Numerical study for multi-strain tuberculosis (TB) model of variable-order fractional derivatives

Area of Science:

• Mathematical modeling
• Epidemiology
• Fractional calculus

Background:

• Tuberculosis (TB) remains a global health challenge, with drug-resistant strains complicating treatment.
• Existing mathematical models often use integer-order derivatives, which may not fully capture complex disease dynamics.
• Previous multi-strain TB models, like Arino and Soliman's 2014 work, provide a foundation for extensions.

Purpose of the Study:

• To develop and analyze a novel multi-strain tuberculosis model using variable-order fractional derivatives.
• To incorporate three distinct TB strains: drug-sensitive, multi-drug resistant (MDR), and extensively drug-resistant (XDR).
• To numerically simulate and compare the performance of two finite difference methods for this fractional model.

Main Methods:

• The study employs a variable-order fractional derivative, defined using the Grünwald-Letnikov definition.
• Two numerical methods, the standard finite difference method (SFDM) and the nonstandard finite difference method (NSFDM), are presented.
• A comparative analysis of SFDM and NSFDM is conducted to evaluate their efficacy.

Main Results:

• Numerical simulations were performed using both SFDM and NSFDM.
• The nonstandard finite difference method (NSFDM) demonstrated superior numerical stability over a wider range of parameters compared to SFDM.
• NSFDM was found to preserve the positivity of the model's solutions, a crucial aspect for epidemiological models.

Conclusions:

• The developed variable-order fractional TB model offers a more nuanced approach to understanding TB transmission dynamics, including drug resistance.
• NSFDM is a robust and reliable numerical technique for simulating fractional differential equation models in epidemiology.
• This research highlights the potential of fractional calculus in modeling infectious diseases and the advantages of NSFDM for such applications.