Esperanza Lozada

In this paper, we investigate a reaction–diffusion system governing the vector-borne and sexual transmission dynamics of the Zika virus between host (human) and vector (mosquito) populations. Adopting the compartmental methodology, the human population is divided into susceptible, exposed, infected, and recovered classes, while the mosquito population is partitioned into susceptible, exposed, and infected compartments. We consider several assumptions for disease transmission; in particular, we consider the transmission from infected to susceptible humans to occur through sexual interaction. The first contribution of this work lies in incorporating spatial displacement for both populations, modeled via Fick’s law with a spatially varying diffusion coefficient to capture environmental heterogeneity. To reflect an epidemiologically closed domain, homogeneous Neumann (zero-flux) boundary conditions are imposed, ensuring total population mass conservation principles. As a second advancement, we establish rigorous mathematical results regarding the well-posedness, boundedness, and positivity of solutions to the resulting reaction–diffusion system. The third innovation of the paper is the fact that an unconditional convergent and biologically consistent finite difference numerical scheme is developed to resolve the coupled non-linear governing equations. Finally, we provide a comprehensive set of numerical simulations focused on the parameter sensitivity.

In this paper, we develop a systematic review of the existing literature on the mathematical modeling of several aspects of pollinators. We selected the MathSciNet and Wos databases and performed a search for the words “pollinator” and “mathematical model”. This search yielded a total of 236 records. After a detailed screening process, we retained 107 publications deemed most relevant to the topic of mathematical modeling in pollinator systems. We conducted a bibliometric analysis and categorized the studies based on the mathematical approaches used as the central tool in the mathematical modeling and analysis. The mathematical theories used to obtain the mathematical models were ordinary differential equations, partial differential equations, graph theory, difference equations, delay differential equations, stochastic equations, numerical methods, and other types of theories, like fractional order differential equations. Meanwhile, the topics were positive bounded solutions, equilibrium and stability analysis, bifurcation analysis, optimal control, and numerical analysis. We summarized the research findings and identified some challenges that could inform the direction of future research, highlighting areas that will aid in the development of future research.

In this article we focus our interest on the study of an inverse problem arising in the mathematical modeling of disease transmission. The mathematical model is given by an initial boundary value problem for a reaction diffusion system. Meanwhile, the inverse problem consists in the determination of the disease and recovery transmission rates from observed measurement of the direct problem solution at some fixed time. The unknowns of the inverse problem are coefficients of the reaction term. We formulate the inverse problem as an optimization problem for an appropriate cost functional. Then, the existence of solutions of the inverse problem is deduced by proving the existence of a minimizer for the cost functional. We establish the uniqueness of identification problem. The uniqueness is a consequence of the first order necessary optimality condition and a stability of the inverse problem unknowns with respect to the observations. Moreover, we develop a numerical approximation and simulations of the inverse problem.

In this article, we propose a mathematical model for one-dimensional heat conduction in a three-layered solid considering that an interfacial condition is present for the temperature and heat flux conditions between the layers. The numerical approach is developed by constructing a finite difference scheme to solve the initial boundary–interface problem. The numerical scheme is designed by considering the accuracy of the model on the inner part of each layer, then extending to the interfaces and boundaries by incorporating the continuous interfacial conditions. The finite difference scheme is unconditionally stable, convergent, and easy to implement since it consists of the solution of two algebraic systems. We provide three numerical examples to confirm that our numerical approximation is consistent with the analytical solution and the physical phenomenon.

We present results that come from analysing students’ processes of solving tasks designed particularly for the learning of differential equations in a situational context. A teaching sequence was implemented during a semester-long course that comprised several tasks to cover the content of the course. Following a qualitative analysis, we show how a contextualised task can favour the construction of meanings associated with differential equations and how the interpretation of the terms present in those equations is developed. We also highlight some difficulties that may arise in students’ interpretations when the task involves meanings from other disciplines.

This article is focused on investigating the mathematical model calibration of a reaction–diffusion system arising in the mathematical model of the spread of an epidemic in a society. We consider that the total population is divided into two classes of individuals, called susceptible and infectious, where a susceptible individual can become infectious, and that upon recovery, an infected individual can become susceptible again. We consider that the population lives in a spatially heterogeneous environment, and that the spread of the dynamics is governed by a reaction–diffusion system consisting of two equations, where the variables of the model are the densities of susceptible and infected individuals. In the reaction term, the coefficients are the rates of disease transmission and the rate of infective recovery. The main contribution of this study is the identification of the reaction coefficients by assuming that the infective and susceptible densities at the end time of the process and on overall spatial domain are observed. We apply the optimal control methodology to prove the main findings: the existence of positive solutions for the state system, the existence of at least one solution for the identification problem, the introduction of first-order necessary conditions, and the local uniqueness of optimal solutions.