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This research looked at nonlinear ordinary differential equations with global differential operators and the Dirac-delta and exponential decay kernels. A recently developed numerical approach based on the repetitive use of the well-known midpoint quadrature approximation. Although no theoretical analysis was offered, the method was applied to solve several nonlinear equations in chaos and epidemiology. The observed findings demonstrate the effect of the chosen function gt, for example, a simple SIR model produced chaotic and crossover behaviors.

Abstract In order to comprehend the dynamics of disease propagation within a society, mathematical formulations are essential. The purpose of this work is to investigate the diagnosis and treatment of lung cancer in persons with weakened immune systems by introducing cytokines ( $$ IL_{2} \& IL_{12}$$ I L 2 & I L 12 ) and anti-PD-L1 inhibitors. To find the stable position of a recently built system TCD $$IL_{2} IL_{12}$$ I L 2 I L 12 Z, a qualitative and quantitative analysis are taken under sensitive parameters. Reliable bounded findings are ensured by examining the generated system’s boundedness, positivity, uniqueness, and local stability analysis, which are the crucial characteristics of epidemic models. The positive solutions with linear growth are shown to be verified by the global derivative, and the rate of impact across every sub-compartment is determined using Lipschitz criteria. Using Lyapunov functions with first derivative, the system’s global stability is examined in order to evaluate the combined effects of cytokines and anti-PD-L1 inhibitors on people with weakened immune systems. Reliability is achieved by employing the Mittag-Leffler kernel in conjunction with a fractal-fractional operator because FFO provide continuous monitoring of lung cancer in multidimensional way. The symptomatic and asymptomatic effects of lung cancer sickness are investigated using simulations in order to validate the relationship between anti-PD-L1 inhibitors, cytokines, and the immune system. Also, identify the actual state of lung cancer control with early diagnosis and therapy by introducing cytokines and anti-PD-L1 inhibitors, which aid in the patients’ production of anti-cancer cells. Investigating the transmission of illness and creating control methods based on our validated results will both benefit from this kind of research.

In order to comprehend the dynamics of disease propagation within a society, mathematical formulations are essential. The purpose of this work is to investigate the diagnosis and treatment of lung cancer in persons with weakened immune systems by introducing cytokines ( I L 2 & I L 12 ) and anti-PD-L1 inhibitors. To find the stable position of a recently built system TCD I L 2 I L 12 Z, a qualitative and quantitative analysis are taken under sensitive parameters. Reliable bounded findings are ensured by examining the generated system’s boundedness, positivity, uniqueness, and local stability analysis, which are the crucial characteristics of epidemic models. The positive solutions with linear growth are shown to be verified by the global derivative, and the rate of impact across every sub-compartment is determined using Lipschitz criteria. Using Lyapunov functions with first derivative, the system’s global stability is examined in order to evaluate the combined effects of cytokines and anti-PD-L1 inhibitors on people with weakened immune systems. Reliability is achieved by employing the Mittag-Leffler kernel in conjunction with a fractal-fractional operator because FFO provide continuous monitoring of lung cancer in multidimensional way. The symptomatic and asymptomatic effects of lung cancer sickness are investigated using simulations in order to validate the relationship between anti-PD-L1 inhibitors, cytokines, and the immune system. Also, identify the actual state of lung cancer control with early diagnosis and therapy by introducing cytokines and anti-PD-L1 inhibitors, which aid in the patients’ production of anti-cancer cells. Investigating the transmission of illness and creating control methods based on our validated results will both benefit from this kind of research. [ABSTRACT FROM AUTHOR]

Mathematical formulations are crucial in understanding the dynamics of disease spread within a community. The objective of this research is to investigate the SEIR model of SARS-COVID-19 (C-19) with the inclusion of vaccinated effects for low immune individuals. A mathematical model is developed by incorporating vaccination individuals based on a proposed hypothesis. The fractal-fractional operator (FFO) is then used to convert this model into a fractional order. The newly developed SEVIR system is examined in both a qualitative and quantitative manner to determine its stable state. The boundedness and uniqueness of the model are examined to ensure reliable findings, which are essential properties of epidemic models. The global derivative is demonstrated to verify the positivity with linear growth and Lipschitz conditions for the rate of effects in each sub-compartment. The system is investigated for global stability using Lyapunov first derivative functions to assess the overall impact of vaccination. In fractal-fractional operators, fractal represents the dimensions of the spread of the disease, and fractional represents the fractional ordered derivative operator. We use combine operators to see real behavior of spread as well as control of COVID-19 with different dimensions and continuous monitoring. Simulations are conducted to observe the symptomatic and asymptomatic effects of the corona virus disease with vaccinated measures for low immune individuals, providing insights into the actual behavior of the disease control under vaccination effects. Such investigations are valuable for understanding the spread of the virus and developing effective control strategies based on justified outcomes.

Mathematical formulations are crucial in understanding the dynamics of disease spread within a community. The objective of this research is to investigate the SEIR model of SARSCOVID-19 (C-19) with the inclusion of vaccinated effects for low immune individuals. A mathematical model is developed by incorporating vaccination individuals based on a proposed hypothesis. The fractal-fractional operator (FFO) is then used to convert this model into a fractional order. The newly developed SEVIR system is examined in both a qualitative and quantitative manner to determine its stable state. The boundedness and uniqueness of the model are examined to ensure reliable findings, which are essential properties of epidemic models. The global derivative is demonstrated to verify the positivity with linear growth and Lipschitz conditions for the rate of effects in each sub-compartment. The system is investigated for global stability using Lyapunov first derivative functions to assess the overall impact of vaccination. In fractal-fractional operators, fractal represents the dimensions of the spread of the disease, and fractional represents the fractional ordered derivative operator. We use combine operators to see real behavior of spread as well as control of COVID-19 with different dimensions and continuous monitoring. Simulations are conducted to observe the symptomatic and asymptomatic effects of the corona virus disease with vaccinated measures for low immune individuals, providing insights into the actual behavior of the disease control under vaccination effects. Such investigations are valuable for understanding the spread of the virus and developing effective control strategies based on justified outcomes. [ABSTRACT FROM AUTHOR]

Mathematical formulations are essential tool to show the dynamics that how various diseases spread in the community. Differential equations with fractional or integer order can be utilized to see the effect of the dynamics direct or indirect Leptospirosis transmission, which are analyzed with different aspects. A mathematical description and dynamical sketch of Leptospirosis with environmental effects have been studied as a result of the successful efforts of various writers. In this study, we analyzed the Leptospirosis model described using a nonlinear fractional-order differential equation that takes the environmental effects into consideration. The proposed fractional order system is investigated qualitatively as well as quantitatively to identify its stable position. Local stability of the Leptospirosis system is verified and test the system with flip bifurcation. Also system is investigated for global stability using Lyapunov first and second derivative functions. The existence, boundedness and positivity of the Leptospirosis is checked, which are the key properties for such of type of epidemic problem to identify reliable findings. Effect of global derivative is demonstrated to verify its rate of effects according to their sub-compartments. Solutions for fractional order system are derived with the help of advanced tool fractal fractional operator for different fractional values. Simulation are carried out to see symptomatic as well as a asymptomatic effects of Leptospirosis in the world wide, also show the actual behavior of Leptospirosis which will be helpful to understand the outbreak of Leptospirosis with environmental effects as well as for future prediction and control strategies.

We have considered a special class of ordinary differential equations in which the differential operators are those of the Caputo fractional global derivative. These equations are generalizations of the well-known differential equations with the Caputo fractional derivative. Due to the various possible applications of these equations to model real-world problems we have first introduced some new inequalities that will be used in all fields of science, technology and engineering where these equations could be applied. We used Nagumo's principles to establish the existence and uniqueness of the solution for this class of equations with additional conditions. We have applied the midpoint principle to obtain a numerical scheme that will be used to solve these equations numerically. Some illustrative examples are presented with excellent results.

Mathematical formulations are essential tool to show the dynamics that how various diseases spread in the community. Differential equations with fractional or integer order can be utilized to see the effect of the dynamics direct or indirect Leptospirosis transmission, which are analyzed with different aspects. A mathematical description and dynamical sketch of Leptospirosis with environmental effects have been studied as a result of the successful efforts of various writers. In this study, we analyzed the Leptospirosis model described using a nonlinear fractional-order differential equation that takes the environmental effects into consideration. The proposed fractional order system is investigated qualitatively as well as quantitatively to identify its stable position. Local stability of the Leptospirosis system is verified and test the system with flip bifurcation. Also system is investigated for global stability using Lyapunov first and second derivative functions. The existence, boundedness and positivity of the Leptospirosis is checked, which are the key properties for such of type of epidemic problem to identify reliable findings. Effect of global derivative is demonstrated to verify its rate of effects according to their sub-compartments. Solutions for fractional order system are derived with the help of advanced tool fractal fractional operator for different fractional values. Simulation are carried out to see symptomatic as well as a asymptomatic effects of Leptospirosis in the world wide, also show the actual behavior of Leptospirosis which will be helpful to understand the outbreak of Leptospirosis with environmental effects as well as for future prediction and control strategies. [ABSTRACT FROM AUTHOR]

We have considered a special class of ordinary differential equations in which the differential operators are those of the Caputo fractional global derivative. These equations are generalizations of the well-known differential equations with the Caputo fractional derivative. Due to the various possible applications of these equations to model real-world problems we have first introduced some new inequalities that will be used in all fields of science, technology and engineering where these equations could be applied. We used Nagumo's principles to establish the existence and uniqueness of the solution for this class of equations with additional conditions. We have applied the midpoint principle to obtain a numerical scheme that will be used to solve these equations numerically. Some illustrative examples are presented with excellent results. [ABSTRACT FROM AUTHOR]

One of the great ability of humans is to use mathematical formulas to replicate what they see in their daily activities, they start with simple models, when these models are unable to achieve their goals and they modify or replace them with more complex mathematical formulas. Chaos has been modelled using classical differential operators, classical with fractional orders. On the other hand, many mathematical mapping have been used to simulate some fractals behaviours. In this paper, in order to capture more fractal and chaos behaviours, we introduce the notion of chaos stochastic models and fractal stochastic mapping. We additionally present the conditions of existence and uniqueness of chaos‐stochastic models. Some numerical simulations are presented to help see the effect of stochastic functions. [ABSTRACT FROM AUTHOR]

Stochastic and fractional differentiation have been developed independently to depicting processes following randomness and power, a declining memory and passage from one process to another respectively. Very recently, fractional stochastic differential equations were suggested with the aim to capture processes following at the same time randomness and memory nonlocality. In this paper to further explore the applicability of this type of differential equations, a SIR model was considered and analyzed analytically and numerically. Some numerical simulations are presented for different values of fractional orders and densities of randomness.

Very recently, the concept of instantaneous change was extended with the aim to accommodate prediction of more complex real world problems that could not be predicted or depicted by the existing rate of change. The extension gave birth to a more general differential operator that to be a derivative associate to the well-known Riemann-Stieltjes integral. In addition to this, using specific functions, one is able to recover all existing local differential operators defined as rate of change. This extended concept is still at its genesis and more works need to be done to establish a Riemann-Stieltjes calculus. In this paper, we aim to present a detailed analysis of an important class of differential equations called stochastic equations with the new classes of differential operators with the global derivative with integer and non-integer orders. We considered many classes as nonlinear Cauchy problems, then we presented existence and the uniqueness of their solutions using the linear growth and the Lipchitz conditions. We derived numerical solutions for each class and presented the error analysis. To show the applicability of these operators, we considered three epidemiological problems, including the zombie virus spread model, the zika virus spread model and Ebola model. We solved each model using the suggested numerical scheme and presented the numerical solutions for different values of fractional order and the global function gt. Our results showed that, more complex real world problems could be depicted using these classes of differential equations.

Very recently an extension of the concept of rate of change was suggested, with the aim to depict more real world problems that could not be depicted due to the limitation of the existing rate of change definition. One of the particular discovery in this is the existence of a derivative associate to Riemann-Stieltjes integral, a connection that has never been presented before. The global differential operator with its associat integral have now constructed a new calculus that is an extension of former calculus based on existing rate of change. To further see the application of this new extension, we investigated in this paper, a lassa fever model using the global differentiation in local and nonlocal sense. Existence and uniqueness conditions were verified using the theorem suggesting the linear growth and Lipschitz conditions. A numerical scheme based on the Newton polynomial was used to solve numerically the obtained models and some simulations presented.

Stochastic and fractional differentiation have been developed independently to depicting processes following randomness and power, a declining memory and passage from one process to another respectively. Very recently, fractional stochastic differential equations were suggested with the aim to capture processes following at the same time randomness and memory nonlocality. In this paper to further explore the applicability of this type of differential equations, a SIR model was considered and analyzed analytically and numerically. Some numerical simulations are presented for different values of fractional orders and densities of randomness. [ABSTRACT FROM AUTHOR]

Very recently, the concept of instantaneous change was extended with the aim to accommodate prediction of more complex real world problems that could not be predicted or depicted by the existing rate of change. The extension gave birth to a more general differential operator that to be a derivative associate to the well-known Riemann-Stieltjes integral. In addition to this, using specific functions, one is able to recover all existing local differential operators defined as rate of change. This extended concept is still at its genesis and more works need to be done to establish a Riemann-Stieltjes calculus. In this paper, we aim to present a detailed analysis of an important class of differential equations called stochastic equations with the new classes of differential operators with the global derivative with integer and non-integer orders. We considered many classes as nonlinear Cauchy problems, then we presented existence and the uniqueness of their solutions using the linear growth and the Lipchitz conditions. We derived numerical solutions for each class and presented the error analysis. To show the applicability of these operators, we considered three epidemiological problems, including the zombie virus spread model, the zika virus spread model and Ebola model. We solved each model using the suggested numerical scheme and presented the numerical solutions for different values of fractional order and the global function g t . Our results showed that, more complex real world problems could be depicted using these classes of differential equations. [ABSTRACT FROM AUTHOR]

Very recently an extension of the concept of rate of change was suggested, with the aim to depict more real world problems that could not be depicted due to the limitation of the existing rate of change definition. One of the particular discovery in this is the existence of a derivative associate to Riemann-Stieltjes integral, a connection that has never been presented before. The global differential operator with its associat integral have now constructed a new calculus that is an extension of former calculus based on existing rate of change. To further see the application of this new extension, we investigated in this paper, a lassa fever model using the global differentiation in local and nonlocal sense. Existence and uniqueness conditions were verified using the theorem suggesting the linear growth and Lipschitz conditions. A numerical scheme based on the Newton polynomial was used to solve numerically the obtained models and some simulations presented. [ABSTRACT FROM AUTHOR]