CLASSICAL PIECEWISE HYBRID WITH A FRACTIONAL DERIVATIVE FOR THE EPIDEMIC MODEL: DYNAMICAL TRANSMISSI

Abstract

ln this research, we developed an epidemic model with a combination of Atangana-Baleanu Caputo derivative and classical operators for the hybrid operator's memory effects, allowing us to observe the dynamics and treatment effects at different time phases of syphilis infection caused by sex. The developed model properties, which take into account linear growth and Lipschitz requirements relating the rate of effects within its many sub-compartments according to the equilibrium points, include positivity, unique solution, exitance, and boundedness in the feasible domain. After conducting sensitivity analysis with various parameters influencing the model for the piecewise fractional operator, the reproductive number RO for the biological viability of the model is determined. Generalized Ulam-Hyers stability results are employed to preserve global stability. The investigated model thus has a unique solution in the specified subinterval in light of the Banach conclusion, and contraction as a consequence holds for the Atangana-Baleanu Caputo derivative with classical operators. The piecewise model that has been suggested has a maximum of one solution. For numerical solutions, piecewise fractional hybrid operators at various fractional order values are solved using the Newton polynomial interpolation method. A comparison is also made between Caputo operator and the piecewise derivative proposed operator. This work improves our knowledge of the dynamics of syphilis and ·offers a solid framework for assessing the effectiveness of interventions for planning and making decisions to manage the illness.

Specification

PREAMBLE TO THE DESCRPTION
THE FIELD OF INVENTION
The field of this invention is epidemic modeling through mathematical tools. Specifically,
it focuses on utilizing piecewise hybrid fractional derivatives to model the transmission
dynamics of infectious diseases, such as syphilis. This model incorporates both classical
and fractional calculus to simulate and understand disease transmission patterns more
effectively.
BACKGROUND OF THE INVENTION
Traditional epidemic models, such as the SEIR model, often rely on classical differential
equations that do not adequately account for memory effects or the inherent variability in
disease progression. The study introduces a fractional order model combined with
piecewise derivatives to address these limitations. This approach allows for a more
accurate reflection of real-world disease dynamics, considering the incubation period,
delayed response to interventions, and variations in treatment effects. The background
highlights the growing need for more precise models in understanding infectious diseases
like.syphilis, which remains a significant global health concern.
SUMMARY OF THE INVENTION
This research presents a novel fractional order epidemic model for syphilis using a
combination of Atangana-Baleanu Caputo derivatives and classical operators. The model
captures disease dynamics by incorporating memory effects and delayed responses. Key
features include:
• Positivity and boundedness of solutions within the model's feasible domain.
•A sensitivity analysis to evaluate the model's performance with various
parameters.
•A piecewise fractional operator that improves model accuracy when simulating
sudden changes in disease dynamics, such as those caused by interventions like
vaccination or treatment campaigns.
•This method provides better insights into disease spread and control strategies,
ensuring improved public health planning and resource allocation.
COMPLETE SPECIFICATION
Specifications
o Focus: The invention is in the field of mathematical epidemiology and focuses on
improving the modelling of infectious diseases, particularly syphilis, by
employing fractional calculus and piecewise derivatives for more accurate
simulations.
o Key Features: Use of Atangana-Baleanu Caputo derivatives to represent memory
effects in disease progression. Piecewise modeling to handle sudden changes, such
as the implementation of public health measures. Sensitivity analysis of the basic
reproduction number (Ro) to assess model stability and effectiveness.
o Practical Application: The model is designed to assist in the management of
syphilis by providing a more robust framework for understanding disease spread.
It supports decision-making in public health by offering a tool for planning
interventions, predicting outcomes, and optimizing resource allocation.
DESCRIPTION
This study investigates the transmission dynamics of syphilis by developing a fractional order
epidemic model using piecewise hybrid derivatives. The model integrates classical and fractional
calculus to capture memory effects and time delays in disease progression. The research
emphasizes the need for more accurate tools in public health planning to understand the dynamics
of diseases, allowing better control and prevention strategies. Key mathematical properties like
stability, boundedness, and sensitivity analysis are employed to ensure the model's reliability.
• Objective: The main objective of this study is to create a mathematical model that better
represents the spread of infectious diseases like syphilis by using fractional derivatives. The
model aims to enhance the accuracy of predictions regarding disease outbreaks and the impact
of various interventions such as vaccination or treatment campaigns. It also seeks to improve
decision-making in public health by providing insights into disease transmission and control.
• Novel Perspective: The study introduces a novel approach by using Atangana-Baleanu
Caputo fractional derivatives in combination with classical operators, which allows for
capturing both memory effects and sudden changes in disease dynamics. This approach is
particularly innovative because it offers a more realistic simulation of how diseases evolve
over time, considering factors like delayed responses and interventions. Additionally, the
piecewise fractional operator provides more flexibility in modeling abrupt transitions, such
as the effects of public health measures, making the model more adaptable to real-world
scenanos.
CLAIM:
I. Improved Epidemic Modeling: We claim that the introduction of fractional order derivatives
into the classical epidemic model significantly enhances the accuracy of predicting the
transmission of infectious diseases, such as syphilis. This improvement arises from the
inclusion of memory effects and delayed responses, which are ahsent in traditional models.
2. Innovative Use of Atangana-Baleanu Caputo Derivatives: The model leverages the AtanganaBaleanu
Caputo fractional derivatives to provide a more accurate description of disease
progression by incorporating the non-local properties of fractional calculus. This allows for
better simulation of disease transmission and intervention outcomes over time.
3. Piecewise Hybrid Operator: We claim that the use of a piecewise hybrid operator enables the
model to handle abrupt changes in disease transmission dynamics, such as those caused by
interventions (e.g., vaccinations or lockdowns). This makes the model adaptable to real-world
scenarios where such changes occur suddenly.
4. Global Stability and Boundedness: The proposed model ensures global stability and
boundedness within the feasible domain, meaning the model behaves predictably and remains
stable under various parameter conditions. This is critical for ensuring reliable simulations
and accurate forecasting of disease spread.
5. Sensitivity Analysis of Basic Reproduction Number (Ro): We claim that ihe model offers a
robust sensitivity analysis of the basic reproduction number (Ro), which provides crucial
insights into the factors that drive disease transmission. This helps in identifying key
parameters that can be targeted for effective control strategies.
6. Broader Applicability to Other Infectious Diseases: The fractional hybrid model, although
applied to syphilis in this study, can be generalized to other infectious diseases. We claim that
the approach is versatile and can be adapted to model diseases with different transmission
mechanisms, enhancing its utility in public health planning and decision-making.

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