On hierarchical competition through reduction of individual growth.

We consider a population organised hierarchically with respect to size in such a way that the growth rate of each individual depends only on the presence of larger individuals. As a concrete example one might think of a forest, in which the incidence of light on a tree (and hence how fast it grows) is affected by shading by taller trees. The classic formulation of a model for such a size-structured population employs a first order quasi-linear partial differential equation equipped with a non-local boundary condition. However, the model can also be formulated as a delay equation, more specifically a scalar renewal equation, for the population birth rate. After discussing the well-posedness of the delay formulation, we analyse how many stationary birth rates the equation can have in terms of the functional parameters of the model. In particular we show that, under reasonable and rather general assumptions, only one stationary birth rate can exist besides the trivial one (associated to the state in which there are no individuals and the population birth rate is zero). We give conditions for this non-trivial stationary birth rate to exist and analyse its stability using the principle of linearised stability for delay equations. Finally, we relate the results to the alternative, partial differential equation formulation of the model.

Assessing the Impact of (Self)-Quarantine through a Basic Model of Infectious Disease Dynamics.

We introduce a system of differential equations to assess the impact of (self-)quarantine of symptomatic infectious individuals on disease dynamics. To this end we depart from using the classic bilinear infection process, but remain within the framework of the mass-action assumption. From the mathematical point of view, the model we propose is interesting due to the lack of continuous differentiability at disease-free steady states, which implies that the basic reproductive number cannot be computed following established mathematical approaches for certain parameter values. However, we parametrise our mathematical model using published values from the COVID-19 literature, and analyse the model simulations. We also contrast model simulations against publicly available COVID-19 test data, focusing on the first wave of the pandemic during March-July 2020 in the UK. Our simulations indicate that actual peak case numbers might have been as much as 200 times higher than the reported positive test cases during the first wave in the UK. We find that very strong adherence to self-quarantine rules yields (only) a reduction of 22% of peak numbers and delays the onset of the peak by approximately 30-35 days. However, during the early phase of the outbreak, the impact of (self)-quarantine is much more significant. We also take into account the effect of a national lockdown in a simplistic way by reducing the effective susceptible population size. We find that, in case of a 90% reduction of the effective susceptible population size, strong adherence to self-quarantine still only yields a 25% reduction of peak infectious numbers when compared to low adherence. This is due to the significant number of asymptomatic infectious individuals in the population.

Modelling evolution of virulence in populations with a distributed parasite load.

Modelling evolution of virulence in host-parasite systems is an actively developing area of research with ever-growing literature. However, most of the existing studies overlook the fact that individuals within an infected population may have a variable infection load, i.e. infected populations are naturally structured with respect to the parasite burden. Empirical data suggests that the mortality and infectiousness of individuals can strongly depend on their infection load; moreover, the shape of distribution of infection load may vary on ecological and evolutionary time scales. Here we show that distributed infection load may have important consequences for the eventual evolution of virulence as compared to a similar model without structuring. Mathematically, we consider an SI model, where the dynamics of the infected subpopulation is described by a von Förster-type equation, in which the infection load plays the role of age. We implement the adaptive dynamics framework to predict evolutionary outcomes in this model. We demonstrate that for simple trade-off functions between virulence, disease transmission and parasite growth rates, multiple evolutionary attractors are possible. Interestingly, unlike in the case of unstructured models, achieving an evolutionary stable strategy becomes possible even for a variation of a single ecological parameter (the parasite growth rate) and keeping the other parameters constant. We conclude that evolution in disease-structured populations is strongly mediated by alterations in the overall shape of the parasite load distribution.

A dynamic model of CT scans for quantifying doubling time of ground glass opacities using histogram analysis.

We quantify a recent five-category CT histogram based classification of ground glass opacities using a dynamic mathematical model for the spatial-temporal evolution of malignant nodules. Our mathematical model takes the form of a spatially structured partial differential equation with a logistic crowding term. We present the results of extensive simulations and validate our model using patient data obtained from clinical CT images from patients with benign and malignant lesions.

Modelling Wolbachia infection in a sex-structured mosquito population carrying West Nile virus.

Wolbachia is possibly the most studied reproductive parasite of arthropod species. It appears to be a promising candidate for biocontrol of some mosquito borne diseases. We begin by developing a sex-structured model for a Wolbachia infected mosquito population. Our model incorporates the key effects of Wolbachia infection including cytoplasmic incompatibility and male killing. We also allow the possibility of reduced reproductive output, incomplete maternal transmission, and different mortality rates for uninfected/infected male/female individuals. We study the existence and local stability of equilibria, including the biologically relevant and interesting boundary equilibria. For some biologically relevant parameter regimes there may be multiple coexistence steady states including, very importantly, a coexistence steady state in which Wolbachia infected individuals dominate. We also extend the model to incorporate West Nile virus (WNv) dynamics, using an SEI modelling approach. Recent evidence suggests that a particular strain of Wolbachia infection significantly reduces WNv replication in Aedes aegypti. We model this via increased time spent in the WNv-exposed compartment for Wolbachia infected female mosquitoes. A basic reproduction number [Formula: see text] is computed for the WNv infection. Our results suggest that, if the mosquito population consists mainly of Wolbachia infected individuals, WNv eradication is likely if WNv replication in Wolbachia infected individuals is sufficiently reduced.

Finite difference approximations for a size-structured population model with distributed states in the recruitment.

We consider a size-structured population model where individuals may be recruited into the population at different sizes. First- and second-order finite difference schemes are developed to approximate the solution of the model. The convergence of the approximations to a unique weak solution is proved. We then show that as the distribution of the new recruits become concentrated at the smallest size, the weak solution of the distributed states-at-birth model converges to the weak solution of the classical Gurtin-McCamy-type size-structured model in the weak* topology. Numerical simulations are provided to demonstrate the achievement of the desired accuracy of the two methods for smooth solutions as well as the superior performance of the second-order method in resolving solution-discontinuities. Finally, we provide an example where supercritical Hopf-bifurcation occurs in the limiting single state-at-birth model and we apply the second-order numerical scheme to show that such bifurcation also occurs in the distributed model.

Pathogen evolution in switching environments: a hybrid dynamical system approach.

We propose a hybrid dynamical system approach to model the evolution of a pathogen that experiences different selective pressures according to a stochastic process. In every environment, the evolution of the pathogen is described by a version of the Fisher-Haldane-Wright equation while the switching between environments follows a Markov jump process. We investigate how the qualitative behavior of a simple single-host deterministic system changes when the stochastic switching process is added. In particular, we study the stability in probability of monomorphic equilibria. We prove that in a "constantly" fluctuating environment, the genotype with the highest mean fitness is asymptotically stable in probability while all others are unstable in probability. However, if the probability of host switching depends on the genotype composition of the population, polymorphism can be stably maintained.

Physiologically structured populations with diffusion and dynamic boundary conditions.

We consider a linear size-structured population model with diffusion in the size-space. Individuals are recruited into the population at arbitrary sizes. We equip the model with generalized Wentzell-Robin (or dynamic) boundary conditions. This approach allows the modelling of populations in which individuals may have distinguished physiological states. We establish existence and positivity of solutions by showing that solutions are governed by a positive quasicontractive semigroup of linear operators on the biologically relevant state space. These results are obtained by establishing dissipativity of a suitably perturbed semigroup generator. We also show that solutions of the model exhibit balanced exponential growth, that is, our model admits a finite-dimensional global attractor. In case of strictly positive fertility we are able to establish that solutions in fact exhibit asynchronous exponential growth.

Structured and unstructured continuous models for Wolbachia infections.

We introduce and investigate a series of models for an infection of a diplodiploid host species by the bacterial endosymbiont Wolbachia. The continuous models are characterized by partial vertical transmission, cytoplasmic incompatibility and fitness costs associated with the infection. A particular aspect of interest is competitions between mutually incompatible strains. We further introduce an age-structured model that takes into account different fertility and mortality rates at different stages of the life cycle of the individuals. With only a few parameters, the ordinary differential equation models exhibit already interesting dynamics and can be used to predict criteria under which a strain of bacteria is able to invade a population. Interestingly, but not surprisingly, the age-structured model shows significant differences concerning the existence and stability of equilibrium solutions compared to the unstructured model.