Homogeneous Branching Processes with Non-Homogeneous Immigration

The stationary immigration has a limited effect over the asymptotic behavior of the underlying branching process. It affects mostly the limiting distribution and the life-period of the process. In contrast, if the immigration rate changes over time, then the asymptotic behavior of the process is significantly different and a variety of new phenomena are observed. In this review we discuss branching processes with time non-homogeneous immigration. Our goal is to help researchers interested in the topic to familiarize themselves with the current state of research.

Modelling for The COVID-19 with The Contacting Distance

The COVID-19, which belongs to the family of Coronaviridae and is large-scale outbreak in the whole world, is a public health emergency for human beings and brings some very harmful consequences in social and economic fields. In order to modelling the COVID-19 and develop the efficient control method corresponding to the contacting distance, this paper proposes an SEIR-type epidemic model with the contacting distance between the healthy individuals and the asymptomatic or symptomatic infected individuals, and the immigration rate of the healthy individuals, since the contacting distance and the immigration rate are two critical factors which determine the transmission of the COVID-19. Firstly, the threshold contacting distance and the threshold immigration rate are obtained by analyze the dynamical behaviors of the proposed SEIR-type epidemic model in order to control the COVID-19. The effect of the contacting distance and the immigration rate on the control of the COVID-19 are revealed based on ecological and epidemiological issues. The results show that the COVID-19 will be controlled while the contacting distance between the healthy individuals and the symptomatic infected individuals is larger than the threshold value A~ and the immigration rate is smaller than the threshold value ~ d. Secondly, the sensitivity analysis is conducted and the results show that the contacting distance and the immigration rate play an important role in controlling the COVID-19. Finally, the numerical test for Wuhan city are conducted and the conclusions show that the extinct lag decreases as the the contacting distance increase or the immigration rate decrease. Our study could give some reasonable suggestions for the health officials and the public.

Modelling for the COVID-19 with the Contacting Distance

The COVID-19, which belongs to the family of Coronaviridae and is large-scale outbreak in the whole world, is a public health emergency for human beings and brings some very harmful consequences in social and economic fields. In order to modelling the COVID-19 and develop the efficient control method corresponding to the contacting distance, this paper proposes an SEIR-type epidemic model with the contacting distance between the healthy individuals and the asymptomatic or symptomatic infected individuals, and the immigration rate of the healthy individuals, since the contacting distance and the immigration rate are two critical factors which determine the transmission of the COVID-19. Firstly, the threshold contacting distance and the threshold immigration rate are obtained by analyze the dynamical behaviors of the proposed SEIR-type epidemic model in order to control the COVID-19. The effect of the contacting distance and the immigration rate on the control of the COVID-19 are revealed based on ecological and epidemiological issues. The results show that the COVID-19 will be controlled while the contacting distance between the healthy individuals and the symptomatic infected individuals is larger than the threshold value $\tilde{d}^*$ and the immigration rate is smaller than the threshold value $\tilde{A}^*$. Secondly, the sensitivity analysis is conducted and the results show that the contacting distance and the immigration rate play an important role in controlling the COVID-19. Finally, the numerical test for Wuhan city are conducted and the conclusions show that the extinct lag decreases as the the contacting distance increase or the immigration rate decrease. Our study could give some reasonable suggestions for the health officials and the public.

A Radical Right-Wing Failure in Canada

This article is a quantitative investigation into why Maxime Bernier’s People’s Party of Canada (PPC), a radical right-wing party (RRP), failed to succeed in the 2019 Canadian federal election. Canada has not witnessed the electoral breakthrough of such a party. I argue the failure of the PPC was the result of a mixture of the stabilization of immigrant inflows and the softening of anti-immigrant sentiment. More favourable conditions for the PPC, including extensive media coverage and increasing support for populist and mildly authoritarian sentiment, may have been necessary, but were not sufficient alone to allow for an RRP breakthrough. RRPs are unlikely to succeed in Canada as long as the immigration rate remains predictable and Canadians continue to hold favourable views towards immigrants.

Stochastic modelling of age-structured population with time and size dependence of immigration rate

A stochastic age-structured population model with immigration of individuals is considered. We assume that the lifespan of each individual is a random variable with a distribution function which may differ fromthe exponential one. The immigration rate of individuals depends on the time and total population size. Upper estimates for the mean and variance of the population size are established based on the theory of branching processes with constant immigration rate. A Monte Carlo simulation algorithm of population dynamics is developed. The results of numerical experiments with the model are presented.

When sinks become sources: adaptive colonization in asexuals

The successful establishment of a population into a new empty habitat outside of its initial niche is a phenomenon akin to evolutionary rescue in the presence of immigration. It underlies a wide range of processes, such as biological invasions by alien organisms, host shifts in pathogens or the emergence of resistance to pesticides or antibiotics from untreated areas.In this study, we derive an analytically tractable framework to describe the coupled evolutionary and demographic dynamics of asexual populations in a source-sink system. In particular, we analyze the influence of several factors — immigration rate, mutational parameters, and harshness of the stress induced by the change of environment — on the establishment success in the sink (i.e. the formation of a self-sufficient population in the sink), and on the time until establishment. To this aim, we use a classic phenotype-fitness landscape (Fisher’s geometrical model in n dimensions) where source and sink habitats determine distinct phenotypic optima. The harshness of stress, in the sink, is determined by the distance between the fitness optimum in the sink and that of the source. The dynamics of the full distribution of fitness and of population size in the sink are analytically predicted under a strong mutation strong immigration limit where the population is always polymorphic.The resulting eco-evolutionary dynamics depend on mutation and immigration rates in a non straightforward way. Below some mutation rate threshold, establishment always occurs in the sink, following a typical four-phases trajectory of the mean fitness. The waiting time to this establishment is independent of the immigration rate and decreases with the mutation rate. Beyond the mutation rate threshold, lethal mutagenesis impedes establishment and the sink population remains so, albeit with an equilibrium state that depends on the details of the fitness landscape. We use these results to get some insight into possible effects of several management strategies.

An island biogeography model for beta diversity and endemism: The roles of speciation, extinction and dispersal

A community composition island biogeography model was developed to explain and predict two community patterns (beta diversity and endemism) with the consideration of speciation, extinction and dispersal processes. Results showed that rate of speciation is positively and linearly associated with beta diversity and endemism, that is, increasing species rates typically could increase the percentage of both endemism and beta diversity. The influences of immigration and extinction rates on beta diversity and endemism are nonlinear, but with numerical simulation, I could observe that increasing extinction rates would lead to decreasing percentage of endemism and beta diversity. The role of immigration rate is very similar to that of speciation rate, having a positive relationship with beta diversity and endemism. Finally, I found that beta diversity is closely related to the percentage of endemism. The slope of this positive relationship is determined jointly by different combinations of speciation, extinction and immigration rates.