Quantitative Biology

The exact analytical solution in closed form of a modified SIR system where recovered individuals are removed from the population is presented. In this dynamical system the populations S(t) and R(t) of susceptible and recovered individuals are found to be generalized logistic functions, while infective ones I(t) are given by a generalized logistic function times an exponential, all of them with the same characteristic time. The dynamics of this modified SIR system is analyzed and the exact computation of some epidemiologically relevant quantities is performed. The main differences between this modified SIR model and original SIR one are presented and explained in terms of the zeroes of their respective conserved quantities. Moreover, it is shown that the modified SIR model with time-dependent transmission rate can be also solved in closed form for certain realistic transmission rate functions.

The site frequency spectrum (SFS) is a popular summary statistic of genomic data. While the SFS of a constant-sized population undergoing neutral mutations has been extensively studied in population genetics, the rapidly growing amount of cancer genomic data has attracted interest in the spectrum of an exponentially growing population. Recent theoretical results have generally dealt with special or limiting cases, such as considering only cells with an infinite line of descent, assuming deterministic tumor growth, or sending the tumor size to infinity. In this work, we we derive exact expressions for the expected SFS of a cell population that evolves according to a stochastic branching process, first for cells with an infinite line of descent and then for the total population, evaluated either at a fixed time (fixed-time spectrum) or at the stochastic time at which the population reaches a certain size (fixed-size spectrum). We find that while the rate of mutation scales the SFS of the total population linearly, the rates of cell birth and cell death change the shape of the spectrum at the small-frequency end, inducing a transition between a 1/ j 2 power-law spectrum and a 1/j spectrum as cell viability decreases. We use this insight to propose a simple estimator for the ratio between the rate of cell death and cell birth, that we show to accurately recover the true ratio from synthetic single-cell sequencing data. Although the discussion is framed in terms of tumor dynamics, our results apply to any exponentially growing population of individuals undergoing neutral mutations.

Objective: To make an estimate of the excess deaths caused by COVID-19 in the non-violent mortality of Peru, controlling for the effect of quarantine. Methods: Analysis of longitudinal data from the departments of Peru using official public information from the National Death Information System and the Ministry of Health of Peru. The analysis is performed between January 1, 2018 and June 23, 2020 (100 days of quarantine). The daily death rate per million inhabitants has been used. The days in which the departments were quarantined with a limit number of accumulated cases of COVID-19 were used to estimate the quarantine impact. Three limits were established for cases: less than 1, 10 and 100 cases. Result: In Peru, the daily death rate per million inhabitants decreased by -1.89 (95% CI: -2.70; -1.07) on quarantine days and without COVID-19 cases. When comparing this result with the total number of non-violent deaths, the excess deaths during the first 100 days of quarantine is 36,230. This estimate is 1.12 times the estimate with data from 2019 and 4.2 times the deaths officers by COVID-19. Conclusion: Quarantine reduced nonviolent deaths; however, they are overshadowed by the increase as a direct or indirect cause of the pandemic. Therefore, the difference between the number of current deaths and that of past years underestimates the real excess of deaths.

For classic Lotka-Volterra systems governing many interacting species, we establish an exclusion principle that rules out the existence of linearly asymptotically stable steady states in subcommunitites of communities that admit a stable state which is internally D-stable. This type of stability is known to be ensured, e.g., by diagonal dominance or Volterra-Lyapunov stability conditions. By consequence, the number of stable steady states of this type is bounded by Sperner's lemma on anti-chains in a poset. The number of stable steady states can nevertheless be very large if there are many groups of species that strongly inhibit outsiders but have weak interactions among themselves. By examples we also show that in general it is possible for a stable community to contain a stable subcommunity consisting of a single species. Thus a recent empirical finding to the contrary, in a study of random competitive systems by Lischke and Löffler (Theo. Pop. Biol. 115 (2017) 24--34), does not hold without qualification.

Microorganisms live in environments that inevitably fluctuate between mild and harsh conditions. As harsh conditions may cause extinctions, the rate at which fluctuations occur can shape microbial communities and their diversity, but we still lack an intuition on how. Here, we build a mathematical model describing two microbial species living in an environment where substrate supplies randomly switch between abundant and scarce. We then vary the rate of switching as well as different properties of the interacting species, and measure the probability of the weaker species driving the stronger one extinct. We find that this probability increases with the strength of demographic noise, and peaks at either low, high, or intermediate switching rates depending on both species' ability to withstand the harsh environment. This complex relationship shows why finding patterns between environmental fluctuations and diversity has historically been difficult: response to fluctuations depends on species' properties. In parameter ranges where the fittest species was most likely to be excluded, however, the beta diversity in larger communities also peaked. In sum, while we find no simple rules on how the frequency of fluctuations shapes the diversity of a community, we show that their effect on interactions between two representative species predicts how diversity in the whole community will change.

We study the evolutionary dynamics of the prisoner's dilemma game in which cooperators and defectors interact with another actor type called exiters. Rather than being exploited by defectors, exiters exit the game in favour of a small payoff. We find that this simple extension of the game allows cooperation to flourish in well-mixed populations when iterations or reputation are added. In networked populations, however, the exit option is less conducive to cooperation. Instead, it enables the coexistence of cooperators, defectors, and exiters through cyclic dominance. Other outcomes are also possible as the exit payoff increases or the network structure changes, including network-wide oscillations in actor abundances that may cause the extinction of exiters and the domination of defectors, although game parameters should favour exiting. The complex dynamics that emerges in the wake of a simple option to exit the game implies that nuances matter even if our analyses are restricted to incentives for rational behaviour.

Stochastic discrete-time SIS and SIR models of endemic diseases are introduced and analyzed. For the deterministic, mean-field model, the basic reproductive number R 0 determines their global dynamics. If R 0 ≤1 , then the frequency of infected individuals asymptotically converges to zero. If R 0 >1 , then the infectious class uniformly persists for all time; conditions for a globally stable, endemic equilibrium are given. In contrast, the infection goes extinct in finite time with probability one in the stochastic models for all R 0 values. To understand the length of the transient prior to extinction as well as the behavior of the transients, the quasi-stationary distributions and the associated mean time to extinction are analyzed using large deviation methods. When R 0 >1 , these mean times to extinction are shown to increase exponentially with the population size N . Moreover, as N approaches ∞ , the quasi-stationary distributions are supported by a compact set bounded away from extinction; sufficient conditions for convergence to a Dirac measure at the endemic equilibrium of the deterministic model are also given. In contrast, when R 0 <1 , the mean times to extinction are bounded above 1/(1−α) where α<1 is the geometric rate of decrease of the infection when rare; as N approaches ∞ , the quasi-stationary distributions converge to a Dirac measure at the disease-free equilibrium for the deterministic model. For several special cases, explicit formulas for approximating the quasi-stationary distribution and the associated mean extinction are given. These formulas illustrate how for arbitrarily small R 0 values, the mean time to extinction can be arbitrarily large, and how for arbitrarily large R 0 values, the mean time to extinction can be arbitrarily large.

Forecasting the effect of COVID-19 is essential to design policies that may prepare us to handle the pandemic. Many methods have already been proposed, particularly, to forecast reported cases and deaths at country-level and state-level. Many of these methods are based on traditional epidemiological model which rely on simulations or Bayesian inference to simultaneously learn many parameters at a time. This makes them prone to over-fitting and slow execution. We propose an extension to our model SIkJ α to forecast deaths and show that it can consider the effect of many complexities of the epidemic process and yet be simplified to a few parameters that are learned using fast linear regressions. We also present an evaluation of our method against seven approaches currently being used by the CDC, based on their two weeks forecast at various times during the pandemic. We demonstrate that our method achieves better root mean squared error compared to these seven approaches during majority of the evaluation period. Further, on a 2 core desktop machine, our approach takes only 3.18s to tune hyper-parameters, learn parameters and generate 100 days of forecasts of reported cases and deaths for all the states in the US. The total execution time for 184 countries is 11.83s and for all the US counties ( > 3000) is 101.03s.

Reproduction numbers, like the basic reproduction number R 0 , play an important role in the analysis and application of dynamic models, including contagion models and ecological population models. One difficulty in deriving these quantities is that they must be computed on a model-by-model basis, since it is typically impractical to obtain general reproduction number expressions applicable to a family of related models, especially if these are of different dimensions. For example, this is typically the case for SIR-type infectious disease models derived using the linear chain trick (LCT). Here we show how to find general reproduction number expressions for such models families (which vary in their number of state variables) using the next generation operator approach in conjunction with the generalized linear chain trick (GLCT). We further show how the GLCT enables modelers to draw insights from these results by leveraging theory and intuition from continuous time Markov chains (CTMCs) and their absorption time distributions (i.e., phase-type probability distributions). To do this, we first review the GLCT and other connections between mean-field ODE model assumptions, CTMCs, and phase-type distributions. We then apply this technique to find reproduction numbers for two sets of models: a family of generalized SEIRS models of arbitrary finite dimension, and a generalized family of finite dimensional predator-prey (Rosenzweig-MacArthur type) models. These results highlight the utility of the GLCT for the derivation and analysis of mean field ODE models, especially when used in conjunction with theory from CTMCs and their associated phase-type distributions.

The decision of whether or not to vaccinate is a complex one. It involves the contribution both to a social good -- herd immunity -- and to one's own well being. It is informed by social influence, personal experience, education, and mass media. In our work, we investigate a situation in which individuals make their choice based on how social neighbourhood responded to previous epidemics. We do this by proposing a minimalistic model using components from game theory, network theory and the modelling of epidemic spreading, and opinion dynamics. Individuals can use the information about the neighbourhood in two ways -- either they follow the majority or the best-performing neighbour. Furthermore, we let individuals learn which of these two decision-making strategies to follow from their experience. Our results show that the flexibility of individuals to chose how to integrate information from the neighbourhood increases the vaccine uptake and decreases the epidemic severity if the following conditions are fulfilled. First, the initial fraction of individuals who imitate the neighbourhood majority should be limited, and second, the memory of previous outbreaks should be sufficiently long. These results have implications for the acceptance of novel vaccines and raising awareness about vaccination, while also pointing to promising future research directions.