Mathematical Model and Analysis of Corruption Dynamics with Optimal Control

In this study, a deterministic mathematical model that explains the transmission dynamics of corruption is proposed and analyzed by considering social influence on honest individuals. Positivity and boundedness of solution of the model are proved and basic reproduction number R 0 is computed using the next-generation matrix method. The analysis shows that corruption-free equilibrium is locally and globally asymptotically stable whenever R 0 < 1 . Also, the endemic equilibrium point is locally and globally asymptotically stable whenever R 0 > 1 . Then, the model was extended to optimal control, and some numerical simulations with and without optimal control are also performed to verify the theoretical analysis using MATLAB. Numerical simulation of optimal control model shows that the prevention and punishment strategy is the most effective strategy to reduce the dynamic transmission of corruption.

Nonstandard Finite Difference Schemes for an SIR Epidemic Model

This paper aims to present two nonstandard finite difference (NFSD) methods to solve an SIR epidemic model. The proposed methods have important properties such as positivity and boundedness and they also preserve conservation law. Numerical comparisons confirm that the accuracy of our method is better than that of other existing standard methods such as the second-order Runge–Kutta (RK2) method, the Euler method and some ready-made MATLAB codes.

Breakdown of a Nonlinear Stochastic Nipah Virus Epidemic Models through Efficient Numerical Methods

Nipah virus (NiV) is a zoonotic virus (transmitted from animals to humans), which can also be transmitted through contaminated food or directly between people. According to a World Health Organization (WHO) report, the transmission of Nipah virus infection varies from animals to humans or humans to humans. The case fatality rate is estimated at 40% to 75%. The most infected regions include Cambodia, Ghana, Indonesia, Madagascar, the Philippines, and Thailand. The Nipah virus model is categorized into four parts: susceptible (S), exposed (E), infected (I), and recovered (R). Methods: The structural properties such as dynamical consistency, positivity, and boundedness are the considerable requirements of models in these fields. However, existing numerical methods like Euler–Maruyama and Stochastic Runge–Kutta fail to explain the main features of the biological problems. Results: The proposed stochastic non-standard finite difference (NSFD) employs standard and non-standard approaches in the numerical solution of the model, with positivity and boundedness as the characteristic determinants for efficiency and low-cost approximations. While the results from the existing standard stochastic methods converge conditionally or diverge in the long run, the solution by the stochastic NSFD method is stable and convergent over all time steps. Conclusions: The stochastic NSFD is an efficient, cost-effective method that accommodates all the desired feasible properties.

Dynamics of Delayed Phytoplankton–Zooplankton System with Disease Spread Among Zooplankton

In this paper, we study the dynamics of phytoplankton–zooplankton system with delay, where delay means that releasing toxin for phytoplankton is not instantaneous. First, we prove the positivity and boundedness of solutions, discuss the Hopf bifurcation caused by delay. Furthermore, we study the property of Hopf bifurcation by center manifold and normal form. Then, we study the global existence of bifurcated periodic solution. Finally by simulation, we show the influence of delay, disease spread and recovery from infected to susceptible on the dynamics of phytoplankton–zooplankton system.

THE IMPACT OF TESTING AND TREATMENT ON THE DYNAMICS OF HEPATITIS B VIRUS

Despite the intervention of WHO on vaccination for reducing the spread of Hepatitis B Virus (HBV), there are records of the high prevalence of HBV in some regions. In this paper, a mathematical model was formulated to analyze the acquisition and transmission process of the virus with the view of identifying the possible way of reducing the menace and mitigating the risk of the virus. The models' positivity and boundedness were demonstrated using well-known theorems. Equating the differential equations to zero demonstrates the equilibria of the solutions i.e., the disease-free and endemic equilibrium. The next Generation Matrix method was used to compute the basic reproduction number for the models. Local and global stabilities of the models were shown via linearization and Lyapunov function methods respectively. The importance of testing and treatment on the dynamics of HBV were fully discussed in this paper. It was discovered that testing at the acute stage of the virus and chronic unaware state helps in better management of the virus.

Dynamical Analysis and Design of Computational Methods for Nonlinear Stochastic Leprosy Epidemic Model

In this article, we present the dynamical analysis of the stochastic leprosy epidemic model. Positivity and boundedness are the criteria used in the deterministic model. A primary technique known as the Euler Maruyama is employed in the solution of the said model. Standard and non-standard computational methods are applied in evaluating the design stability and efficiency based on the chosen criteria. The standard computational methods like the Stochastic Euler and the Stochastic Runge Kutta fail to restore the essential features of biological problems. However, our proposed approach, the stochastic non-standard finite difference (NSFD), is used and found to be efficient, cost-effective, and accommodates all the desired feasible properties. Our method achieves all-time convergence against the backdrop of other classical techniques that perform conditionally or fail over a long period. In the end, a comparison between this scheme and the existing ones reviews the novelty of our approach.

A Fractional Epidemic Model with Mittag-Leffler Kernel for COVID-19

The aim is to explore a COVID-19 SEIR model involving Atangana-Baleanu Caputo type (ABC) fractional derivatives. Existence, uniqueness, positivity, and boundedness of the solutions for the model are established. Some stability results of the proposed system are also presented. Numerical simulations results obtained in this paper, according to the real data, show that the model is more suitable for the disease evolution.

Stability Analysis and Optimal Control of a Fractional Order Synthetic Drugs Transmission Model

In this work, a fractional-order synthetic drugs transmission model with psychological addicts has been proposed along with psychological treatment. The effects of synthetic drugs are deadly and sometimes even violent. We have studied the local and global stability of the model with different criterion. The existence and uniqueness criterion along with positivity and boundedness of the solutions have also been established. The local and global stabilities are decided by the basic reproduction number R0. We have also analyzed the sensitivity of parameters. An optimal control problem has been formulated by controlling psychological addiction and analyzed by the help of Pontryagin maximum principle. These results are verified by numerical simulations.

Modeling and analysis of fractional order Zika model

<abstract> <p>We propose mathematical model for the transmission of the Zika virus for humans spread by mosquitoes. We construct a scheme for the Zika virus model with Atangna-Baleanue Caputo sense and fractal fractional operator by using generalized Mittag-Leffler kernel. The positivity and boundedness of the model are also calculated. The existence of uniquene solution is derived and stability analysis has been made for the model by using the fixed point theory. Numerical simulations are made by using the Atangana-Toufik scheme and fractal fractional operator with a different dimension of fractional values which support the theoretical outcome of the proposed system. Developed scheme including simulation will provide better understanding in future analysis and for control strategy regarding Zika virus.</p> </abstract>

Low temperatures or high isolation delay increases the average COVID-19 infections in India : A Mathematical modeling approach

The dynamics of COVID-19 in India are captured using a set of delay differential equations by dividing a population into five compartments. The Positivity and Boundedness of the system is shown. The Existence and Uniqueness condition for the solution of system of equations is presented. The equilibrium points are calculated and stability analysis is performed. Sensitivity analysis is performed on the parameters of the model. Bifurcation analysis is performed and the critical delay is calculated. By formulating the spread parameter as a function of temperature, the impact of temperature on the population is studied. We concluded that with the decrease in temperature, the average infections in the population increases. In view of the coming winter season in India, there will be an increase in new infections. This model falls in line with the characteristics that increase in isolation delay increases average infections in the population.