A common non-pharmaceutical intervention (NPI) during the COVID-19 pandemic has been group size limits. Furthermore, educational settings of schools and universities have either fully closed or reduced their class sizes. As countries begin to reopen classrooms, a key question will be how large classes can be while still preventing local outbreaks of disease. Here, we develop and analyse a simple, stochastic epidemiological model where individuals (considered as students) live in fixed households and are assigned to a fixed class for daily lessons. We compare key measures of the epidemic—the peak infected, the total infected by day 180 and the calculated R 0 —as the size of class is varied. We find that class sizes of 10 could largely restrict outbreaks and often had overlapping inter-quartile ranges with our most cautious case of classes of five. However, class sizes of 30 or more often result in large epidemics. Reducing the class size from 40 to 10 can reduce R 0 by over 30%, as well as significantly reducing the numbers infected. Intermediate class sizes show considerable variation, with the total infected varying by as much as from 10% to 80% for the same class size. We show that additional in-class NPIs can limit the epidemic still further, but that reducing class sizes appears to have a larger effect on the epidemic. We do not specifically tailor our model for COVID-19, but our results stress the importance of small class sizes for preventing large outbreaks of infectious disease.

While experimental studies have demonstrated within-population variation in host tolerance to parasitism, theoretical studies rarely predict for polymorphism to arise. However, most theoretical models do not consider the crucial distinction between tolerance to the effects of infection-induced deaths (mortality tolerance) and tolerance to the parasite-induced reduction in the reproduction of infected hosts (sterility tolerance).While some studies have examined trade-offs between host tolerance and resistance mechanisms, none has considered a correlation within different tolerance mechanisms.We assume that sterility tolerance and mortality tolerance are directly traded-off in a host population subjected to a pathogen and use adaptive dynamics to study their evolutionary behaviour. We find that such a trade-off between the two tolerance strategies can drive the host population to branch into dimorphic strains, leading to coexistence of strains with sterile hosts that have low mortality and fully fertile with high mortality rates. Further, we find a wider range of trade-off shapes allows branching at intermediate or high infected population size. Our other significant finding is that sterility tolerance is maximised (and mortality tolerance minimised) at an intermediate diseaseinduced mortality rate. Additionally, evolution entirely reverses the disease prevalence pattern corresponding to the recovery rate, compared to when no strategies evolve.We provide novel predictions on the evolutionary behaviour of two tolerance strategies concerning such a trade-off.

Tolerance and resistance are two modes of defence mechanisms used by hosts when faced with parasites. Here, we assume tolerance reduces infection‐induced mortality rate and resistance reduces the susceptibility of getting infected. Importantly, a negative association between these two strategies has often been found experimentally. We study the simultaneous evolution of resistance and tolerance in a host population where they are related by such a trade‐off. Using evolutionary invasion theory, we examine the patterns of optimal investment in each defence strategy, under different ecological scenarios. Our focus is on predicting which of the two strategies is favoured under various epidemiological and ecological conditions. Our key findings surround the impact of recovery and sterility of infected hosts. As the rate at which infected hosts recover from the infection, that is the recovery rate increases, the investment in tolerance increases (resistance decreases) when infected hosts are sterile, but this pattern reverses when infected hosts can reproduce. We further found that a change in the parameter determining the intraspecies competition for resources leading to a reduction in birth rate, that is the crowding factor affects investments in tolerance and resistance only when infected hosts can reproduce. These results emphasize the role of fecundity in driving the evolutionary dynamics of a host. We also find that disease prevalence can increase or decrease depending on whether or not the host evolves: prevalence is highest at low recovery rates when the host does not evolve, but the feedback of a change in tolerance and resistance reverses this pattern, leading to lower prevalence at low recovery rates as host evolves.

No abstract

A common non-pharmaceutical intervention (NPI) during the Covid-19 pandemic has been group size limits. Further, educational settings of schools and universities have either fully closed or reduced their class sizes. As countries begin to reopen classrooms, a key question will be how large classes can be while still preventing local outbreaks of disease. Here we develop and analyse a simple, stochastic epidemiological model where individuals (considered as students) live in fixed households and are assigned to a fixed class for daily lessons. We compare key measures of the epidemic - the peak infected, the total infected by day 180 and the calculated R0 - as the size of class is varied. We find that class sizes of 10 could largely restrict outbreaks and often had overlapping inter-quartile ranges with our most cautious case of classes of 5. However, class sizes of 30 or more often result in large epdiemics. Reducing the class size from 40 to 10 can reduce R0 by as much as 30%, as well as signficantly reducing the numbers infected. Intermediate class sizes show considerable variation, with the total infected varying as much as from 20% to 80% for the same class size. We show that additional in-class NPIs can limit the epidemic still further, but that reducing class sizes appears to have a larger effect on the epidemic. We do not specifically tailor our model for Covid-19, but our results stress the importance of small class sizes for preventing large outbreaks of infectious disease.