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We analyze an epidemiological model with treatment and recruitment considering the risk perception. In this model, we consider an exponential function as a recruitment rate. We have found that this model undergoes the transcritical Bogdanov-Takens bifurcation with boundary, where the system experiences the transcritical bifurcation between the disease-free equilibrium point and the endemic equilibrium point. The Hopf bifurcation also arises at the endemic equilibrium point, this is, the appearance or disappearance of a limit cycle, and finally, the Homoclinic bifurcation which transforms the limit cycle into a homoclinic cycle, starting and ending at the disease-free equilibrium point.

Since the start of the still ongoing COVID-19 pandemic, there have been many modeling efforts to assess several issues of importance to public health. In this work, we review the theory behind some important mathematical models that have been used to answer questions raised by the development of the pandemic. We start revisiting the basic properties of simple Kermack-McKendrick type models. Then, we discuss extensions of such models and important epidemiological quantities applied to investigate the role of heterogeneity in disease transmission e.g. mixing functions and superspreading events, the impact of non-pharmaceutical interventions in the control of the pandemic, vaccine deployment, herd-immunity, viral evolution and the possibility of vaccine escape. From the perspective of mathematical epidemiology, we highlight the important properties, findings, and, of course, deficiencies, that all these models have., Comment: 42 pages, 4 figures

In this work we look at several mathematical models that have been constructed during the present pandemic to address dfferent issues of importance to public health policies about epidemic scenarios and thier causes. We start by briefly reviewing the most basic properties of the classic Kermack-McKendrick models and then proceed to look at some generalizations and their applications to contact structures, cocirculation of viral infections, growth patterns of epidemic curves, characterization of probability distributions and passage times necessary to parametrize the compartments that are added to the basic Kermack-McKendrick model. All of these examples revolve around the idea that a system of differential equations constructed for an specifc epidemiological problem, has as central and main theoretical and conceptual support the epidemiological, medical and biological context that motivates its construction and analysis. Fitting differential models to data without a sound perspective on the biological feasibility of the mathematical model will provide an statistical significant fit but will also fall short of providing useful information for the management, mitigation and eventual control of the epidemic of SARS-CoV-2.

Epidemiological models contain a set of parameters that must be adjusted based on available observations. Once a model has been calibrated, it can be used as a forecasting tool to make predictions and to evaluate contingency plans. It is customary to employ only point estimators for such predictions. However, some models may fit the same data reasonably well for a broad range of parameter values, and this flexibility means that predictions stemming from such models will vary widely, depending on the particular parameter values employed within the range that give a good fit. When data are poor or incomplete, model uncertainty widens further. A way to circumvent this problem is to use Bayesian statistics to incorporate observations and use the full range of parameter estimates contained in the posterior distribution to adjust for uncertainties in model predictions. Specifically, given the epidemiological model and a probability distribution for observations, we use the posterior distribution of model parameters to generate all possible epidemiological curves via the posterior predictive distribution. From the envelope of all curves one can extract the worst-case scenario and study the impact of implementing contingency plans according to this assessment. We apply this approach to the potential evolution of COVID-19 in Mexico City and assess whether contingency plans are being successful and whether the epidemiological curve has flattened., Comment: 6 pages, 9 figures. The draft has been rewritten and the SI has been included as an ancillary file. The calibration of the model has been redone using the updated public data on May 7th

In this paper we introduce a novel method to conduct inference with models defined through a continuous-time Markov process, and we apply these results to a classical stochastic SIR model as a case study. Using the inverse-size expansion of van Kampen we obtain approximations for first and second moments for the state variables. These approximate moments are in turn matched to the moments of an inputed generic discrete distribution aimed at generating an approximate likelihood that is valid both for low count or high count data. We conduct a full Bayesian inference to estimate epidemic parameters using informative priors. Excellent estimations and predictions are obtained both in a synthetic data scenario and in two Dengue fever case studies., Comment: 30 pages, 5 figures, submitted for peer review