Conformable pertussis dynamics with waning immunity and natural boosting: stability regions and a structure-preserving NSFD scheme

Abstract

Pertussis recurrence is influenced by the interaction of vaccine-induced immunity, waning post-infection immunity, natural immune boosting, and secondary susceptibility. A six-compartment model is formulated using the difference–quotient representation of the conformable relaxation equation, with primary susceptible S, vaccinated V, infected I, recovered R, waning immunity W, and secondary-susceptible F populations. For differentiable state variables, the conformable operator is equivalent to an ordinary derivative under the nonlinear time transformation \(u=t^\alpha /\alpha\). Hence, the conformable order changes the timing of transient dynamics without altering the equilibrium equations or the reproduction threshold. Existence and uniqueness, non-negativity, boundedness, disease-free and endemic equilibria, the control reproduction number, and local stability are established. Two-parameter stability regions for the disease-free regime \(\mathscr {R}_{c}<1\) and the endemic regime \(\mathscr {R}_{c}>1\) are obtained numerically from the spectral abscissa of the corresponding Jacobian. A nonstandard finite difference (NSFD) scheme is then constructed using the DQR/ESDDFD denominator obtained from the conformable relaxation structure. The discrete scheme preserves non-negativity, while boundedness on finite computational intervals is established by mathematical induction. Numerical simulations for several conformable orders support the analytical threshold behavior: below the threshold, infection approaches the disease-free state, whereas above the threshold, the solution approaches a positive endemic state. The conformable order changes the transient time scale but not the limiting equilibrium coordinates. The analysis therefore provides a consistent link between conformable pertussis dynamics and structure-preserving numerical approximation.\query{Please check the edit made in the article title.}