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We consider spatial population dynamics on a lattice, following a type of a contact (birth–death) stochastic process. We show that simple mathematical approximations for the density of cells can be obtained in a variety of scenarios. In the case of a homogeneous cell population, we derive the cellular density for a two-dimensional (2D) spatial lattice with an arbitrary number of neighbors, including the von Neumann, Moore, and hexagonal lattice. We then turn our attention to evolutionary dynamics, where mutant cells of different properties can be generated. For disadvantageous mutants, we derive an approximation for the equilibrium density representing the selection–mutation balance. For neutral and advantageous mutants, we show that simple scaling (power) laws for the numbers of mutants in expanding populations hold in 2D and 3D, under both flat (planar) and range population expansion. These models have relevance for studies in ecology and evolutionary biology, as well as biomedical applications including the dynamics of drug-resistant mutants in cancer and bacterial biofilms. 

To study viral evolutionary processes within patients, mathematical models have been instrumental. Yet, the need for stochastic simulations of minority mutant dynamics can pose computational challenges, especially in heterogeneous systems where very large and very small sub-populations coexist. Here, we describe a hybrid stochastic-deterministic algorithm to simulate mutant evolution in large viral populations, such as acute HIV-1 infection, and further include the multiple infection of cells. We demonstrate that the hybrid method can approximate the fully stochastic dynamics with sufficient accuracy at a fraction of the computational time, and quantify evolutionary end points that cannot be expressed by deterministic models, such as the mutant distribution or the probability of mutant existence at a given infected cell population size. We apply this method to study the role of multiple infection and intracellular interactions among different virus strains (such as complementation and interference) for mutant evolution. Multiple infection is predicted to increase the number of mutants at a given infected cell population size, due to a larger number of infection events. We further find that viral complementation can significantly enhance the spread of disadvantageous mutants, but only in select circumstances: it requires the occurrence of direct cell-to-cell transmission through virological synapses, as well as a substantial fitness disadvantage of the mutant, most likely corresponding to defective virus particles. This, however, likely has strong biological consequences because defective viruses can carry genetic diversity that can be incorporated into functional virus genomes via recombination. Through this mechanism, synaptic transmission in HIV might promote virus evolvability. 

Epidemiological data about SARS-CoV-2 spread indicate that the virus is not transmitted uniformly in the population. The transmission tends to be more effective in select settings that involve exposure to relatively high viral dose, such as in crowded indoor settings, assisted living facilities, prisons or food processing plants. To explore the effect on infection dynamics, we describe a new mathematical model where transmission can occur (i) in the community at large, characterized by low-dose exposure and mostly mild disease, and (ii) in so-called transmission hot zones, characterized by high-dose exposure that can be associated with more severe disease. The model yields different types of epidemiological dynamics, depending on the relative importance of hot zone and community transmission. Interesting dynamics occur if the rate of virus release/deposition from severely infected people is larger than that of mildly infected individuals. Under this assumption, we find that successful infection spread can hinge upon high-dose hot zone transmission, yet the majority of infections are predicted to occur in the community at large with mild disease. In this regime, residual hot zone transmission can account for continued virus spread during community lockdowns, and the suppression of hot zones after community interventions are relaxed can cause a prolonged lack of infection resurgence following the reopening of society. This gives rise to the notion that targeted interventions specifically reducing virus transmission in the hot zones have the potential to suppress overall infection spread, including in the community at large. Epidemiological trends in the USA and Europe are interpreted in light of this model. 

Abstract Recombination has been shown to contribute to human immunodeficiency virus-1 (HIV-1) evolution in vivo, but the underlying dynamics are extremely complex, depending on the nature of the fitness landscapes and of epistatic interactions. A less well-studied determinant of recombinant evolution is the mode of virus transmission in the cell population. HIV-1 can spread by free virus transmission, resulting largely in singly infected cells, and also by direct cell-to-cell transmission, resulting in the simultaneous infection of cells with multiple viruses. We investigate the contribution of these two transmission pathways to recombinant evolution, by applying mathematical models to in vitro experimental data on the growth of fluorescent reporter viruses under static conditions (where both transmission pathways operate), and under gentle shaking conditions, where cell-to-cell transmission is largely inhibited. The parameterized mathematical models are then used to extrapolate the viral evolutionary dynamics beyond the experimental settings. Assuming a fixed basic reproductive ratio of the virus (independent of transmission pathway), we find that recombinant evolution is fastest if virus spread is driven only by cell-to-cell transmission and slows down if both transmission pathways operate. Recombinant evolution is slowest if all virus spread occurs through free virus transmission. This is due to cell-to-cell transmission 1, increasing infection multiplicity; 2, promoting the co-transmission of different virus strains from cell to cell; and 3, increasing the rate at which point mutations are generated as a result of more reverse transcription events. This study further resulted in the estimation of various parameters that characterize these evolutionary processes. For example, we estimate that during cell-to-cell transmission, an average of three viruses successfully integrated into the target cell, which can significantly raise the infection multiplicity compared to free virus transmission. In general, our study points towards the importance of infection multiplicity and cell-to-cell transmission for HIV evolution. 

We have analysed the COVID-19 epidemic data of more than 174 countries (excluding China) in the period between 22 January and 28 March 2020. We found that some countries (such as the USA, the UK and Canada) follow an exponential epidemic growth, while others (like Italy and several other European countries) show a power law like growth. Regardless of the best fitting law, many countries can be shown to follow a common trajectory that is similar to Italy (the epicentre at the time of analysis), but with varying degrees of delay. We found that countries with ‘younger’ epidemics, i.e. countries where the epidemic started more recently, tend to exhibit more exponential like behaviour, while countries that were closer behind Italy tend to follow a power law growth. We hypothesize that there is a universal growth pattern of this infection that starts off as exponential and subsequently becomes more power law like. Although it cannot be excluded that this growth pattern is a consequence of social distancing measures, an alternative explanation is that it is an intrinsic epidemic growth law, dictated by a spatially distributed community structure, where the growth in individual highly mixed communities is exponential but the longer term, local geographical spread (in the absence of global mixing) results in a power law. This is supported by computer simulations of a metapopulation model that gives rise to predictions about the growth dynamics that are consistent with correlations found in the epidemiological data. Therefore, seeing a deviation from straight exponential growth may be a natural progression of the epidemic in each country. On the practical side, this indicates that (i) even in the absence of strict social distancing interventions, exponential growth is not an accurate predictor of longer term infection spread, and (ii) a deviation from exponential spread and a reduction of estimated doubling times do not necessarily indicate successful interventions, which are instead indicated by a transition to a reduced power or by a deviation from power law behaviour. 

Mutant evolution in spatially structured systems is important for a range of biological systems, but aspects of it still require further elucidation. Adding to previous work, we provide a simple derivation of growth laws that characterize the number of mutants of different relative fitness in expanding populations in spatial models of different dimensionalities. These laws are universal and independent of "microscopic" modeling details. We further study the accumulation of mutants and find that with advantageous and neutral mutants, more of them are present in spatially structured, compared to well-mixed colonies of the same size. The behavior of disadvantageous mutants is subtle: if they are disadvantageous through a reduction in division rates, the result is the same, and it is the opposite if the disadvantage is due to a death rate increase. Finally, we show that in all cases, the same results are observed in fragmented, non-spatial patch models. This suggests that the patterns observed are the consequence of population fragmentation, and not spatial restrictions per se. We provide an intuitive explanation for the complex dependence of disadvantageous mutant evolution on spatial restriction, which relies on desynchronized dynamics in different locations/patches, and plays out differently depending on whether the disadvantage is due to a lower division rate or a higher death rate. Implications for specific biological systems, such as the evolution of drug-resistant cell mutants in cancer or bacterial biofilms, are discussed.