A Well-Posed Fractional Order Cholera Model with Saturated Incidence Rate.

Area of Science:

• Epidemiology
• Mathematical Biology
• Fractional Calculus

Background:

• Cholera transmission dynamics are complex and require sophisticated modeling approaches.
• Traditional Susceptible-Infected-Recovered (SIR) models may not fully capture disease dynamics, especially with saturated incidence rates.
• Fractional-order differential equations offer a powerful tool for modeling memory effects in biological systems.

Purpose of the Study:

• To construct and analyze a fractional-order cholera model using the Caputo sense.
• To investigate the impact of saturated incidence rates on disease transmission.
• To determine the conditions for disease eradication versus persistence based on the basic reproduction ratio (R0).

Main Methods:

• Development of a fractional-order SIR model incorporating a saturated incidence rate.
• Analysis of the model's solutions, including positivity, boundedness, existence, and uniqueness.
• Computation and stability analysis of equilibrium points (disease-free and endemic) using R0.
• Numerical simulations to validate analytical findings and explore the role of fractional order and public awareness.

Main Results:

• The model's solutions are proven to be positive, bounded, and possess unique existence.
• The basic reproduction ratio (R0) is identified as the critical threshold for disease dynamics.
• If R0 < 1, the disease-free equilibrium is locally asymptotically stable, indicating disease eradication.
• If R0 > 1, an endemic equilibrium exists and is locally asymptotically stable, signifying disease persistence.
• Numerical simulations confirm analytical results and demonstrate the biological significance of the fractional order and public awareness.

Conclusions:

• Fractional-order modeling provides a more nuanced understanding of cholera transmission dynamics.
• The basic reproduction ratio (R0) is a key determinant of cholera's epidemiological outcome.
• Public awareness campaigns can significantly influence disease control, as shown through numerical analysis.