Asymptotic analysis of the SIR model and the Gompertz distribution

The SIR (Susceptible–Infected–Removed) is one of the simplest models for epidemic outbreaks. The present paper derives a novel, simple, analytical asymptotic solution for the I-variable, which is valid on the entire real line. Connections with the Gompertz and Gumbel distributions are also demonstrated. The approach is applied to the ongoing coronavirus disease 2019 (COVID-19) pandemic in four European countries — Belgium, Italy, Sweden, and Bulgaria. The reported raw incidence data from the outbreaks in 2020–2021 have been fitted using constrained least squares. It is demonstrated that the asymptotic solution can be used successfully for parametric estimation either in stand-alone mode or as a preliminary step in the parametric estimation using numerical inversion of the exact parametric solution. [ FROM AUTHOR]

Pseudo Steady-State Period in Non-Stationary Infinite-Server Queue with State Dependent Arrival Intensity

An infinite-server queueing model with state-dependent arrival process and exponential distribution of service time is analyzed. It is assumed that the difference between the value of the arrival rate and total service rate becomes positive starting from a certain value of the number of customers in the system. In this paper, time until reaching this value by the number of customers in the system is called the pseudo steady-state period (PSSP). Distribution of duration of PSSP, its raw moments and its simple approximation under a certain scaling of the number of customers in the system are analyzed. Novelty of the considered problem consists of an arbitrary dependence of the rate of customer arrival on the current number of customers in the system and analysis of time until reaching from below a certain level by the number of customers in the system. The relevant existing papers focus on the analysis of time interval since exceeding a certain level until the number of customers goes down to this level (congestion period). Our main contribution consists of the derivation of a simple approximation of the considered time distribution by the exponential distribution. Numerical examples are presented, which confirm good quality of the proposed approximation.

OPTIMAL SWITCHING BETWEEN LOCKING DOWN AND OPENING THE ECONOMY BECAUSE OF AN INFECTION

We consider a two-regime switching model with the goal of minimizing the expected discounted cumulative combination of the number of infections together with an inverse economical indicator. We assume the two regime choices are between opening and locking down the economy, and the choice affects the infection rate. We also assume that the economy level also has a small influence on both the infection rate and the cumulative function being minimized. We then asymptotically find the value function and the boundaries of the stopping regions and perform a numerical calibration to draw conclusions about optimal lockdown in a pandemic. © 2021 Society for Industrial and Applied Mathematics

Effect of Human Mobility on the Spatial Spread of Airborne Diseases: An Epidemic Model with Indirect Transmission.

We extended a class of coupled PDE-ODE models for studying the spatial spread of airborne diseases by incorporating human mobility. Human populations are modeled with patches, and a Lagrangian perspective is used to keep track of individuals' places of residence. The movement of pathogens in the air is modeled with linear diffusion and coupled to the SIR dynamics of each human population through an integral of the density of pathogens around the population patches. In the limit of fast diffusion pathogens, the method of matched asymptotic analysis is used to reduce the coupled PDE-ODE model to a nonlinear system of ODEs for the average density of pathogens in the air. The reduced system of ODEs is used to derive the basic reproduction number and the final size relation for the model. Numerical simulations of the full PDE-ODE model and the reduced system of ODEs are used to assess the impact of human mobility, together with the diffusion of pathogens on the dynamics of the disease. Results from the two models are consistent and show that human mobility significantly affects disease dynamics. In addition, we show that an increase in the diffusion rate of pathogen leads to a lower epidemic.

The asymptotic analysis of novel coronavirus disease via fractional-order epidemiological model

We develop a model and investigate the temporal dynamics of the transmission of the novel coronavirus. The main sources of the coronavirus disease were bats and unknown hosts, which left the infection in the seafood market and became the major cause of the spread among the population. Evidence shows that the infection spiked due to the interaction between humans. Hence, the formulation of the model proposed in this study is based on human-to-human and reservoir-to-human interaction. We formulate the model by keeping in view the esthetic of the novel disease. We then fractionalize it with the application of fractional calculus. Particularly, we will use the Caputo-Fabrizio operator for fractionalization. We analyze the existence and uniqueness of the well-known fixed point theory. Moreover, it will be proven that the considered model is biologically and mathematically feasible. We also calculate the threshold quantity (reproductive number) to discuss steady states and to show that the particular epidemic model is stable asymptotically under some restrictions. We also discuss the sensitivity analysis of the threshold quantity to find the relative impact of every epidemic parameter on the transmission of the coronavirus disease. Both the global and local properties of the proposed model will be analyzed for the developed model using the mean value theorem, Barbalat's lemma, and linearization. We also performed some numerical simulations to verify the theoretical work via some graphical representations. © 2022 Author(s).

Applying Laplace Adomian decomposition method (LADM) for solving a model of Covid-19.

In this study, we apply the Laplace Adomian decomposition method (LADM) for the mathematical model of Covid-19. The mathematical model includes a system of nonlinear ordinary differential equations. Therefore, the model cannot be solved analytically but only by approximation. The application of LADM approximates the solution profiles of the dynamical variables of the Covid-19 model by an analytical power series. The conventional way to calculate the expressions of the approximation solutions is complicated both in terms of mathematical calculations and in terms of computer run time. In this paper, we propose a new algorithm for implementing the LADM method combined with the singularly perturbed vector field (SPVF) method. The new algorithm we offer is significantly reducing the running time of both the computer and the mathematical calculations. We compared the results obtained from the LADM to the numerical simulations. Some plots are presented to show the reliability and simplicity of the new algorithm.

To quarantine, or not to quarantine: A theoretical framework for disease control via contact tracing.

Contact tracing via smartphone applications is expected to be of major importance for maintaining control of the COVID-19 pandemic. However, viable deployment demands a minimal quarantine burden on the general public. That is, consideration must be given to unnecessary quarantining imposed by a contact tracing policy. Previous studies have modeled the role of contact tracing, but have not addressed how to balance these two competing needs. We propose a modeling framework that captures contact heterogeneity. This allows contact prioritization: contacts are only notified if they were acutely exposed to individuals who eventually tested positive. The framework thus allows us to address the delicate balance of preventing disease spread while minimizing the social and economic burdens of quarantine. This optimal contact tracing strategy is studied as a function of limitations in testing resources, partial technology adoption, and other intervention methods such as social distancing and lockdown measures. The framework is globally applicable, as the distribution describing contact heterogeneity is directly adaptable to any digital tracing implementation.

Analytic solution of the SEIR epidemic model via asymptotic approximant.

An analytic solution is obtained to the SEIR Epidemic Model. The solution is created by constructing a single second-order nonlinear differential equation in ln S and analytically continuing its divergent power series solution such that it matches the correct long-time exponential damping of the epidemic model. This is achieved through an asymptotic approximant (Barlow et al., 2017) in the form of a modified symmetric Padé approximant that incorporates this damping. The utility of the analytical form is demonstrated through its application to the COVID-19 pandemic.

Accurate closed-form solution of the SIR epidemic model.

An accurate closed-form solution is obtained to the SIR Epidemic Model through the use of Asymptotic Approximants (Barlow et al., 2017). The solution is created by analytically continuing the divergent power series solution such that it matches the long-time asymptotic behavior of the epidemic model. The utility of the analytical form is demonstrated through its application to the COVID-19 pandemic.