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In this article, we have studied the dynamics of the Ebola virus disease by means of fractional differentiation combined with a fractal dimension. It has been shown explicitly that the Ebola model is positively invariant. The fixed point theorem procedures check the solution’s existence and uniqueness to the model using the Mittag-Leffler kernel. The stability of the Ebola fractal–fractional model is studied using the Hyers–Ulam stable analysis. The numerical simulation for the trajectories of the proposed model is obtained using Lagrangian interpolation. Also, the numerical sensitivity analysis of the model is presented. For instance, a reduction in the rate of human-to-human contact results in a decline in the rate of infection, implying that the disease at a point will die out. Again, an increase in the loss of immunity rate to unity leads to a massive spread of the Ebola disease in the population. The findings in our work depict that we can achieve an Ebola-free state should the health sector give more education on maintaining a strong immune system and reducing human-to-human contact, especially during outbreaks of the Ebola disease, using measures like isolation and quarantine. Finally, using the fractal–fractional operator, we observed that any amount of memory variation and repetition in the outbreak of the disease influences the spread of the Ebola disease.

Numerous organisations across the globe have significant challenges about corruption, characterised by a systematic, endemic, and pervasive nature that permeates various societal establishments. Hence, we propose the fractional order model of corruption, which encompasses the involvement of corrupt individuals across various stages of education and employment. Specifically, we examine the presence of corruption among children in elementary schools, youths in tertiary institutions, adults in civil services, adults in government and public offices, and individuals who have renounced their involvement in corrupt practices. The basic reproduction number of the system was determined by utilising the next-generation matrix. The strength number was obtained by calculating the second derivative of the corruption-related compartments. The examined model solution’s existence, uniqueness, and stability were established using the Krasnoselski fixed point theorem, the Banach contraction principle, and the Ulam–Hyers theorem, respectively. Based on the numerous figures presented, our simulations indicate a positive correlation between the decline in fractal–fractional order and the increase in the number of individuals susceptible to corruption. This phenomenon results in an increase in the prevalence of corruption among designated sectors of the general population. The persistence of corruption in society is a significant challenge to its eradication, as individuals who see personal gains from engaging in corrupt practices tend to exhibit a recurring inclination towards such behaviour. Nevertheless, it is recommended that to mitigate corruption within various corruption-prone subcategories, there is a need to enhance the level of consciousness and promotion of anti-corruption measures throughout all societal establishments.

We consider an epidemiological model that takes into account pathogens in the environment and social distancing. Such model with six classes has been updated with fractional differential operators and piecewise differential operators. Equilibrium points and reproductive number for the model have been obtained using next generation approach. Necessary optimality conditions for the model modified by addition of some control variables have been presented. For the numerical solution of the model, a numerical method based on Newton polynomial has been considered. The graphical representations for the model with fractional and piecewise derivatives have been presented to better understand the dynamic of model incorporating environment and social distancing.

It has been discovered that fractal-fractional differential operators with constant orders make good mathematical models for more challenging problems observed in nature. This idea has lately been expanded to include variable orders, which seem to be better suited for anomalous processes than their fractional counterparts were. In order to establish a numerical system for resolving such issues, we have employed two straightforward methods in this study. We provided a thorough theory on the existence and uniqueness of exact solution for each case. With several examples, we described the consistency and stability requirements in each situation.

A fractional system for transmission dynamics behavior of influenza infection with two strains is investigated and applied to the qualitative assess of crossing immunities on the expansion of the disease. The generalized mathematical model is significantly more informative regarding the infection outbreak and gives a deep description of the infection’s dynamic process. We firstly take the integer order system given in current literature and then we globalize it by applying the Caputo operator of fractional derivatives. Moreover, we present the rudimentary results of the Caputo operator for the examination of the system, comprising the basic reproduction number R0 and equilibrium points stabilities with the help of the Routh–Hurwitz and the Lyapunov operators techniques. We also focus on the approximate root. For simulations, we apply the famous generalize predictor–corrector Adams–Bashforth Moulton method, which shows the impact of two strains of influenza disease.

Scientists and researchers are increasingly interested in mathematical modelling of infectious diseases with non-integer order. It is self-evident that a fixed order can only characterize classical models in epidemiology, but models with fractional-order derivatives are not fixed order. When a derivative has a non-fixed order, it becomes more potent in representing real-world problems. Different fresh techniques concerning fractional operators, such as the exponential decay and the Mittag-Leffler kernel, have been proposed in recent years, which transcend the constraints of prior fractional-order derivatives. These novel operators are functional in modelling problems in science and engineering—a more recent operator in fractional calculus, known as the fractal–fractional operator. Fractal–fractional operators have not been frequently utilized to explore complex dynamics of infectious disease spread in an ecosystem. In this paper, a novel fractal–fractional operator is used to study the dynamics of Q fever in livestock and ticks. Fixed point theorem is used to prove the model’s existence and uniqueness under the Atangana–Baleanu (Mittag-Leffler kernel) fractal–fractional operator. A non-linear analysis is used to determine the model’s Hyers–Ulam stability. A numerical scheme for the Q fever disease using the newly constructed numerical scheme based on Newton polynomial is presented. The numerical simulation of the model shows some exciting dynamics of the disease in an ecosystem. The fractal–fractional dynamism shows that a change in the geometric pattern of nature also affects the spread of Q fever. For example, it is noticed that the fractal dimensions and fractional order produces different asymptotic stabilities for symptomatic livestock, Coxiella burnetii in the environment, susceptible and infected ticks. It is of a view that this study will also contribute to the rich dynamics of the fractal and the fractional perspective of disease spread in nature.

This article deals with the Fitzhugh–Nagumo equation in the presence of stochastic function. A numerical scheme has been developed for the solution of such equations which preserves the certain structure of the unknown functions, also we have given the stability analysis, consistency of the problem, and explicitly optimal a priori estimates for the existence of solutions of such equations. A unique solution has been guaranteed. The corresponding explicit estimates in the function spaces are formulated in the form of theorems. Lastly, one important feature of the article is the simulation of the proposed numerical scheme in the form of the 2D and 3D plots which shows the efficacy of the stochastic analysis of such nonlinear partial differential equations.

The current work is of interest to introduce a detailed analysis of the novel fractional COVID-19 model. Non-local fractional operators are one of the most efficient tools in order to understand the dynamics of the disease spread. For this purpose, we intend as an attempt at investigating the fractional COVID-19 model through Caputo operator with order χ∈(0,1). Employing the fixed point theorem, it is shown that the solutions of the proposed fractional model are determined to satisfy the existence and uniqueness conditions under the Caputo derivative. On the other hand, its iterative solutions are indicated by making use of the Laplace transform of the Caputo fractional operator. Also, we establish the stability criteria for the fractional COVID-19 model via the fixed point theorem. The invariant region in which all solutions of the fractional model under investigation are positive is determined as the non-negative hyperoctant R+7. Moreover, we perform the parameter estimation of the COVID-19 model by utilizing the non-linear least squares curve fitting method. The sensitivity analysis of the basic reproduction number R0c is carried out to determine the effects of the proposed fractional model’s parameters on the spread of the disease. Numerical simulations show that all results are in good agreement with real data and all theoretical calculations about the disease.

Mathematical models are mainly used to depict real world problems that humans encounter in their daily explorations, investigations and activities. However, these mathematical models have some limitations as indeed the big challenges are the conversion of observations into mathematical formulations. If this conversion is inefficient, then mathematical models will provide some predictions with deficiencies. A specific real-world problem could then have more than one mathematical model, each model with its advantages and disadvantages. In the last months, the spread of covid-19 among humans have become fatal, destructive and have paralyzed activities across the globe. The lockdown regulations and many other measures have been put in place with the hope to stop the spread of this deathly disease that have taken several souls around the globe. Nevertheless, to predict the future behavior of the spread, humans rely on mathematical models and their simulations. While many models, have been suggested, it is important to point out that all of them have limitations therefore newer models can still be suggested. In this paper, we examine an alternative model depicting the spread behavior of covid-19 among humans. Different differential and integral operators are used to get different scenarios.

Numerous organisations across the globe have significant challenges about corruption, characterised by a systematic, endemic, and pervasive nature that permeates various societal establishments. Hence, we propose the fractional order model of corruption, which encompasses the involvement of corrupt individuals across various stages of education and employment. Specifically, we examine the presence of corruption among children in elementary schools, youths in tertiary institutions, adults in civil services, adults in government and public offices, and individuals who have renounced their involvement in corrupt practices. The basic reproduction number of the system was determined by utilising the next-generation matrix. The strength number was obtained by calculating the second derivative of the corruption-related compartments. The examined model solution's existence, uniqueness, and stability were established using the Krasnoselski fixed point theorem, the Banach contraction principle, and the Ulam–Hyers theorem, respectively. Based on the numerous figures presented, our simulations indicate a positive correlation between the decline in fractal–fractional order and the increase in the number of individuals susceptible to corruption. This phenomenon results in an increase in the prevalence of corruption among designated sectors of the general population. The persistence of corruption in society is a significant challenge to its eradication, as individuals who see personal gains from engaging in corrupt practices tend to exhibit a recurring inclination towards such behaviour. Nevertheless, it is recommended that to mitigate corruption within various corruption-prone subcategories, there is a need to enhance the level of consciousness and promotion of anti-corruption measures throughout all societal establishments. • We propose the fractional order model of corruption. • We examine the presence of corruption among children in elementary schools. • The strength number was obtained by calculating the second derivative of the corruption-related compartments. • The examined model solution's existence, uniqueness, and stability. • Corrupt practices tend to exhibit a recurring inclination towards such behaviour. [ABSTRACT FROM AUTHOR]

In this article, we have studied the dynamics of the Ebola virus disease by means of fractional differentiation combined with a fractal dimension. It has been shown explicitly that the Ebola model is positively invariant. The fixed point theorem procedures check the solution's existence and uniqueness to the model using the Mittag-Leffler kernel. The stability of the Ebola fractal–fractional model is studied using the Hyers–Ulam stable analysis. The numerical simulation for the trajectories of the proposed model is obtained using Lagrangian interpolation. Also, the numerical sensitivity analysis of the model is presented. For instance, a reduction in the rate of human-to-human contact results in a decline in the rate of infection, implying that the disease at a point will die out. Again, an increase in the loss of immunity rate to unity leads to a massive spread of the Ebola disease in the population. The findings in our work depict that we can achieve an Ebola-free state should the health sector give more education on maintaining a strong immune system and reducing human-to-human contact, especially during outbreaks of the Ebola disease, using measures like isolation and quarantine. Finally, using the fractal–fractional operator, we observed that any amount of memory variation and repetition in the outbreak of the disease influences the spread of the Ebola disease. • We studied the dynamics of the Ebola virus disease by means of fractal-fractional. • The stability of the Ebola fractal–fractional model is studied. • The trajectories of the proposed model is obtained using Lagrangian interpolation. • A reduction in the rate of human-to-human contact results in a decline in the rate of infection. • Memory variation and repetition in the outbreak of the disease influences the spread of the Ebola disease. [ABSTRACT FROM AUTHOR]

We consider an epidemiological model that takes into account pathogens in the environment and social distancing. Such model with six classes has been updated with fractional differential operators and piecewise differential operators. Equilibrium points and reproductive number for the model have been obtained using next generation approach. Necessary optimality conditions for the model modified by addition of some control variables have been presented. For the numerical solution of the model, a numerical method based on Newton polynomial has been considered. The graphical representations for the model with fractional and piecewise derivatives have been presented to better understand the dynamic of model incorporating environment and social distancing. • The effect of environment and social distance to Covid-19 spread. • A Covid-19 model with fractional and fractal–fractional operators. • Covid-19 model with piecewise differential and integral operators. • The calculation of basic reproduction number and equilibrium points. • Necessary optimality conditions for Covid-19 model under some control strategies. [ABSTRACT FROM AUTHOR]

• Existence and uniqueness results for Cauchy problems with fractal variable order and constant fractional order. • Some novel inequalities related to fractal-fractional integrals. • Numerical schemes for Cauchy problems with fractal-fractional nonlinear ordinary differential equations. It has been discovered that fractal-fractional differential operators with constant orders make good mathematical models for more challenging problems observed in nature. This idea has lately been expanded to include variable orders, which seem to be better suited for anomalous processes than their fractional counterparts were. In order to establish a numerical system for resolving such issues, we have employed two straightforward methods in this study. We provided a thorough theory on the existence and uniqueness of exact solution for each case. With several examples, we described the consistency and stability requirements in each situation. [ABSTRACT FROM AUTHOR]

A fractional system for transmission dynamics behavior of influenza infection with two strains is investigated and applied to the qualitative assess of crossing immunities on the expansion of the disease. The generalized mathematical model is significantly more informative regarding the infection outbreak and gives a deep description of the infection's dynamic process. We firstly take the integer order system given in current literature and then we globalize it by applying the Caputo operator of fractional derivatives. Moreover, we present the rudimentary results of the Caputo operator for the examination of the system, comprising the basic reproduction number R 0 and equilibrium points stabilities with the help of the Routh–Hurwitz and the Lyapunov operators techniques. We also focus on the approximate root. For simulations, we apply the famous generalize predictor–corrector Adams–Bashforth Moulton method, which shows the impact of two strains of influenza disease. • Formulation of the fractional model. • Existence and uniqueness of the global positive solution are obtained. • Deterministic analysis of the model. • Stability and approximate of the results. • Numerical simulation of the model. [ABSTRACT FROM AUTHOR]

Scientists and researchers are increasingly interested in mathematical modelling of infectious diseases with non-integer order. It is self-evident that a fixed order can only characterize classical models in epidemiology, but models with fractional-order derivatives are not fixed order. When a derivative has a non-fixed order, it becomes more potent in representing real-world problems. Different fresh techniques concerning fractional operators, such as the exponential decay and the Mittag-Leffler kernel, have been proposed in recent years, which transcend the constraints of prior fractional-order derivatives. These novel operators are functional in modelling problems in science and engineering—a more recent operator in fractional calculus, known as the fractal–fractional operator. Fractal–fractional operators have not been frequently utilized to explore complex dynamics of infectious disease spread in an ecosystem. In this paper, a novel fractal–fractional operator is used to study the dynamics of Q fever in livestock and ticks. Fixed point theorem is used to prove the model's existence and uniqueness under the Atangana–Baleanu (Mittag-Leffler kernel) fractal–fractional operator. A non-linear analysis is used to determine the model's Hyers–Ulam stability. A numerical scheme for the Q fever disease using the newly constructed numerical scheme based on Newton polynomial is presented. The numerical simulation of the model shows some exciting dynamics of the disease in an ecosystem. The fractal–fractional dynamism shows that a change in the geometric pattern of nature also affects the spread of Q fever. For example, it is noticed that the fractal dimensions and fractional order produces different asymptotic stabilities for symptomatic livestock, Coxiella burnetii in the environment, susceptible and infected ticks. It is of a view that this study will also contribute to the rich dynamics of the fractal and the fractional perspective of disease spread in nature. [ABSTRACT FROM AUTHOR]

This article deals with the Fitzhugh–Nagumo equation in the presence of stochastic function. A numerical scheme has been developed for the solution of such equations which preserves the certain structure of the unknown functions, also we have given the stability analysis, consistency of the problem, and explicitly optimal a priori estimates for the existence of solutions of such equations. A unique solution has been guaranteed. The corresponding explicit estimates in the function spaces are formulated in the form of theorems. Lastly, one important feature of the article is the simulation of the proposed numerical scheme in the form of the 2D and 3D plots which shows the efficacy of the stochastic analysis of such nonlinear partial differential equations. • We investigate the Fitzhugh–Nagumo equation in the presence of stochastic function. • We construct a numerical scheme for the solution of such equations which preserves the certain structure of the unknown functions. • We discussed a unique solution. • We demonstrated the numerical simulations. [ABSTRACT FROM AUTHOR]

The current work is of interest to introduce a detailed analysis of the novel fractional COVID-19 model. Non-local fractional operators are one of the most efficient tools in order to understand the dynamics of the disease spread. For this purpose, we intend as an attempt at investigating the fractional COVID-19 model through Caputo operator with order χ ∈ (0 , 1). Employing the fixed point theorem, it is shown that the solutions of the proposed fractional model are determined to satisfy the existence and uniqueness conditions under the Caputo derivative. On the other hand, its iterative solutions are indicated by making use of the Laplace transform of the Caputo fractional operator. Also, we establish the stability criteria for the fractional COVID-19 model via the fixed point theorem. The invariant region in which all solutions of the fractional model under investigation are positive is determined as the non-negative hyperoctant R + 7 . Moreover, we perform the parameter estimation of the COVID-19 model by utilizing the non-linear least squares curve fitting method. The sensitivity analysis of the basic reproduction number R 0 c is carried out to determine the effects of the proposed fractional model's parameters on the spread of the disease. Numerical simulations show that all results are in good agreement with real data and all theoretical calculations about the disease. [ABSTRACT FROM AUTHOR]

Mathematical models are mainly used to depict real world problems that humans encounter in their daily explorations, investigations and activities. However, these mathematical models have some limitations as indeed the big challenges are the conversion of observations into mathematical formulations. If this conversion is inefficient, then mathematical models will provide some predictions with deficiencies. A specific real-world problem could then have more than one mathematical model, each model with its advantages and disadvantages. In the last months, the spread of covid-19 among humans have become fatal, destructive and have paralyzed activities across the globe. The lockdown regulations and many other measures have been put in place with the hope to stop the spread of this deathly disease that have taken several souls around the globe. Nevertheless, to predict the future behavior of the spread, humans rely on mathematical models and their simulations. While many models, have been suggested, it is important to point out that all of them have limitations therefore newer models can still be suggested. In this paper, we examine an alternative model depicting the spread behavior of covid-19 among humans. Different differential and integral operators are used to get different scenarios. [ABSTRACT FROM AUTHOR]