Stability, Bifurcation, and a Pair of Conserved Quantities in a Simple Epidemic System with Reinfection for the Spread of Diseases Caused by Coronaviruses

In this paper, we study a modified SIRI model without vital dynamics, based on a system of nonlinear ordinary differential equations, for epidemics that exhibit partial immunity after infection, reinfection, and disease-induced death. This model can be applied to study epidemics caused by SARS-CoV, MERS-CoV, and SARS-CoV-2 coronaviruses, since there is the possibility that, in diseases caused by these pathogens, individuals recovered from the infection have a decrease in their immunity and can be reinfected. On the other hand, it is known that, in populations infected by these coronaviruses, individuals with comorbidities or older people have significant mortality rates or deaths induced by the disease. By means of qualitative methods, we prove that such system has an endemic equilibrium and an infinite line of nonhyperbolic disease-free equilibria, we determine the local and global stability of these equilibria, and we also show that it has no periodic orbits. Furthermore, we calculate the basic reproductive number R 0 and find that the system exhibits a forward bifurcation: disease-free equilibria are stable when R 0 < 1 / σ and unstable when R 0 > 1 / σ , while the endemic equilibrium consist of an asymptotically stable upper branch that appears from R 0 > 1 / σ , σ being the rate that quantifies reinfection. We also show that this system has two conserved quantities. Additionally, we show some of the most representative numerical solutions of this system.

A Mathematical Model of Universal Basic Income and Its Numerical Simulations

In this paper, an elementary mathematical model describing the introduction of a universal basic income in a closed market society is constructed. The model is formulated in terms of a system of nonlinear ordinary differential equations, each of which gives account of how the number of individuals in a certain income class changes in time. Societies ruled by different fiscal systems (with no taxes, with taxation and redistribution, with a welfare system) are considered and the effect of the presence of a basic income in the various cases is analysed by means of numerical simulations. The main findings are that basic income effectively acts as a tool of poverty alleviation: indeed, in its presence the portion of individuals in the poorest classes and economic inequality diminish. Of course, the issue of a universal basic income in the real world is more complex and involves a variety of aspects. The goal here is simply to show how mathematical models can help in forecasting scenarios resulting from one or the other policy.

Solving Nonlinear Boundary Value Problems Using the Higher Order Haar Wavelet Method

The current study is focused on development and adaption of the higher order Haar wavelet method for solving nonlinear ordinary differential equations. The proposed approach is implemented on two sample problems—the Riccati and the Liénard equations. The convergence and accuracy of the proposed higher order Haar wavelet method are compared with the widely used Haar wavelet method. The comparison of numerical results with exact solutions is performed. The complexity issues of the higher order Haar wavelet method are discussed.

Analytical solutions to the M-derivative resonant Davey–Stewartson equations

In this paper, the (2+1)-dimensional resonant Davey–Stewartson equations are solved by using two methods; namely, [Formula: see text]-expansion and [Formula: see text]-expansion methods. A wave transform is used to convert the (2+1)-dimensional resonant Davey–Stewartson (RDS) equations with M-derivative into a system of nonlinear ordinary differential equations. Different forms of solutions, such as dark, bright, singular and periodic singular solutions are successfully constructed. The obtained solutions are plotted in 3D for both M- derivative and classical derivative to more understand the effect of M-derivative on the studied equation.

Numerical method for parameter inference of systems of nonlinear ordinary differential equations with partial observations

Parameter inference of dynamical systems is a challenging task faced by many researchers and practitioners across various fields. In many applications, it is common that only limited variables are observable. In this paper, we propose a method for parameter inference of a system of nonlinear coupled ordinary differential equations with partial observations. Our method combines fast Gaussian process-based gradient matching and deterministic optimization algorithms. By using initial values obtained by Bayesian steps with low sampling numbers, our deterministic optimization algorithm is both accurate, robust and efficient with partial observations and large noise.

Derivative-Free Iterative Methods with Some Kurchatov-Type Accelerating Parameters for Solving Nonlinear Systems

Some Kurchatov-type accelerating parameters are used to construct some derivative-free iterative methods with memory for solving nonlinear systems. New iterative methods are developed from an initial scheme without memory with order of convergence three. New methods have the convergence order 2+5≈4.236 and 5, respectively. The application of new methods can solve standard nonlinear systems and nonlinear ordinary differential equations (ODEs) in numerical experiments. Numerical results support the theoretical results.

A Mathematical Method for Predicting the Design Performance of Single and Multi-stage Rockets

Methods for predicting the performance of rockets are not new, however they often exist only within private organizations and in order to ensure competitive advantage, organizations tend to not share any details about their inner performance models. This open-source method gives students, design-teams and hobbyists a method to obtain baseline approximations for the performance of both single and multi-stage tandem rockets and provides a method which can easily be modified to meet the end-user’s requirements. The method solves for the mass, flight-path angle, velocity, altitude, and down-range distance using a numerical integrator to solve a set of nonlinear ordinary differential equations.