Analytical properties of a three-compartmental dynamical demographic model

Hamiltonian structure of compartmental epidemiological models

Any epidemiological compartmental model with constant population is shown to be a Hamiltonian dynamical system in which the total population plays the role of the Hamiltonian function. Moreover, some particular cases within this large class of models are shown to be bi-Hamiltonian. New interacting compartmental models among different populations, which are endowed with a Hamiltonian structure, are introduced. The Poisson structures underlying the Hamiltonian description of all these dynamical systems are explicitly presented, and their associated Casimir functions are shown to provide an effic…

Robust parameter determination in epidemic models with analytical descriptions of uncertainties

Compartmental equations are primary tools in disease spreading studies. Their predictions are accurate for large populations but disagree with empirical and simulated data for finite populations, where uncertainties become a relevant factor. Starting from the agent-based approach, we investigate the role of uncertainties and autocorrelation functions in SIS epidemic model, including their relationship with epidemiological variables. We find new differential equations that take uncertainties into account. The findings provide improved predictions to the SIS model and it can offer new insights f…

Extinction in a self-regulating population with demographic and environmental noise

We present an explicit unified stochastic model of fluctuations in population size due to random birth, death, density-dependent competition and environmental fluctuations. Stochastic dynamics provide insight into small populations, including processes such as extinction, that cannot be correctly treated by deterministic methods. We present exact analytical and simulation-based results for extinction times of our stochastic model and compare the different effects of environmental stochasticity and intrinsic demographic stochasticity. We use both the discrete master equation approach and an exa…

A reductive analysis of a compartmental model for COVID-19: data assimilation and forecasting for the United Kingdom

We introduce a deterministic model that partitions the total population into the susceptible, infected, quarantined, and those traced after exposure, the recovered and the deceased. We hypothesize 'accessible population for transmission of the disease' to be a small fraction of the total population, for instance when interventions are in force. This hypothesis, together with the structure of the set of coupled nonlinear ordinary differential equations for the populations, allows us to decouple the equations into just two equations. This further reduces to a logistic type of equation for the to…

具有继发性感染的 SIRS 模型的几何分析

A geometric analysis of the SIRS model with secondary infections

本文构建含暂时免疫与继发性感染的三时间尺度 SIRS 仓室模型，分析平衡点稳定性、$\mathcal{R}_0<1$ 时双地方病平衡及尺度相互作用，并以模拟和分岔分析验证。

Intrinsic noise in game dynamical learning

Demographic noise has profound effects on evolutionary and population dynamics, as well as on chemical reaction systems and models of epidemiology. Such noise is intrinsic and due to the discreteness of the dynamics in finite populations. We here show that similar noise-sustained trajectories arise in game dynamical learning, where the stochasticity has a different origin: agents sample a finite number of moves of their opponents in-between adaptation events. The limit of infinite batches results in deterministic modified replicator equations, whereas finite sampling leads to a stochastic dyna…