Integer Versus Fractional Order SEIR Deterministic and Stochastic Models of Measles.

Md Rafiul Islam,Tamer Oraby,Angela Peace,Daniel Medina

doi:10.3390/ijerph17062014

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.

Highlights

Modeling the spread of infectious diseases before the introduction of vaccines, as well as the validation of these models, has been widely studied since the works of Reference [1,2,3,4,5,6,7,8,9,10]
We solve the system of fractional differential equations (FDEs) (Equation (14)) using Algorithm 1 and the systems of ordinary differential equations (ODEs)
In order to study the qualitative aspects of the FDE and its numerical solution, we carried out a number of simulations

Summary

Introduction

Modeling the spread of infectious diseases before the introduction of vaccines, as well as the validation of these models, has been widely studied since the works of Reference [1,2,3,4,5,6,7,8,9,10]. Deterministic models using ordinary differential equations (ODEs) have received great attention [12,13,14,15,16] and wide assimilation by health sciences [17]. Other deterministic models, such as difference equations, are used to model the spread of diseases; for instance, see Fisman et al [18]. Integer order derivatives of ordinary differential equations are special cases of fractional order derivatives It was noted in more than one paper, e.g., Reference [26], that FDEs give a better depiction of the courses of epidemics and natural phenomena than ODEs. Few researchers have

Methods

Results

Discussion

Conclusion