Shumaila Azam1, Jorge E Macías-Díaz2, Nauman Ahmed3, Ilyas Khan4, Muhammad S Iqbal1, Muhammad Rafiq5, Kottakkaran S Nisar6, Muhammad O Ahmad1. 1. Department of Mathematics and Statistics, University of the Lahore, Lahore, Pakistan. 2. Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, Aguascalientes, Ags. 20131, Mexico. Electronic address: jemacias@correo.uaa.mx. 3. Department of Mathematics and Statistics, University of the Lahore, Lahore, Pakistan. Electronic address: nauman.ahmed@math.uol.edu.pk. 4. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City 72915, Vietnam. Electronic address: ilyaskhan@tdtu.edu.vn. 5. Faculty of Engineering, University of Central Punjab, Lahore, Pakistan. Electronic address: m.rafiq@ucp.edu.pk. 6. Department of Mathematics, College of Arts and Sciences, Prince Sattam Bin Abdulaziz University, Wadi Aldawasir, Kingdom of Saudi Arabia. Electronic address: sooppy@psau.edu.sa.
Abstract
BACKGROUND AND OBJECTIVE: Epidemic models are used to describe the dynamics of population densities or population sizes under suitable physical conditions. In view that population densities and sizes cannot take on negative values, the positive character of those quantities is an important feature that must be taken into account both analytically and numerically. In particular, susceptible-infected-recovered (SIR) models must also take into account the positivity of the solutions. Unfortunately, many existing schemes to study SIR models do not take into account this relevant feature. As a consequence, the numerical solutions for these systems may exhibit the presence of negative population values. Nowadays, positivity (and, ultimately, boundedness) is an important characteristic sought for in numerical techniques to solve partial differential equations describing epidemic models. METHOD: In this work, we will develop and analyze a positivity-preserving nonstandard implicit finite-difference scheme to solve an advection-reaction nonlinear epidemic model. More concretely, this discrete model has been proposed to approximate consistently the solutions of a spatio-temporal nonlinear advective dynamical system arising in many infectious disease phenomena. RESULTS: The proposed scheme is capable of guaranteeing the positivity of the approximations. Moreover, we show that the numerical scheme is consistent, stable and convergent. Additionally, our finite-difference method is capable of preserving the endemic and the disease-free equilibrium points. Moreover, we will establish that our methodology is stable in the sense of von Neumann. CONCLUSION: Comparisons with existing techniques show that the technique proposed in this work is a reliable and efficient structure-preserving numerical model. In summary, the present approach is a structure-preserving and efficient numerical technique which is easy to implement in any scientific language by any scientist with minimal knowledge on scientific programming.
BACKGROUND AND OBJECTIVE: Epidemic models are used to describe the dynamics of population densities or population sizes under suitable physical conditions. In view that population densities and sizes cannot take on negative values, the positive character of those quantities is an important feature that must be taken into account both analytically and numerically. In particular, susceptible-infected-recovered (SIR) models must also take into account the positivity of the solutions. Unfortunately, many existing schemes to study SIR models do not take into account this relevant feature. As a consequence, the numerical solutions for these systems may exhibit the presence of negative population values. Nowadays, positivity (and, ultimately, boundedness) is an important characteristic sought for in numerical techniques to solve partial differential equations describing epidemic models. METHOD: In this work, we will develop and analyze a positivity-preserving nonstandard implicit finite-difference scheme to solve an advection-reaction nonlinear epidemic model. More concretely, this discrete model has been proposed to approximate consistently the solutions of a spatio-temporal nonlinear advective dynamical system arising in many infectious disease phenomena. RESULTS: The proposed scheme is capable of guaranteeing the positivity of the approximations. Moreover, we show that the numerical scheme is consistent, stable and convergent. Additionally, our finite-difference method is capable of preserving the endemic and the disease-free equilibrium points. Moreover, we will establish that our methodology is stable in the sense of von Neumann. CONCLUSION: Comparisons with existing techniques show that the technique proposed in this work is a reliable and efficient structure-preserving numerical model. In summary, the present approach is a structure-preserving and efficient numerical technique which is easy to implement in any scientific language by any scientist with minimal knowledge on scientific programming.