Fuzzy fractional mathematical model of COVID-19 epidemic

In this paper, we develop a mathematical model with a Caputo fractional derivative under fuzzy sense for the prediction of COVID-19. We present numerical results of the mathematical model for COVID-19 of most three infected countries such as the USA, India and Italy. Using the proposed model, we estimate predicting future outbreaks, the effectiveness of preventive measures and potential control strategies of the infection. We provide a comparative study of the proposed model with Ahmadian’s fuzzy fractional mathematical model. The results demonstrate that our proposed fuzzy fractional model gives a nearer forecast to the actual data. The present study can confirm the efficiency and applicability of the fractional derivative under uncertainty conditions to mathematical epidemiology.

Epidemiological waves - types, drivers and modulators in the COVID-19 pandemic

Introduction: A discussion of 'waves' of the COVID-19 epidemic in different countries is a part of the national conversation for many, but there is no hard and fast means of delineating these waves in the available data and their connection to waves in the sense of mathematical epidemiology is only tenuous. Methods: We present an algorithm which processes a general time series to identify substantial, significant and sustained periods of increase in the value of the time series, which could reasonably be described as 'observed waves'. This provides an objective means of describing observed waves in time series. Results: The output of the algorithm as applied to epidemiological time series related to COVID-19 corresponds to visual intuition and expert opinion. Inspecting the results of individual countries shows how consecutive observed waves can differ greatly with respect to the case fatality ratio. Furthermore, in large countries, a more detailed analysis shows that consecutive observed waves have different geographical ranges. We also show how waves can be modulated by government interventions and find that early implementation of non-pharmaceutical interventions correlates with a reduced number of observed waves and reduced mortality burden in those waves. Conclusion: It is possible to identify observed waves of disease by algorithmic methods and the results can be fruitfully used to analyse the progression of the epidemic.

Stochastic simulation of successive waves of COVID-19 in the province of Barcelona

Analytic compartmental models are currently used in mathematical epidemiology to forecast the COVID-19 pandemic evolution and explore the impact of mitigation strategies. In general, such models treat the population as a single entity, losing the social, cultural and economical specifici- ties. We present a network model that uses socio-demographic datasets with the highest available granularity to predict the spread of COVID-19 in the province of Barcelona. The model is flexible enough to incorporate the effect of containment policies, such as lockdowns or the use of protec- tive masks, and can be easily adapted to future epidemics. We follow a stochastic approach that combines a compartmental model with detailed individual microdata from the population census, including social determinants and age-dependent strata, and time-dependent mobility information. We show that our model reproduces the dynamical features of the disease across two waves and demonstrate its capability to become a powerful tool for simulating epidemic events.

Two new compartmental epidemiological models and their equilibria

Compartmental models have long served as important tools in mathematical epidemiology, with their usefulness highlighted by the recent COVID-19 pandemic. However, most of the classical models fail to account for certain features of this disease and others like it, such as the ability of exposed individuals to recover without becoming infectious, or the possibility that asymptomatic individuals can indeed transmit the disease but at a lesser rate than the symptomatic. Furthermore, the rise of new disease variants and the imperfection of vaccines suggest that concept of endemic equilibrium is perhaps more pertinent than that of herd immunity. Here we propose a new compartmental epidemiological model and study its equilibria, characterizing the stability of both the endemic and disease-free equilibria in terms of the basic reproductive number. Moreover, we introduce a second compartmental model, generalizing our first, which accounts for vaccinated individuals, and begin an analysis of its equilibria.

Structure of epidemic models: toward further applications in economics

In this paper, we review the structure of various epidemic models in mathematical epidemiology for the future applications in economics. The heterogeneity of population and the generalization of nonlinear terms play important roles in making more elaborate and realistic models. The basic, effective, control and type reproduction numbers have been used to estimate the intensity of epidemic, to evaluate the effectiveness of interventions and to design appropriate interventions. The advanced epidemic models includes the age structure, seasonality, spatial diffusion, mutation and reinfection, and the theory of reproduction numbers has been generalized to them. In particular, the existence of sustained periodic solutions has attracted much interest because they can explain the recurrent waves of epidemic. Although the theory of epidemic models has been developed in decades and the development has been accelerated through COVID-19, it is still difficult to completely answer the uncertainty problem of epidemic models. We would have to mind that there is no single model that can solve all questions and build a scientific attitude to comprehensively understand the results obtained by various researchers from different backgrounds.

Monitoring and Analysis of COVID-19 Pandemic: The Need for an Empirical Approach

The current worldwide pandemic produced by coronavirus disease 2019 (COVID-19) has changed the paradigm of mathematical epidemiology due to the high number of unknowns of this new disease. Thus, the empirical approach has emerged as a robust tool to analyze the actual situation carried by the countries and also allows us to predict the incoming scenarios. In this paper, we propose three empirical indexes to estimate the state of the pandemic. These indexes quantify both the propagation and the number of estimated cases, allowing us to accurately determine the real risk of a country. We have calculated these indexes' evolution for several European countries. Risk diagrams are introduced as a tool to visualize the evolution of a country and evaluate its current risk as a function of the number of contagious individuals and the empiric reproduction number. Risk diagrams at the regional level are useful to observe heterogeneity on COVID-19 penetration and spreading in some countries, which is essential during deconfinement processes. During the pandemic, there have been significant differences seen in countries reporting case criterion and detection capacity. Therefore, we have introduced estimations about the real number of infectious cases that allows us to have a broader view and to better estimate the risk. These diagrams and indexes have been successfully used for the monitoring of European countries and regions during the COVID-19 pandemic.

A Review of Matrix SIR Arino Epidemic Models

Many of the models used nowadays in mathematical epidemiology, in particular in COVID-19 research, belong to a certain subclass of compartmental models whose classes may be divided into three “(x,y,z)” groups, which we will call respectively “susceptible/entrance, diseased, and output” (in the classic SIR case, there is only one class of each type). Roughly, the ODE dynamics of these models contains only linear terms, with the exception of products between x and y terms. It has long been noticed that the reproduction number R has a very simple Formula in terms of the matrices which define the model, and an explicit first integral Formula is also available. These results can be traced back at least to Arino, Brauer, van den Driessche, Watmough, and Wu (2007) and to Feng (2007), respectively, and may be viewed as the “basic laws of SIR-type epidemics”. However, many papers continue to reprove them in particular instances. This motivated us to redraw attention to these basic laws and provide a self-contained reference of related formulas for (x,y,z) models. For the case of one susceptible class, we propose to use the name SIR-PH, due to a simple probabilistic interpretation as SIR models where the exponential infection time has been replaced by a PH-type distribution. Note that to each SIR-PH model, one may associate a scalar quantity Y(t) which satisfies “classic SIR relations”, which may be useful to obtain approximate control policies.