Integer Versus Fractional Order SEIR Deterministic and Stochastic Models of Measles

Abstract

In this paper, we compare the performance between systems of ordinary and (Caputo) fractional differential equations depicting the susceptible-exposed-infectious-recovered (SEIR) models of diseases. In order to understand the origins of both approaches as mean-field approximations of integer and fractional stochastic processes, we introduce the fractional differential equations (FDEs) as approximations of some type of fractional nonlinear birth and death processes. Then, we examine validity of the two approaches against empirical courses of epidemics; we fit both of them to case counts of three measles epidemics that occurred during the pre-vaccination era in three different locations. While ordinary differential equations (ODEs) are commonly used to model epidemics, FDEs are more flexible in fitting empirical data and theoretically offer improved model predictions. The question arises whether, in practice, the benefits of using FDEs over ODEs outweigh the added computational complexities. While important differences in transient dynamics were observed, the FDE only outperformed the ODE in one of out three data sets. In general, FDE modeling approaches may be worth it in situations with large refined data sets and good numerical algorithms. Laplace transform of a function f (t) is defined as L( f )(s) =f (s) = ∞ 0 e −st f (t)dt.

A Study of Some Generalized Results of Neutral Stochastic Differential Equations in the Framework of Caputo–Katugampola Fractional Derivatives

“…By employing Equations ( 25) through (30) in Equation ( 24), consequently, we obtain the following results:…”

Differential and Time-Discrete SEIRS Models with Vaccination: Local Stability, Validation and Sensitivity Analysis Using Bulgarian COVID-19 Data

“…By default, the use of such non-local models requires larger refined datasets. They are also quite computationally expensive, which is why well-specialized numerical algorithms are required [29]. Currently, integer-order explicit or implicit timediscrete models have been used to the model COVID-19 epidemics.…”

Modeling the Evolution of SARS-CoV-2 Using a Fractional-Order SIR Approach

“…Given the difficulties of the Susceptible-Infectious-Recovered (SIR) integer model to suit complex behaviors using few coefficients (such as those that depend on complex interactions among the community regarding sociological, economic, and political aspects [18]), we propose the use of a fractional SIR model to describe SARS-CoV-2 propagation in two different countries (France and Colombia) under two government policy scenarios: ruling during lockdown and ruling before it. France and Colombia were selected as case studies from Europe and Latin-America, respectively.…”