Unifying incidence and prevalence under a time-varying general branching process.

Renewal equations are a popular approach used in modelling the number of new infections, i.e., incidence, in an outbreak. We develop a stochastic model of an outbreak based on a time-varying variant of the Crump-Mode-Jagers branching process. This model accommodates a time-varying reproduction number and a time-varying distribution for the generation interval. We then derive renewal-like integral equations for incidence, cumulative incidence and prevalence under this model. We show that the equations for incidence and prevalence are consistent with the so-called back-calculation relationship. We analyse two particular cases of these integral equations, one that arises from a Bellman-Harris process and one that arises from an inhomogeneous Poisson process model of transmission. We also show that the incidence integral equations that arise from both of these specific models agree with the renewal equation used ubiquitously in infectious disease modelling. We present a numerical discretisation scheme to solve these equations, and use this scheme to estimate rates of transmission from serological prevalence of SARS-CoV-2 in the UK and historical incidence data on Influenza, Measles, SARS and Smallpox.

Survival phenotype, selfish individual versus Darwinian phenotype.

Consider and infinitely large asexual population without mutations and direct interactions. The activities of an individual determine the fecundity and the survival probability of individuals, moreover each activity takes time. We view this population model as a simple combination of life history and optimal foraging models. The phenotypes are given by probability distributions on these activities. We concentrate on the following phenotypes defined by optimization of different objective functions: selfish individual (maximizes the average offspring number during life span), survival phenotype (maximizes the probability of non-extinction of descendants) and Darwinian phenotype (maximizes the phenotypic growth rate). We find that the objective functions above can achieve their maximum at different activity distributions, in general. We find that the objective functions above can achieve their maximum at different activity distributions, in general. The novelty of our work is that we let natural selection act on the different objective functions. Using the classical Darwinian reasoning, we show that in our selection model the Darwinian phenotype outperforms all other phenotypes.