Selected topics on reaction-diffusion-advection models from spatial ecology

We discuss the effects of movement and spatial heterogeneity on population dynamics via reaction-diffusion-advection models, focusing on the persistence, competition, and evolution of organisms in spatially heterogeneous environments. Topics include Lokta-Volterra competition models, river models, evolution of biased movement, phytoplankton growth, and spatial spread of epidemic disease. Open problems and conjectures are presented

Global dynamics of a dengue fever model incorporating transmission seasonality

The changes of seasons cause that the transmission of dengue fever is characterized by periodicity. We develop a dengue fever transmission model incorporating seasonal periodicity and spatial heterogeneity. Based on the well-posedness of solution for this model, we propose its basic reproduction number R0, and we discuss the properties of this number including its limiting form when the diffusion coefficients change. Moreover, the dynamical behavior of this model infers that if R0 ⩽ 1, then the disease-free periodic solution is globally asymptotically stable, and if R0 > 1, then the model possesses a positive periodic solution, which is globally asymptotically stable. These theoretical findings are further illustrated by the final numerical simulations. Additionally, we add that the similar problem has been investigated by M. Zhu and Y. Xu [A time-periodic dengue fever model in a heterogeneous environment, Math. Comput. Simul., 155:115–129, 2019] in which some dynamical results have been studied only on the cases R0 1. Our results not only include the scenario on the case R0 = 1, but also involve the more succinct conditions on the cases R0 1

Fractional derivative models for the spread of diseases

This thesis considers the mathematical modelling of disease, using fractional differential equations in order to provide a tool for the description of memory effects. In Chapter 3 we illustrate a commensurate fractional order tumor model, and we find a critical value of the fractional derivative dependent on the parameter values of the model. For fractional derivatives of orders less than the critical value an unstable equilibrium point of the system becomes stable. In order to show changes in the observed areas of attraction of two stable points in the system, we then consider a fractional order SIR epidemic model and investigate the change from a monostable to a bistable system.;Chapter 4 considers a model for virus dynamics where the fractional orders for populations are different, called an incommensurate system. An approximate analytical solution for the characteristic equation of the incommensurate model is found when the different fractional orders are similar and close to the critical value of the fractional order of the commensurate system. In addition, the instability boundary is found as a function of both parameters. A comparison between analytical and numerical results shows the high accuracy of this approximation.;Chapter 5 consists of two parts, in the first part we generalise the integer Fisher's equation to be a space-time fractional differential equation and consider travelling wave solutions. In the second part we generalise an integer SIR model with spatial heterogeneity, which was studied by Murray [117], to a space-time fractional derivative model. We apply the (G0/G)-expansion method and find travelling wave solutions, although in this case we must consider the Jumarie's modified Riemann-Liouville fractional derivative. Finally, we consider the effect of changing the orders of time and space fractional derivatives on the location and speed of the travelling wave solution.This thesis considers the mathematical modelling of disease, using fractional differential equations in order to provide a tool for the description of memory effects. In Chapter 3 we illustrate a commensurate fractional order tumor model, and we find a critical value of the fractional derivative dependent on the parameter values of the model. For fractional derivatives of orders less than the critical value an unstable equilibrium point of the system becomes stable. In order to show changes in the observed areas of attraction of two stable points in the system, we then consider a fractional order SIR epidemic model and investigate the change from a monostable to a bistable system.;Chapter 4 considers a model for virus dynamics where the fractional orders for populations are different, called an incommensurate system. An approximate analytical solution for the characteristic equation of the incommensurate model is found when the different fractional orders are similar and close to the critical value of the fractional order of the commensurate system. In addition, the instability boundary is found as a function of both parameters. A comparison between analytical and numerical results shows the high accuracy of this approximation.;Chapter 5 consists of two parts, in the first part we generalise the integer Fisher's equation to be a space-time fractional differential equation and consider travelling wave solutions. In the second part we generalise an integer SIR model with spatial heterogeneity, which was studied by Murray [117], to a space-time fractional derivative model. We apply the (G0/G)-expansion method and find travelling wave solutions, although in this case we must consider the Jumarie's modified Riemann-Liouville fractional derivative. Finally, we consider the effect of changing the orders of time and space fractional derivatives on the location and speed of the travelling wave solution

International Conference on Mathematical Analysis and Applications in Science and Engineering – Book of Extended Abstracts

The present volume on Mathematical Analysis and Applications in Science and Engineering - Book of Extended Abstracts of the ICMASC’2022 collects the extended abstracts of the talks presented at the International Conference on Mathematical Analysis and Applications in Science and Engineering – ICMA2SC'22 that took place at the beautiful city of Porto, Portugal, in June 27th-June 29th 2022 (3 days). Its aim was to bring together researchers in every discipline of applied mathematics, science, engineering, industry, and technology, to discuss the development of new mathematical models, theories, and applications that contribute to the advancement of scientific knowledge and practice. Authors proposed research in topics including partial and ordinary differential equations, integer and fractional order equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, probability, computational mathematics, amongst others. The conference was designed to maximize the involvement of all participants and will present the state-of- the-art research and the latest achievements.info:eu-repo/semantics/publishedVersio

Novel codynamics of the HIV-1/HTLV-Ⅰ model involving humoral immune response and cellular outbreak: A new approach to probability density functions and fractional operators

Both human immunodeficiency virus type 1 (HIV-1) and human T-lymphotropic virus type Ⅰ (HTLV-Ⅰ) are retroviruses that afflict CD4 T cells. In this article, the codynamics of within-host HIV-1 and HTLV-Ⅰ are presented via piecewise fractional differential equations by employing a stochastic system with an influential strategy for biological research. It is demonstrated that the scheme is mathematically and biologically feasible by illustrating that the framework has positive and bounded global findings. The necessary requirements are deduced, ensuring the virus's extinction. In addition, the structure is evaluated for the occurrence of an ergodic stationary distribution and sufficient requirements are developed. A deterministic-stochastic mechanism for simulation studies is constructed and executed in MATLAB to reveal the model's long-term behavior. Utilizing rigorous analysis, we predict that the aforesaid model is an improvement of the existing virus-to-cell and cell-to-cell interactions by investigating an assortment of behaviour patterns that include cross-over to unpredictability processes. Besides that, the piecewise differential formulations, which can be consolidated with integer-order, Caputo, Caputo-Fabrizio, Atangana-Baleanu and stochastic processes, have been declared to be exciting opportunities for researchers in a spectrum of disciplines by enabling them to incorporate distinctive features in various temporal intervals. As a result, by applying these formulations to difficult problems, researchers can achieve improved consequences in reporting realities with white noise. White noise in fractional HIV-1/HTLV-Ⅰ codynamics plays an extremely important function in preventing the proliferation of an outbreak when the proposed flow is constant and disease extermination is directly proportional to the magnitude of the white noise

On Honey Bee Colony Dynamics and Disease Transmission

The work herein falls under the umbrella of mathematical modeling of disease transmission. The majority of this document focuses on the extent to which infection undermines the strength of a honey bee colony. These studies extend from simple mass-action ordinary differential equations models, to continuous age-structured partial differential equation models and finally a detailed agent-based model which accounts for vector transmission of infection between bees as well as a host of other influences and stressors on honey bee colony dynamics. These models offer a series of predictions relevant to the fate of honey bee colonies in the presence of disease and the nonlinear effects of disease, seasonality and the complicated dynamics of honey bee colonies. We are also able to extract from these models metrics that preempt colony failure. The analysis of disease dynamics in age-structured honey bee colony models required the study of next generation operators (NGO) and the basic reproduction number, $R_0$, for partial differential equations. This led us to the development of a coherent path from the NGO to its discrete compartmental counterpart, the next generation matrix (NGM) as well as the derivation of new closed-form formulae for the NGO for specific classes of disease models

Malaria Models with Spatial Effects

Malaria, a vector-borne infectious disease caused by the Plasmodium parasite, is still endemic in more than 100 countries in Africa, Southeast Asia, the Eastern Mediterranean, Western Pacific, Americas, and Europe. In 2010 there were about 219 million malaria cases, with an estimated 660,000 deaths, mostly children unde