Quantitative Biology

As Europe is facing the second wave of the CoViD-19 pandemic, each country should carefully review how it dealt with the first wave of outbreak. Lessons from the first experience should be useful to avoid indiscriminate closures and, above all, to determine universal (understandable) parameters to guide the introduction of containment measures to reduce the spreading of the virus. The use of few (effective) parameters is indeed of extreme importance to create a link between authorities and population, allowing the latter to understand the reason for some restrictions and, consequently, to allow an active participation in the fight against the pandemic. Testing strategies, fitting skew parameters (as mean, mode, standard deviation, and skewness), mortality rates, and weekly CoViD-19 spreading data, as more people are getting infected, were used to compare the first wave of the outbreak in the Italian regions and to determine which parameters have to be checked before introducing restrictive containment measures. We propose few \textit{universal} parameters that, once appropriately weighed, could be useful to correctly differentiate the pandemic situation in the national territory and to rapidly assign the properly pandemic risk to each region.

We present a modified age-structured SIR model based on known patterns of social contact and distancing measures within Washington, USA. We find that population age-distribution has a significant effect on disease spread and mortality rate, and contribute to the efficacy of age-specific contact and treatment measures. We consider the effect of relaxing restrictions across less vulnerable age-brackets, comparing results across selected groups of varying population parameters. Moreover, we analyze the mitigating effects of vaccinations and examine the effectiveness of age-targeted distributions. Lastly, we explore how our model can applied to other states to reflect social-distancing policy based on different parameters and metrics.

This paper is devoted to the multidisciplinary modelling of a pandemic initiated by an aggressive virus, specifically the so-called \textit{SARS--CoV--2 Severe Acute Respiratory Syndrome, corona virus n.2}. The study is developed within a multiscale framework accounting for the interaction of different spatial scales, from the small scale of the virus itself and cells, to the large scale of individuals and further up to the collective behaviour of populations. An interdisciplinary vision is developed thanks to the contributions of epidemiologists, immunologists and economists as well as those of mathematical modellers. The first part of the contents is devoted to understanding the complex features of the system and to the design of a modelling rationale. The modelling approach is treated in the second part of the paper by showing both how the virus propagates into infected individuals, successfully and not successfully recovered, and also the spatial patterns, which are subsequently studied by kinetic and lattice models. The third part reports the contribution of research in the fields of virology, epidemiology, immune competition, and economy focused also on social behaviours. Finally, a critical analysis is proposed looking ahead to research perspectives.

Contact tracing and quarantine are well established non-pharmaceutical epidemic control tools. The paper aims to clarify the impact of these measures in COVID-19 epidemic. A new deterministic model is introduced (SEIRQ: susceptible, exposed, infectious, removed, quarantined) with Q compartment capturing individuals and releasing them with delay. We obtain a simple rule defining the reproduction number R in terms of quarantine parameters, ratio of diagnosed cases and transmission parameters. The model is applied to the epidemic in Poland in March - April 2020, when social distancing measures were in place. We investigate 3 scenarios corresponding to different ratios of diagnosed cases. Our results show that depending on the scenario contact tracing could have prevented from 50\% to over 90\% of cases. The effects of quarantine are limited by fraction of undiagnosed cases. Taking into account the transmission intensity in Poland prior to introduction of social restrictions it is unlikely that the control of the epidemic could be achieved without any social distancing measures.

The paradigm for compartment models in epidemiology assumes exponentially distributed incubation and removal times, which is not realistic in actual populations. Commonly used variations with multiple exponentially distributed variables are more flexible, yet do not allow for arbitrary distributions. We present a new formulation, focussing on the SEIR concept that allows to include general distributions of incubation and removal times. We compare the solution to two types of agent-based model simulations, a spatially homogeneous one where infection occurs by proximity, and a model on a scale-free network with varying clustering properties, where the infection between any two agents occurs via their link if it exists. We find good agreement in both cases. Furthermore a family of asymptotic solutions of the equations is found in terms of a logistic curve, which after a non-universal time shift, fits extremely well all the microdynamical simulations. The formulation allows for a simple numerical approach; software in Julia and Python is provided.

Parrondo's paradox indicates a paradoxical situation in which a winning expectation may occur in sequences of losing games. There are many versions of the original Parrondo's games in the literature, but the games are played by two players in all of them. We introduce a new extended version of games played by three players and a three-sided biased dice instead of two players and a biased coin in this work. In the first step, we find the part of the parameters space where the games are played fairly. After adding noise to fair probabilities, we combine two games randomly, periodically, and nonlinearly and obtain the conditions under which the paradox can occur. This generalized model can be applied in all science and engineering fields. It can also be used for genetic switches. Genetic switches are often made by two reactive elements, but the existence of more elements can lead to more existing decisions for cells. Each genetic switch can be considered a game in which the reactive elements compete to increase their molecular concentrations. We present three genetic networks based on a new generalized Parrondo's games model, consisting of two noisy genetic switches. The combination of them can increase network robustness to noise. Each switch can also be used as an initial pattern to construct a synthetic switch to change undesirable cells' fate.

In a given country, the cumulative death toll of the first wave of the COVID-19 epidemic follows a sigmoid curve as a function of time. In most cases, the curve is well described by the Gompertz function, which is characterized by two essential parameters, the initial growth rate and the decay rate as the first epidemic wave subsides. These parameters are determined by socioeconomic factors and the countermeasures to halt the epidemic. The Gompertz model implies that the total death toll depends exponentially, and hence very sensitively, on the ratio between these rates. The remarkably different epidemic curves for the first epidemic wave in Sweden and Norway and many other countries are classified and discussed in this framework, and their usefulness for the planning of mitigation strategies is discussed.

Raw data on the cumulative number of deaths at a country level generally indicate a spatially variable distribution of the incidence of COVID-19 disease. An important issue is to determine whether this spatial pattern is a consequence of environmental heterogeneities, such as the climatic conditions, during the course of the outbreak. Another fundamental issue is to understand the spatial spreading of COVID-19. To address these questions, we consider four candidate epidemiological models with varying complexity in terms of initial conditions, contact rates and non-local transmissions, and we fit them to French mortality data with a mixed probabilistic-ODE approach. Using standard statistical criteria, we select the model with non-local transmission corresponding to a diffusion on the graph of counties that depends on the geographic proximity, with time-dependent contact rate and spatially constant parameters. This original spatially parsimonious model suggests that in a geographically middle size centralized country such as France, once the epidemic is established, the effect of global processes such as restriction policies, sanitary measures and social distancing overwhelms the effect of local factors. Additionally, this modeling approach reveals the latent epidemiological dynamics including the local level of immunity, and allows us to evaluate the role of non-local interactions on the future spread of the disease. In view of its theoretical and numerical simplicity and its ability to accurately track the COVID-19 epidemic curves, the framework we develop here, in particular the non-local model and the associated estimation procedure, is of general interest in studying spatial dynamics of epidemics.

Developing feasible strategies and setting realistic targets for disease prevention and control depends on representative models, whether conceptual, experimental, logistical or mathematical. Mathematical modelling was established in infectious diseases over a century ago, with the seminal works of Ross and others. Propelled by the discovery of etiological agents for infectious diseases, and Koch's postulates, models have focused on the complexities of pathogen transmission and evolution to understand and predict disease trends in greater depth. This has led to their adoption by policy makers; however, as model-informed policies are being implemented, the inaccuracies of some predictions are increasingly apparent, most notably their tendency to overestimate the impact of control interventions. Here, we discuss how these discrepancies could be explained by methodological limitations in capturing the effects of heterogeneity in real-world systems. We suggest that improvements could derive from theory developed in demography to study variation in life-expectancy and ageing. Using simulations, we illustrate the problem and its impact, and formulate a pragmatic way forward.

Countries highly exposed to incoming traffic from China were expected to be at the highest risk of COVID-19 spread. However, COVID-19 case numbers (infection levels) are negatively correlated with incoming traffic-level. Moreover, infection levels are positively correlated with population-size, while the latter should only affect infection-level once herd immunity is reached. These could be explained if a low-virulence strain (LVS) began spreading a few months earlier from China, providing immunity from the later emerging known SARS-CoV-2 high-virulence strain (HVS). We find that the dynamics of the COVID-19 pandemic depend on the LVS and HVS spread doubling-times and the delay between their initial onsets. We find that LVS doubling-time to be T L ∼1.59±0.17 times slower than the HVS ( T H ), but its earlier onset allowed its global wide-spread to the levels required for herd-immunity. In countries exposed earlier to the LVS and/or having smaller population-size, the LVS achieved herd-immunity earlier, allowing less time for the spread of the HVS, and giving rise to lower HVS-infection levels. Such model accurately predicts a country's infection-level ({\rm R^{2}=0.74}; p-value of {\rm 5.2\times10^{-13}}), given only its population-size and incoming-traffic from China. It explains the negative correlation with incoming-traffic ( c exp ), the positive correlation with the population size (n_{pop}) and their specific relations ( N cases ∝ n T L / T H pop × c T L / T H −1 exp ). We find that most countries should have already achieved herd-immunity. Further COVID-19-spread in these countries is limited and is not expected to rise by more than a factor of 2-3. We suggest tests/predictions to further verify the model and biologically identify the LVS, and discuss the implications.