We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

Dynamic optimization: The calculus of variations and optimal control in economics and management: Morton I. KAMIEN and Nancy L. SCHWARTZ Volume 4 in: Dynamic Economics: Theory and Applications, North-Holland, New York, 1981, xi + 331 pages, Dfl.90.00

Non-pharmaceutical interventions during the COVID-19 pandemic: A review.

Infectious diseases and human behavior are intertwined. On one side, our movements and interactions are the engines of transmission. On the other, the unfolding of viruses might induce changes to our daily activities. While intuitive, our understanding of such feedback loop is still limited. Before COVID-19 the literature on the subject was mainly theoretical and largely missed validation. The main issue was the lack of empirical data capturing behavioral change induced by diseases. Things have dramatically changed in 2020. Non-pharmaceutical interventions (NPIs) have been the key weapon against the SARS-CoV-2 virus and affected virtually any societal process. Travels bans, events cancellation, social distancing, curfews, and lockdowns have become unfortunately very familiar. The scale of the emergency, the ease of survey as well as crowdsourcing deployment guaranteed by the latest technology, several Data for Good programs developed by tech giants, major mobile phone providers, and other companies have allowed unprecedented access to data describing behavioral changes induced by the pandemic. Here, I aim to review some of the vast literature written on the subject of NPIs during the COVID-19 pandemic. In doing so, I analyze 347 articles written by more than 2518 of authors in the last $12$ months. While the large majority of the sample was obtained by querying PubMed, it includes also a hand-curated list. Considering the focus, and methodology I have classified the sample into seven main categories: epidemic models, surveys, comments/perspectives, papers aiming to quantify the effects of NPIs, reviews, articles using data proxies to measure NPIs, and publicly available datasets describing NPIs. I summarize the methodology, data used, findings of the articles in each category and provide an outlook highlighting future challenges as well as opportunities

Non-pharmaceutical interventions during the COVID-19 pandemic: a rapid review

Mumps is the most common cause of acquired unilateral deafness in children, in which hearing loss occurs at all auditory frequencies. We use a box model to model hearing loss in children caused by the mumps virus, and since the fractional-order derivative retains the effect of system memory, we use the Caputo–Fabrizio fractional derivative in this modeling. In the beginning, we compute the basic reproduction number R 0 and equilibrium points of the system and investigate the stability of the system at the equilibrium point. By utilizing the Picard–Lindelof technique, we prove the existence an unique solution for given fractional CF -system of hearing loss model and investigate the stability of iterative method by fixed point theory. The optimal control of the system is determined by considering the treatment as a control strategy to reduce the number of infected people. Using the Euler method for the fractional-order Caputo–Fabrizio derivative, the approximate solution of the system is calculated. We present a numerical simulation for the transmission of disease with respect to the transmission rate and the basic reproduction number in two cases R 0 1 and R 0 > 1 . To investigate the effect of fractional order derivative on the behavior and value of each of the variables in Model 2, we calculate the results for several fractional order derivatives and compare the results. Also, considering the importance of reproduction number in the continuation of disease transmission, we analyze the sensitivity of R 0 respect to each of the model parameters and determine the impact of each parameter.

A theoretical study of the Caputo-Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control

In this work, we study the Fractional Order (FO) model HIV/AIDS involving the Liouville–Caputo and Atangana–Baleanu–Caputo derivatives. The generalized HIV/AIDS model enable and indicates that some infected specific move from symptomatic phase to the asymptomatic phase in all kind of analysis. Special iterative solutions were obtained by the use of Laplace and Sumudu transform. Existence, uniqueness of the solution and stability criteria for the FO model were obtained by fixed point theorem. For the numerical treatment of generalized HIV/AIDS model, we using Adams methods. Furthermore, the convergency of the numerical solutions were analyzed in detail. Finally, for results illustration numerical simulations are presented.

Stability analysis and numerical solutions of fractional order HIV/AIDS model

Number of well-known contagious diseases exist around the world that mainly include HIV, Hepatitis B, influenzas etc., among these, a recently contested coronavirus (COVID-19) is a serious class of such transmissible syndromes. Abundant scientific evidence the wild animals are believed to be the primary hosts of the virus. Majority of such cases are considered to be human-to-human transmission, while a few are due to wild animals-to-human transmission and substantial burdens on healthcare system following this spread. To understand the dynamical behavior such diseases, we fitted a susceptible-infectious-quarantined model for human cases with constant proportions. We proposed a model that provide better constraints on understanding the climaxes of such unseen disastrous spread, relevant consequences, and suggesting future imperative strategies need to be adopted. The main features of the work include the positivity, boundedness, existence and uniqueness of solution of the model. The conditions were derived under which the COVID-19 may extinct or persist in the population. Sensitivity and estimation of those important parameters have been carried out that plays key role in the transmission mechanism. To optimize the spread of such disease, we present a control problem for further analysis using two control measures. The necessary conditions have been derived using the Pontryagin's maximum principle. Parameter values have been estimated from the real data and experimental numerical simulations are presented for comparison as well as verification of theoretical results. The obtained numerical results also present the verification, accuracy, validation, and robustness of the proposed scheme.

Mathematical analysis of spread and control of the novel corona virus (COVID-19) in China.

In this article, we present the transmission dynamic of the acute and chronic hepatitis B epidemic problem and develop an optimal control strategy to control the spread of hepatitis B in a community. In order to do this, first we present the model formulation and find the basic reproduction number R0. We show that if R0≤1, then the disease-free equilibrium is both locally as well as globally asymptotically stable. Then, we prove that the model is locally and globally asymptotically stable, if R0>1. To control the spread of this infection, we develop a control strategy by applying three control variables such as isolation of infected and non-infected individuals, treatment and vaccination to minimize the number of acute infected, chronically infected with hepatitis B individuals and maximize the number of susceptible and recovered individuals. Finally, we present numerical simulation to illustrate the feasibility of the control strategy.

https://www.tandfonline.com/doi/pdf/10.1080/17513758.2016.1256441

The transmission dynamic and optimal control of acute and chronic hepatitis B

We discuss the dynamic of a stochastic hepatitis B epidemic model. A stochastic hepatitis B model is formulated with a varying population environment for a long term behavior. The proposed model consists of three classes, namely the susceptible individuals in which the transmission rate is distributed by white noise, the infected individuals in which the same perturbation occurs and the recovered individuals. We derive sufficient conditions for the extinction and the persistence. Finally, we carry out the numerical simulations to support our analytical results.

The extinction and persistence of the stochastic hepatitis B epidemic model

In this paper, we present an alternative representation of the advection–reaction diffusion model involving fractional-order derivatives with Mittag-Leffler kernel. The study includes three main aspects: existence and uniqueness of solutions, Hyers–Ulam stability, and numerical simulations. For the existence and uniqueness of solutions, we use fixed point approach; also, we presents the Hyers–Ulam stability. For the numerical simulations, a new numerical scheme that involve Lagrange interpolation, Laplace transform and forward Euler technique is considered. Numerical simulations were obtained for some specific parameters.

Tahir Khan

Papers

Stability analysis and numerical solutions of fractional order HIV/AIDS model

Mathematical analysis of spread and control of the novel corona virus (COVID-19) in China.

The transmission dynamic and optimal control of acute and chronic hepatitis B

The extinction and persistence of the stochastic hepatitis B epidemic model

Stability analysis for fractional order advection–reaction diffusion system