Ruin problems for epidemic insurance

The paper discusses the risk of ruin in insurance coverage of an epidemic in a closed population. The model studied is an extended susceptible–infective–removed (SIR) epidemic model built by Lefèvre and Simon (Methodology Comput. Appl. Prob.22, 2020) as a block-structured Markov process. A fluid component is then introduced to describe the premium amounts received and the care costs reimbursed by the insurance. Our interest is in the risk of collapse of the corresponding reserves of the company. The use of matrix-analytic methods allows us to determine the distribution of ruin time, the probability of ruin, and the final amount of reserves. The case where the reserves are subjected to a Brownian noise is also studied. Finally, some of the results obtained are illustrated for two particular standard SIR epidemic models.

Bifurcation analysis of an SIR epidemic model with saturated treatment function

In this thesis we investigate the dynamics and bifurcation of SIR epidemic models with horizontal and vertical transmissions and saturated treatment rate. It is proved that such SIR epidemic models always have positive disease free equilibria and also have three positive epidemic equilibria. The ranges of the parameters related in the model were found under which the equilibria of the models are positive. By applying the qualitative theory of planar systems, it is shown the disease free equilibria is a saddle, stable node and globally asymptotically stable. Furthermore, it is also shown that the interior equilibria are saddle, saddle node or saddle point.

Bifurcation analysis of an SIR epidemic model with saturated treatment function

In this thesis we investigate the dynamics and bifurcation of SIR epidemic models with horizontal and vertical transmissions and saturated treatment rate. It is proved that such SIR epidemic models always have positive disease free equilibria and also have three positive epidemic equilibria. The ranges of the parameters related in the model were found under which the equilibria of the models are positive. By applying the qualitative theory of planar systems, it is shown the disease free equilibria is a saddle, stable node and globally asymptotically stable. Furthermore, it is also shown that the interior equilibria are saddle, saddle node or saddle point.

Phase portraits of SIR epidemic models with vertical transmissions and linear treatment

Susceptible-infective-removed epidemic models with horizontal and vertical transmissions and linear treatment rates are investigated. All the ranges of the parameters involved in the models for the infection-free equilibrium and the epidemic equilibrium to be positive are found. Like the previous results on the models without vertical transmissions or the linear treatments, we study stability of these equilibria. The novelty is that we justify that these positive equilibria are stable focuses or stable nodes under suitable conditions on the parameters. These results provide more detailed descriptions of behaviours of the epidemic diseases near the equilibria. Our results will exhibit the effect of the vertical transmissions and the linear treatment rates on the epidemic models. Some simulations results are provided to understand the phase portraits near the equilibria.

Phase portraits of SIR epidemic models with vertical transmissions and linear treatment

Susceptible-infective-removed epidemic models with horizontal and vertical transmissions and linear treatment rates are investigated. All the ranges of the parameters involved in the models for the infection-free equilibrium and the epidemic equilibrium to be positive are found. Like the previous results on the models without vertical transmissions or the linear treatments, we study stability of these equilibria. The novelty is that we justify that these positive equilibria are stable focuses or stable nodes under suitable conditions on the parameters. These results provide more detailed descriptions of behaviours of the epidemic diseases near the equilibria. Our results will exhibit the effect of the vertical transmissions and the linear treatment rates on the epidemic models. Some simulations results are provided to understand the phase portraits near the equilibria.

The Fractional Epidemics Theory: Managing and Ending an Epidemic

Susceptible–infectious–recovered (SIR) models are widely used for estimating the dynamics of epidemics and project that social distancing “flattens the curve”, i.e., reduces but delays the peak in daily infections, causing a longer epidemic. Based on these projections, individuals and governments have advocated lifting containment measures such as social distancing to shift the peak forward and limit societal and economic disruption. Paradoxically, the COVID-19 pandemic data exhibits phenomenology opposite to the SIR models’ projections. Here, we present a new model that replicates the observed phenomenology and quantitates pandemic dynamics with simple and actionable analytical tools for policy makers. Specifically, it offers a prescription of achievable and economically palatable measures for ending an epidemic.One Sentence SummaryThe SIR epidemic models are wrong; a new model offers achievable and economically viable measures for ending an epidemic.

Inverse Problem for Identification of Infectivity and Recovery Rates in SIR Epidemic Models as Functions of Time Illustrated with Corona Virus Dynamics up to July 09, 2020

This work deals with the inverse problem in epidemiology based on a SIR model with time-dependent infectivity and recovery rates, allowing for a better prediction of the long term evolution of a pandemic. The method is used for investigating the COVID-19 spread by first solving an inverse problem for estimating the infectivity and recovery rates from real data. Then, the estimated rates are used to compute the evolution of the disease. The time-depended parameters are estimated for the World and several countries (The United States of America, Canada, Italy, France, Germany, Sweden, Russia, Brazil, Bulgaria, Japan, South Korea, New Zealand) and used for investigating the COVID-19 spread in these countries.

Identifying the number of unreported cases in SIR epidemic models

An SIR epidemic model is analysed with respect to the identification of its parameters and initial values, based upon reported case data from public health sources. The objective of the analysis is to understand the relationship of unreported cases to reported cases. In many epidemic diseases the reported cases are a small fraction of the unreported cases. This fraction can be estimated by the identification of parameters for the model from reported case data. The analysis is applied to the Hong Kong seasonal influenza epidemic in New York City in 1968–1969.