The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.

The Ensemble Kalman Filter: Theoretical formulation and practical implementation

Fractional finite difference methods are useful to solve the fractional differential equations. The aim of this article is to prove the stability and convergence of the fractional Euler method, the fractional Adams method and the high order methods based on the convolution formula by using the generalized discrete Gronwall inequality. Numerical experiments are also presented, which verify the theoretical analysis.

The finite difference methods for fractional ordinary differential equations

In this work, we formulate and mathematically study integer and fractional models of typhoid fever transmission dynamics. The models include vaccination as a control measure. After recalling some preliminary results for the integer model (determination of the epidemiological threshold denoted by Rc, asymptotic stability of the equilibrium point without disease whenever Rc 1), we replace the integer derivative with the Caputo derivative. We perform a stability analysis of the disease-free equilibrium and prove the existence and uniqueness of the solution of the fractional model using fixed point theory. We construct the numerical scheme and prove its stability. Simulation results show that when the fractional-order η decreases, the peak of infected humans is delayed. To reduce the proliferation of the disease, mass vaccination combined with environmental sanitation is recommended. We then extend the previous model by replacing the mass action incidences with standard incidences. We compute the corresponding epidemiological threshold denoted by Rc⋆ and ensure the uniform stability of the disease-free equilibrium, for both new models, when Rc⋆<1. A new calibration of the new model is conducted with real data of Mbandjock, Cameroon, to estimate Rc⋆=1.4348. We finally perform several numerical simulations that permit us to conclude that such diseases can possibly be tackled through vaccination combined with environmental sanitation.

Fractional Dynamics of Typhoid Fever Transmission Models with Mass Vaccination Perspectives

Biomathematics has become one of the most significant areas of research as a result of interdisciplinary study. Chronic diseases sometimes referred to as non-communicable and communicable diseases, are conditions that develop over an extended period as a result of different factors like genetics, lifestyle, and environment. The most important common types of disease are cardiovascular, alcohol, cancer, and diabetes. More than three-quarters of the world’s (31.4 million) deaths occur in low- and middle-income nations, which are disproportionately affected by different infections. Fractional Calculus is a prominent topic for research within the discipline of Applied Mathematics due to its usefulness in solving problems in many different branches of science, engineering, and medicine. Recent researchers have identified the importance of mathematical tools in various disease models as being very useful to study the dynamics with the help of fractional and integer calculus modeling. Due to the complexity of the underlying connections, both deterministic and stochastic epidemiological models are founded on an inadequate understanding of the infectious network. Over the past several years, the use of different fractional operators to model the problem has grown, and it is now a common way to study how epidemics spread. Recently, researchers have actively considered fractional calculus to study different diseases like COVID-19, cancer, TB, HIV, dengue fever, diabetes, cholera, pine welts, smoking and heart attacks, etc. With the help of fractional operator, we modified a mathematical model for the dynamical transmission, analysis, treatment, vaccination, and precaution leveling necessary to mitigate the negative impact of illness on society in the long run, overcoming the memory effect without defining or considering others parameters. In this review paper, we considered all the recent studies based on the fractional modeling of infectious and non-infectious diseases with different fractional operators such as Caputo, Caputo Fabrizio, ABC, and constant proportional with Caputo, etc. This review paper aims to bring all the information together by considering different fractional operators and their uses in the field of infectious disease modeling. The steps taken to accomplish the goal were developing a mathematical model, identifying the equilibrium point, figuring out the minimal reproductive number, and assessing the stability around the equilibrium point. For future direction, we consider the cancer model to study the growth cells of cancer and the impact of therapy to control infections. An equilibrium solution and an analysis of the behavior dynamics of the cell spread with treatment in the form of chemotherapy were obtained. The simulation shows that the population of cancer cells is influenced by the pace of cancer cell growth with the Caputo fractional derivative. The acquired results show how effective and precise the suggested approach is in helping to better understand how chemotherapy works. Chemotherapy medications have been found to increase immunity against particular cancer by reducing the number of tumor cells. Further, we suggested some future work directions with the help of the new hybrid fractional operator. Our innovative methodology might have significant effects on global stakeholders, policymakers, and national health systems. The current strategies for controlling outbreaks and the vaccination and prevention policies that have been implemented would benefit from a more accurate representation of the dynamics of contagious diseases, which necessitates the development of highly complex mathematical models. Microorganisms, interactions between individuals or groups, and environmental, social, economic, and demographic factors on a broader scale are all examples. 

A review on epidemic models in sight of fractional calculus

In this work, we formulate a mathematical model with a non-integer order derivative to investigate typhoid fever transmission dynamics. To combat the spread of this disease in the human community, control measures like vaccination are included in the proposed model. We calculate the epidemiological threshold called the control reproduction number, R c , and perform the asymptotic stability of the typhoid-free equilibrium point. We prove that the typhoid-free equilibrium for both integer and non-integer models is locally and globally asymptotically stable whenever R c is less than one. We also prove that both models admit only one endemic equilibrium point which is globally asymptotically stable whenever R c > 1 and no endemic equilibrium point otherwise. This means that the backward bifurcation phenomenon does not occur. In absence of vaccination, R c is equal to the basic reproduction number R 0 . We found out that R c R 0 which means that vaccination can permit to decrease of the spread of typhoid fever in the human community. Using fixed point theory, we perform existence and uniqueness analysis of solutions of the fractional model. We use the Adams-Bashforth-Moulton method to construct a numerical scheme of the fractional model. We prove the stability of the proposed numerical scheme. To calibrate our model, we estimate model parameters on clinical data of Mbandjock District Hospital in the Centre Region of Cameroon, using the Non-linear Least-Square method. This permits us to find R c = 1.3722 , which means that we are in an endemic state (since R c > 1 ), and then to predict new cases of typhoid fever per month at Mbandjock in the next new year. To determine model parameters that are responsible for disease spread in the human community, we perform sensitivity analysis (SA). This analysis shows that the vaccination rate, the human-bacteria contact rate, as well as the recovery rate, are the most important parameters in the disease spread. To validate our analytical results, and to see the impact of some control measures in the spread of typhoid fever in the human community, as well as the impact of the fractional-order on typhoid transmission dynamics, we perform several numerical simulations.

Adamou Dang Koko

Papers

Projections and fractional dynamics of the typhoid fever: A case study of Mbandjock in the Centre Region of Cameroon