Deterministic Seirs Epidemic Model for Modeling Vital Dynamics, Vaccinations, and Temporary Immunity

Marek Trawicki

doi:10.3390/math5010007

Abstract

In this paper, the author proposes a new SEIRS model that generalizes several classical deterministic epidemic models (e.g., SIR and SIS and SEIR and SEIRS) involving the relationships between the susceptible S, exposed E, infected I, and recovered R individuals for understanding the proliferation of infectious diseases. As a way to incorporate the most important features of the previous models under the assumption of homogeneous mixing (mass-action principle) of the individuals in the population N, the SEIRS model utilizes vital dynamics with unequal birth and death rates, vaccinations for newborns and non-newborns, and temporary immunity. In order to determine the equilibrium points, namely the disease-free and endemic equilibrium points, and study their local stability behaviors, the SEIRS model is rescaled with the total time-varying population and analyzed according to its epidemic condition R0 for two cases of no epidemic (R0 ≤ 1) and epidemic (R0 > 1) using the time-series and phase portraits of the susceptible s, exposed e, infected i, and recovered r individuals. Based on the experimental results using a set of arbitrarily-defined parameters for horizontal transmission of the infectious diseases, the proportional population of the SEIRS model consisted primarily of the recovered r (0.7–0.9) individuals and susceptible s (0.0–0.1) individuals (epidemic) and recovered r (0.9) individuals with only a small proportional population for the susceptible s (0.1) individuals (no epidemic). Overall, the initial conditions for the susceptible s, exposed e, infected i, and recovered r individuals reached the corresponding equilibrium point for local stability: no epidemic (DFE X ¯ D F E ) and epidemic (EE X ¯ E E ).

Highlights

Over the past decades, mathematical models have been developed and implemented to study the spread of infectious diseases since the early 20th century in the field of mathematical epidemiology [1–4]
In order to differentiate between the possibilities for the epidemic condition R0, the two cases of no epidemic (R0 ≤ 1) and epidemic (R0 > 1)
As a way to examine the relationships between the different rescaled variables, phase portraits were utilized to trace the solution of the system of ordinary differential equations for the rescaled SEIRS model in (18)

Summary

Introduction

Mathematical models have been developed and implemented to study the spread of infectious diseases since the early 20th century in the field of mathematical epidemiology [1–4]. The stochastic and deterministic epidemic models allow researchers to gain valuable insights into numerous infectious diseases and investigate strategies for combating them. The transition rates from one compartment to another compartment are mathematically expressed with derivatives. Based on the assorted compartments for the population and derivatives for the transition rates, the system of ordinary differential equations serves to describe the changes in population as a function of time. From the seminal work in 1927, Kermack and McKendrick constructed a simple deterministic compartment model that still today acts as the fundamental model for developing and implementing even more complicated mathematical epidemic models [5].

Methods

Results

Conclusion