On a three-dimensional and two four-dimensional oncolytic viro-therapy models

We revisit here and carry out further works on tumor-virotherapy compartmental models of [Tian, 2011, Wang et al., 2013, Phan and Tian, 2017, Guo et al., 2019]. The results of these papers are only slightly pushed further. However, what is new is the fact that we make public our electronic notebooks, since we believe that easy electronic reproducibility is crucial in an era in which the role of the software becomes very important.Comment: 41 pages, 15 figure

Permanence in seasonal ecological models with spatial heterogeneity

Reaction-Diffusion systems are frequently used to model the population dynamics of interacting biological species in bounded spatial regions. The local population growth law for the density of each species in such a system depends in general on time, spatial location and the density of each species. This work examines the question of longterm coexistence in such models in the case in which the growth laws are periodic in time. The criterion for longterm coexistence we employ is the existence of positive (in a sense appropriate to the boundary conditions we impose on the system) asymptotic upper and lower bounds on each component of the system, a criterion we call permanence which is related to a dynamical systems concept called Abstract Permanence . In this work we establish sufficient conditions for permanence phenomena in several cases. Our approach is to reformulate the system of differential equations as a semidynamical system, apply machinery available in this context to obtain abstract permanence and then to show that abstract permanence implies the aforementioned upper and lower asymptotic bounds on the components of the system. The conditions that we ultimately derive for permanence are expressed in Quantifiable ways in terms of the spectra of linear differential operators associated with the original reaction-diffusion system. In so doing, we connect asymptotic coexistence in such a system to the underlying biological assumptions about the model which are expressed in the parameters and coefficients of these operators. We illustrate our results via two species predator-prey models and three-species competition models. We also show that permanence implies the existence of a componentwise positive periodic orbit (which is not necessary stable)

Modeling the Transmission of the SARS-CoV-2 Delta Variant in a Partially Vaccinated Population

In a population with ongoing vaccination, the trajectory of a pandemic is determined by how the virus spreads in unvaccinated and vaccinated individuals that exhibit distinct transmission dynamics based on different levels of natural and vaccine-induced immunity. We developed a mathematical model that considers both subpopulations and immunity parameters, including vaccination rates, vaccine effectiveness, and a gradual loss of protection. The model forecasted the spread of the SARS-CoV-2 delta variant in the US under varied transmission and vaccination rates. We further obtained the control reproduction number and conducted sensitivity analyses to determine how each parameter may affect virus transmission. Although our model has several limitations, the number of infected individuals was shown to be a magnitude greater (~10×) in the unvaccinated subpopulation compared to the vaccinated subpopulation. Our results show that a combination of strengthening vaccine-induced immunity and preventative behavioral measures like face mask-wearing and contact tracing will likely be required to deaccelerate the spread of infectious SARS-CoV-2 variants

Asymptotic Behavior of a Competition-Diffusion System with Variable Coefficients and Time Delays

A class of time-delay reaction-diffusion systems with variable coefficients which arise from the model of two competing ecological species is discussed. An asymptotic global attractor is established in terms of the variable coefficients, independent of the time delays and the effect of diffusion by the upper-lower solutions and iteration method