75. (C) The time-averaged basic reproductive ratio minus one () is plotted versus the maturation time (τ) for a fixed drug efficacy and the simple on-off model of drug levels (black line).
is independent of maturation time when immature cells do not die (dw = 0 here) and weakly dependent for dw ≪ m.
versus
is unable to explain the observed persistence versus extinction of viral strains (e.g. Fig 2). We derived a new quantity,
(blue line), which works in the presence of synchronization to describe the observed behavior.
was obtained via numerical solution of Equation (S.99) for
and substitution of
into Equation (S.100). The equilibrium infection level for a single strain (red line) is scaled to match
at τ = 1.8. The drug efficacy is set to f = 0.9, and the death rate of immature infected cells is zero (dw = 0). (D) Same as (C), except the drug efficacy is set to f = 0.5, and the death rate of immature infected cells is non-zero (dw = 0.1). The equilibrium infection level is scaled to match
at τ = 0. Connecting lines between points are drawn as a guide for the eye.