<p>Tuberculosis (TB) and pneumonia continue to be significant sources of infectious disease mortality globally. However, the epidemiological interaction between the two infections produces a transmission dynamics problem which cannot be fully described by conventional integer order models because of their incapacity to model memory. This paper proposes a new fractional order compartmental model of co-infection between TB and pneumonia using the Caputo fractional order differential operator. The six compartmental model, <InlineEquation ID="IEq1"><EquationSource Format="TEX">\({\text{S}}{\text{E}}_{\text{T}}{\text{I}}_{\text{T}}{\text{I}}_{\text{P}}{\text{I}}_{\text{T}\text{P}}\)</EquationSource></InlineEquation>, adopts bilinear incidences and mass action terms of co-infections and takes into consideration the impact of previous disease history on current transmission dynamics. It proves the positivity and boundedness of solutions, the uniqueness and the existence of solutions, and obtaining <InlineEquation ID="IEq2"><EquationSource Format="TEX">\({\text{R}}_{0}=\text{m}\text{a}\text{x}\left\{{\text{R}}_{0\text{T}},{\text{R}}_{0\text{P}}\right\}\)</EquationSource></InlineEquation> using the next generation matrix method. Results show that the stability analysis holds that the disease-free equilibrium is locally and globally asymptotically stable in the case where <InlineEquation ID="IEq3"><EquationSource Format="TEX">\({\text{R}}_{0}\)</EquationSource></InlineEquation> < 1 while in the case of <InlineEquation ID="IEq4"><EquationSource Format="TEX">\({\text{R}}_{0}\)</EquationSource></InlineEquation> > 1, it is disease persistence in the population. The sensitivity analysis shows that the TB and pneumonia transmission rates (<InlineEquation ID="IEq5"><EquationSource Format="TEX">\({\upbeta}_{\text{T}}\)</EquationSource></InlineEquation> and <InlineEquation ID="IEq6"><EquationSource Format="TEX">\({\upbeta}_{\text{P}}\)</EquationSource></InlineEquation>) and the TB active-latent conversion rate (σ) are the most sensitive parameters responsible for the transmission of disease and intervention purposes. The numerical results obtained show that the memory effect significantly affects the transient and asymptotic behavior of the system by increasing the order of memory, delaying the peak infection, delaying convergence, and extending the persistence of co-infection. The results show that interaction between TB and pneumonia leads to an increase in the burden of disease under certain regimes of parameters. The validation with 6 years' data from Ethiopia shows that the fractional-order model (α = 0.925) achieves a 6.6% reduction in mean absolute residual (MAR) compared with the integer-order model (<i>α</i> = 1.00) and a 22.2% reduction compared with the OLS baseline, confirming the empirical benefit of incorporating memory effects.</p>

Memory effects in tuberculosis–pneumonia co-dynamics: a Caputo fractional-order differential equation framework validated against Ethiopian data

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2026年8月3日

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Mideksa Tola Jiru,
Sathish Kumar Kumaravel

摘要

Tuberculosis (TB) and pneumonia continue to be significant sources of infectious disease mortality globally. However, the epidemiological interaction between the two infections produces a transmission dynamics problem which cannot be fully described by conventional integer order models because of their incapacity to model memory. This paper proposes a new fractional order compartmental model of co-infection between TB and pneumonia using the Caputo fractional order differential operator. The six compartmental model, \({\text{S}}{\text{E}}_{\text{T}}{\text{I}}_{\text{T}}{\text{I}}_{\text{P}}{\text{I}}_{\text{T}\text{P}}\), adopts bilinear incidences and mass action terms of co-infections and takes into consideration the impact of previous disease history on current transmission dynamics. It proves the positivity and boundedness of solutions, the uniqueness and the existence of solutions, and obtaining \({\text{R}}_{0}=\text{m}\text{a}\text{x}\left\{{\text{R}}_{0\text{T}},{\text{R}}_{0\text{P}}\right\}\) using the next generation matrix method. Results show that the stability analysis holds that the disease-free equilibrium is locally and globally asymptotically stable in the case where \({\text{R}}_{0}\) < 1 while in the case of \({\text{R}}_{0}\) > 1, it is disease persistence in the population. The sensitivity analysis shows that the TB and pneumonia transmission rates (\({\upbeta}_{\text{T}}\) and \({\upbeta}_{\text{P}}\)) and the TB active-latent conversion rate (σ) are the most sensitive parameters responsible for the transmission of disease and intervention purposes. The numerical results obtained show that the memory effect significantly affects the transient and asymptotic behavior of the system by increasing the order of memory, delaying the peak infection, delaying convergence, and extending the persistence of co-infection. The results show that interaction between TB and pneumonia leads to an increase in the burden of disease under certain regimes of parameters. The validation with 6 years' data from Ethiopia shows that the fractional-order model (α = 0.925) achieves a 6.6% reduction in mean absolute residual (MAR) compared with the integer-order model (α = 1.00) and a 22.2% reduction compared with the OLS baseline, confirming the empirical benefit of incorporating memory effects.