Fig 1.
Simulation of the pathogen spread and treatment on a random host network.
The different background colors indicate absence (green) and presence (blue) of treatment. The phases in absence of treatment are the initial spread of the wild-type strain (light green) and the potential reversion back to wild type after treatment has stopped (dark green). Treatment is divided into the phases before (dark blue) and after (light blue) the de novo emergence of resistance. The solid lines indicate the fraction of hosts infected with either the wild type (red) or the resistant strain (purple). The black line gives the total fraction of infected hosts. The bottom panels illustrate the corresponding phases on the level of the individual hosts connected to each other. During the initial spread there are only two types of hosts: susceptibles (grey) and wild-type infecteds (red). The arrows indicate the transmission events from an infected to a susceptible host. During treatment two new types of hosts arise: In the resistance-free phase we only need to distinguish between treated (green with pill) and untreated (red) infecteds. In the resistance phase we additionally have resistant infected hosts (purple). Finally, in the reversion phase we loose the treated class of hosts and remain with susceptibles, wild-type and resistant infecteds. We generally assume that after recovery individuals are susceptible again and that treatment of resistant infecteds has no effect.
Fig 2.
Effect of treatment halt, network size and density on reversion probability.
a,b. Probability of reversion after a gradual treatment halt as a function of the relative fitness of the resistant strain for critical resistance fractions fr = 0.5 and 0.1, respectively. Almost certain reversion happens for sufficiently large fitness disadvantages of the resistant strain: relative fitness sA < 0.975 for a resistant fraction of fr = 0.5 and sA < 0.985 for fr = 0.1. As a reference, the analytical probability, using a Moran model [50], of reversion in a random host population with homogeneous degree is shown (black lines). The model for disease spread (coloured lines) shows generally a smoother transition as compared to the Moran model. In networks with higher variance in degree, the probability of reversion changes more gradually with relative fitness. c,d. Difference of the reversion probability between immediate and gradual treatment halt for fr = 0.5 and 0.1, respectively. In panels a-d color gradient indicate increasing variance of the degree distribution of the network. e. Probability of reversion as a function of the relative fitness of the resistant strain for host populations with zero degree variance and various system sizes. f. Reversion probability as a function of the relative fitness of the resistant strain for host populations with various densities (i.e. mean degrees) and zero variance. The change in connection density is compensated by adapting the transmission rate of the pathogen such that the epidemic threshold is kept constant, R0 = 3. In all simulations, treatment coverage is complete (c = 1) and drug efficacy is half maximal (e = 0.5).
Fig 3.
Characterization of the effect of degree variance on reversion probability.
a-c. The magnitude of the effect of variance in degree of the network relative to zero variance (gray lines) as a function of the relative fitness of the resistant strain, for a fraction of resistant infecteds of fr = 0.1 at a gradual treatment halt. Panel a illustrates the impact of degree variance on the probability of reversion. For values of the relative fitness of the resistant type close to but slightly smaller than 1, an increase in degree variance leads to a substantially lower probability of reversion. Panel b illustrates the effect of network occupancy. It reports the magnitude in effect of the variance in degree on the reversion probability in case of a shuffled distribution of the infection type (resistant versus wild type) among all infecteds at the end of treatment. To assess the effect of network occupancy within the infecteds at the end of treatment, panel c shows the difference between treatment halt without and treatment halt with shuffling of the infection type. We see that network occupancy has a slightly positive effect on the probability of reversion. d. The effect of variance (σ2 = 24) in degree relative to zero variance as a function of the relative fitness of the resistant strain, for a range of host population sizes. e. Impact of host network density on the relative effect of variance (σ2 = 24) as a function of the relative fitness of the resistant strain. f. Relative prevalence of the wild-type strain during the post treatment phase for a relative fitness of the mutant strain of sA = 0.995. Solid lines show the analytical solution of a two-strain pair approximation. The mean relative prevalence (dotted lines) is lower for network with a higher degree variance. The standard deviation of the mean relative prevalence (outlined with according color gradient) increases with the degree variance, indicating an increase in the magnitude of stochastic noise with increasing variance in degree of the host network.
Fig 4.
Description of infected population at the end of treatment.
a. Pairwise contacts at the end of treatment for all possible combinations of pairs involving at least one infected individual. For each infected individual its abundance in a host structures with σ2 = 24 and zero variance is reported both as observed at the end of treatment (obs.) and after randomisation of the infecteds (rand.). The abundance is given relative to the total number of pairs with at least one infected individual. b-e. Average degree of all infected (black), wild-type infected (red), and resistant strain infected (purple) individuals at the end of the treatment phase. Panel b shows the mean degree as a function of the degree variance in the network. Increasing variance in the degree distribution of the host network leads to a higher mean degree in the infected individuals. Individuals infected with the resistant strain show an even higher mean degree. Panel c illustrates the dependence of the mean degree on the treatment coverage. Panel d shows the mean degree as a function of the critical fraction of resistant infected individuals, fr. The advantage of a higher mean degree of the resistant strain vanishes with increasing critical fraction. In panel e we show the mean degree as a function of the rate of de novo emergence during the treatment phase. The trend of a higher mean degree of infected individuals with an even stronger signal for resistant infecteds is robust throughout a large range of de novo emergence rates, breaking down only for unrealistically high rates. If not specified otherwise, all simulations assume a degree variance of σ2 = 24 of the host network, full treatment coverage, c = 1, a drug efficacy of e = 0.5, and a de novo emergence rate r = 0.0001.