Numerical analysis of polio model: A mathematical approach to epidemiological model using derivative with Mittag–Leffler Kernel

Abstract

The goal of this study is to analyze and obtain a new numerical approach to an important mathematical model called polio, which is one of the highly infectious and dangerous diseases challenging many lives in most developing nations, most especially in Africa, Latin America, and Asia. A number of research outputs have proven beyond doubt that modeling with noninteger-order derivative is much more accurate and reliable when compared with the integer-order counterparts. In the present case, an extension is given to the polio model by replacing the classical time derivative with the newly defined operator known as the Atangana-Baleanu fractional derivative which has its formulation based on the noble Mittag-Leffler kernel. This derivative has been tested and applied in number of ways to model a range of physical phenomena in science and engineering.The Picard-Lindelöf theorem is applied to determine the condition under which the proposed model has a solution, also to show that the solution exists and unique. Local stability analysis of the disease-free equilibrium and endemic state is also discussed. A novel approximation based on the Adams-Bashforth method is formulated to numerically approximate the fractional derivative operator. To justify the theoretical findings, some numerical results obtained for different instances of fractional order are presented.

Some sufficient conditions for global asymptotic stability of a hepatitis C virus mathematical model considering host immune system

“…In addition, some recent research work suggests that more complex or meaningful dynamical properties may arise for fractional order differential equations. [17][18][19][20] Therefore, we will expand model (1) according to the actual situation and leave it for our further research work.…”

“…In fact, depending on the actual situation, it is also very interesting to introduce time delay in model () or consider a nonautonomous model with drug therapy. In addition, some recent research work suggests that more complex or meaningful dynamical properties may arise for fractional order differential equations 17–20 . Therefore, we will expand model () according to the actual situation and leave it for our further research work.…”

“…In addition, by (17), it is not difficult to show that, for t ≥ t(𝜀 1 ), the second derivative of L(t) is also bounded. This implies that .…”

A computational technique for the Caputo fractal-fractional diabetes mellitus model without genetic factors

“…Nowadays, mathematical modeling is considered an effective and important tool for describing the cause and transmission dynamics of many common infectious diseases such as the HIV/AIDS [ 29 , 30 ], COVID-19 [ 27 , 31 , 48 ], tuberculosis (TB) [ 3 , 37 ], Lassa fever [ 4 ], Ebola, syphilis [ 9 ] cancer cells [ 36 ], polio [ 21 ] and many others which are classified in [ 4 ]. In recent years, a fractional-order derivative which is being regarded as an extension of the integer-order derivative has gained a lot of scholars’ attention due to its nonlocal properties and memory effects [ 22 , 41 ].…”

Mathematical analysis and numerical simulation of time-fractional host-parasitoid system with Caputo operator

“…What's more, several problems in biology, finance, physics, chemistry, and applied sciences are mathematically modeled by systems of ordinary and fractional differential equations [7][8][9][10]. Based on the successful application of the fractional order derivative to model a number real-life phenomena [11][12][13][14][15], we are motivated to consider a coupled two-component differential system of equations…”