Mathematical modeling of infectious diseases using ordinary and fractional differential equations



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Infectious diseases can spread and turn into epidemics, taking thousands of lives within a matter of just a few days. Mathematical models can help us to gain insights into the dynamics of diseases and their control strategies. Besides ordinary differential equations (ODE), fractional differential equations (FDE) have been used, especially in the last decade to model the course of epidemics. Three major goals of this dissertation are: 1) For an epidemic model, which are the important parameters and do how these influence output measures? 2) Which transmission pathway is dominant? 3) Whether an ODE or FDE modeling approach fits better to data?

We developed Susceptible-Infected-Recovered-Susceptible (SIRS) models of a recently discovered fungal pathogen, \textit{Batrachochytrium salamandrivorans} (Bsal) transmission using a system of ordinary differential equations. Our models included two routes of pathogen transmission: direct transmission via contact between infected and susceptible individuals and environmental transmission via shed zoospores in the water. Unlike previous models, we categorized individuals into multiple stages of infection. We found the invasion probability for Bsal (i.e., the basic reproductive number, R0) into a population of the Eastern Newt adults. We performed numerical simulations and parameter sensitivity analysis using Latin hypercube sampling with partial rank coefficient correlation. We identified the Bsal dominant transmission pathway and suggested disease control strategies.

Finally, we compare the performance of systems of ordinary and (Caputo) fractional differential equations depicting Susceptible-Exposed-Infected-Recovered (SEIR) models of diseases. In order to understand the origins of both approaches as mean-field approximations of fractional stochastic processes, we introduced the ordinary and fractional differential equations as approximations of some type of fractional nonlinear birth-death processes. We examined the validity of the two approaches against empirical courses of epidemics. We fit both models to case counts of three measles epidemics that occurred during the pre-vaccination era in three different locations. Our results suggest that data might guide the choice of modeling framework (ODE vs. FDE). Overall, this dissertation highlights the fact that data can play an important role in multiple stages of the model process.



Infectious diseases, Ordinary differential equations, Fractional differential equations, Batrachochytrium salamandrivorans