A diffusive SEIR model for community transmission of Covid-19 epidemics: application to Brazil

A mathematical model incorporating diffusion is developed to describe the spatial spread of COVID-19 epidemics in geographical regions. The dynamics of the spatial spread are based on community transmission of the virus. The model is applied to the outbreak of the COVID-19 epidemic in Brazil.

An extended SEIARD model for COVID-19 vaccination in Mexico: analysis and forecast

In this study, we propose and analyse an extended SEIARD model with vaccination. We compute the control reproduction number $\mathcal{R}_c$ of our model and study the stability of equilibria. We show that the set of disease-free equilibria is locally asymptotically stable when $\mathcal{R}_c<1$ and unstable when $\mathcal{R}_c>1$, and we provide a sufficient condition for its global stability. Furthermore, we perform numerical simulations using the reported data of COVID-19 infections and vaccination in Mexico to study the impact of different vaccination, transmission and efficacy rates on the dynamics of the disease.

Modeling SARS-CoV-2 spread with dynamic isolation

Background: The SARS-CoV-2 pandemic is spreading with a greater intensity across the globe. The synchrony of public health interventions and epidemic waves signify the importance of evaluation of the underline interventions. Method: We developed a mathematical model to present the transmission dynamics of SARS-CoV-2 and to analyze the impact of key nonpharmaceutical interventions such as isolation and screening program on the disease outcomes to the people of New Jersey, USA. We introduced a dynamic isolation of susceptible population with a constant (imposed) and infection oriented interventions. Epidemiological and demographic data are used to estimate the model parameters. The baseline case was explored further to showcase several critical and predictive scenarios. Results and analysis: The model simulations are in good agreement with the infection data for the period of 5 March 2020 to 31 January 2021. Dynamic isolation and screening program are found to be potential measures that can alter the course of epidemic. A 7% increase in isolation rate may result in a 31% reduction of epidemic peak whereas a 3 times increase in screening rate may reduce the epidemic peak by 35%. The model predicts that nearly 9.7% to 12% of the total population of New Jersey may become infected within the middle of July 2021 along with 24.6 to 27.3 thousand cumulative deaths. Within a wide spectrum of probable scenarios, there is a possibility of third wave Conclusion: Our findings could be informative to the public health community to contain the pandemic in the case of economy reopening under a limited or no vaccine coverage. Additional epidemic waves can be avoided by appropriate screening and isolation plans.

Cluster solutions in networks of weakly coupled oscillators on a 2D square torus

We consider a model for an N × N lattice network of weakly coupled neural oscilla- tors with periodic boundary conditions (2D square torus), where the coupling between neurons is assumed to be within a von Neumann neighborhood of size r, denoted as von Neumann r-neighborhood. Using the phase model reduction technique, we study the existence of cluster solutions with constant phase differences (Ψh, Ψv) between adjacent oscillators along the horizontal and vertical directions in our network, where Ψh and Ψv are not necessarily to be identical. Applying the Kronecker production representation and the circulant matrix theory, we develop a novel approach to analyze the stability of cluster solutions with constant phase difference (i.e., Ψh,Ψv are equal). We begin our analysis by deriving the precise conditions for stability of such cluster solutions with von Neumann 1-neighborhood and 2 neighborhood couplings, and then we generalize our result to von Neumann r-neighborhood coupling for arbitrary neighborhood size r ≥ 1. This developed approach for the stability analysis indeed can be extended to an arbitrary coupling in our network. Finally, numerical simulations are used to validate the above analytical results for various values of N and r by considering an inhibitory network of Morris-Lecar neurons.

Inverse Reconstruction of Cell Proliferation Laws in Cancer Invasion Modelling

The process of local cancer cell invasion of the surrounding tissue is key for the overall tumour growth and spread within the human body, the past 3 decades witnessing intense mathematical modelling efforts in these regards. However, for a deep understanding of the cancer invasion process these modelling studies require robust data assimilation approaches. While being of crucial importance in assimilating potential clinical data, the inverse problems approaches in cancer modelling are still in their early stages, with questions regarding the retrieval of the characteristics of tumour cells motility, cells mutations, and cells population proliferation, remaining widely open. This study deals with the identification and reconstruction of the usually unknown cancer cell proliferation law in cancer modelling from macroscopic tumour snapshot data collected at some later stage in the tumour evolution. Considering two basic tumour configurations, associated with the case of one cancer cells population and two cancer cells subpopulations that exercise their dynamics within the extracellular matrix, we combine Tikhonov regularisation and gaussian mollification approaches with finite element and finite differences approximations to reconstruct the proliferation laws for each of these sub-populations from both exact and noisy measurements. Our inverse problem formulation is accompanied by numerical examples for the reconstruction of several proliferation laws used in cancer growth modelling.

$m$th-order Fisher-KPP equation with free boundaries and time-aperiodic advection

In this paper, we investigate a $m$th-order Fisher-KPP equation with free boundaries and time-aperiodic advection. Considering the influence of advection term and initial conditions on the long time behavior of solutions, we obtain spreading-vanishing dichotomy, spreading-transition-vanishing trichotomy, and vanishing happens with the coefficient of advection term in small amplitude, medium-sized amplitude and large amplitude, respectively. Then, the appropriate parameters are selected in the simulation to intuitively show the corresponding theoretical results. Moreover, the wave-spreading and wave-vanishing cases of the solutions are observed in our study.

A robust phenomenological approach to investigate COVID-19 data for France

We provide a new method to analyze the COVID-19 cumulative reported cases data based on a two-step process: first we regularize the data by using a phenomenological model which takes into account the endemic or epidemic nature of the time period, then we use a mathematical model which reproduces the epidemic exactly. This allows us to derive new information on the epidemic parameters and to compute the effective basic reproductive ratio on a daily basis. Our method has the advantage of identifying robust trends in the number of new infectious cases and produces an extremely smooth reconstruction of the epidemic. The number of parameters required by the method is parsimonious: for the French epidemic between February 2020 and January 2021 we use only 11 parameters in total.

Global properties of a virus dynamics model with self-proliferation of CTLs

A viral infection model with self-proliferation of cytotoxic T lymphocytes (CTLs) is proposed and its global dynamics is obtained. When the per capita self-proliferation rate of CTLs is sufficient large, an infection-free but immunity-activated equilibrium always exists and is globally asymptotically stable if the basic reproduction number of virus is less than a threshold value, which means that the immune effect still exists though virus be eliminated. Qualitative numerical simulations further indicate that the increase of per capita self-proliferation rate may lead to more severe infection outcome, which may provide insight into the failure of immune therapy.

The replicator dynamics of generalized Nash games

Generalized Nash Games are a powerful modelling tool, first introduced in the 1950's. They have seen some important developments in the past two decades. Separately, Evolutionary Games were introduced in the 1960's and seek to describe how natural selection can drive phenotypic changes in interacting populations. In this paper, we show how the dynamics of these two independently formulated models can be linked under a common framework and how this framework can be used to expand Evolutionary Games. At the center of this unified model is the Replicator Equation and the relationship we establish between it and the lesser known Projected Dynamical System.

A COVID-19 epidemic model predicting the effectiveness of vaccination

A model of a COVID-19 epidemic is developed to predict the effectiveness of vaccination. The model incorporates key features of COVID-19 epidemics: asymptomatic and symptomatic infectiousness, reported and unreported cases, and social measures that decrease infection transmission. The model incorporates key features of vaccination: vaccination efficiency, vaccination scheduling, and relaxation of socialmeasures that decrease infection transmission as vaccination is implemented. The model is applied to predict vaccination effectiveness in the United Kingdom.