Simple Mathematics on Covid-19 Expansion

We review the Kermack and McKendrick model of epidemics and apply it to Covid-19. Despite the simplicity of this model, solid conclusions are extracted that can assist potential decisions on the strategy to combat the outbreak, essentially configuring a scenario ranging from short-term suppression to long-term mitigation depending on the achieved reduction in the contact number.

Relations of parameters for describing the epidemic of COVID―19 by the Kermack―McKendrick model

In order to quantitatively characterize the epidemic of COVID-19, useful relations among parameters describing an epidemic in general are derived based on the Kermack–McKendrick model. The first relation is 1/τgrow =1/τtrans−1/τinf, where τgrow is the time constant of the exponential growth of an epidemic, τtrans is the time for a pathogen to be transmitted from one patient to uninfected person, and the infectious time τinf is the time during which the pathogen keeps its power of transmission. The second relation p(∞) ≈ 1−exp(−(R0−1)/0.60) is the relation between p(∞), the final size of the disaster defined by the ratio of the total infected people to the population of the society, and the basic reproduction number, R0, which is the number of persons infected by the transmission of the pathogen from one infected person during the infectious time. The third relation 1/τend = 1/τinf−(1−p(∞))/τtrans gives the decay time constant τend at the ending stage of the epidemic. Derived relations are applied to influenza in Japan in 2019 for characterizing the epidemic.

Stability and Energy-Casimir Mapping for Integrable Deformations of the Kermack-McKendrick System

Integrable deformations of a Hamilton-Poisson system can be obtained altering its constants of motion. These deformations are integrable systems that can have various dynamical properties. In this paper, we give integrable deformations of the Kermack-McKendrick model for epidemics, and we analyze a particular integrable deformation. More precisely, we point out two Poisson structures that lead to infinitely many Hamilton-Poisson realizations of the considered system. Furthermore, we study the stability of the equilibrium points, we give the image of the energy-Casimir mapping, and we point out some of its properties.

Complexities of Epidemic Modeling

The prior chapter derived and simulated the most basic epidemic model, assuming that people can be in only one of three states (susceptible, infected, or recovered) and that people mix homogeneously throughout the population. In this chapter, the author examines how the Kermack-McKendrick model can be extended to simulate a wide variety of complex diseases and circumstances and be adapted to incorporate the complex ways that people contact each other. Once we leave the context of the Kermack-McKendrick model, the calculation of R0 becomes complicated, so that the researcher must resort to simulation to identify what effect a disease will have in a population and to measure the potential impact of a public health intervention on the disease. The author additionally describes methods for simulating individual behavior in response to an epidemic.

The deterministic Kermack‒McKendrick model bounds the general stochastic epidemic

We prove that, for Poisson transmission and recovery processes, the classic susceptible→infected→recovered (SIR) epidemic model of Kermack and McKendrick provides, for any given time t>0, a strict lower bound on the expected number of susceptibles and a strict upper bound on the expected number of recoveries in the general stochastic SIR epidemic. The proof is based on the recent message passing representation of SIR epidemics applied to a complete graph.