A CaputoFabrizio Fractional-Order SEIHRVN Model for Human Metapneumovirus (hMPV): Vaccination, Stability and Sensitivity Analysis
DOI:
https://doi.org/10.48165/jmmfc.2025.2204Keywords:
Human metapneumovirus (hMPV), Fractional order epidemic model, Caputo fabrizio derivative, Immune evasion, Stability, Basic Reproduction Number Sensitivity analysisAbstract
Human metapneumovirus (hMPV) is a major cause of acute respiratory infections across all age groups with no licensed vaccine currently available. To capture the disease dynamics and assess potential control strategies we propose a fractional-order CF–SEIHRVN model incorporating seasonal transmission, hospitalization, waning immunity and vaccination. The model stratifies the population into Susceptible (S), Exposed (E), Infectious (I), Hospitalized/Severe (H), Recovered (R), Vaccinated (V) and total population size (N). Caputo–Fabrizio fractional derivatives are employed to account for memory effects in transmission and recovery processes. Key epidemiological parameters are drawn from demographic and clinical data for China and India. Seasonal forcing in β(t) reflects climatic in fluences on transmission. The framework enables computation of the basic reproduction number R0, stability analysis of equilibria, Hopf bifurcation investigation, and optimal control formulation. Sen sitivity analysis highlights key parameters affecting R0. Numerical simulations illustrate intervention strategies providing valuable insights into hMPV control under different demographic conditions.
References
Anderson RM, May RM. Infectious diseases of humans: Dynamics and control. Oxford: Oxford University Press; 1991.
Caputo M. Linear models of dissipation whose Q is almost frequency independent. Geophys J Int. 1967;13(5):529–539.
Hilfer R, editor. Applications of fractional calculus in physics. World Scientific; 2000.
Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. Elsevier; 2006.
Magin RL. Fractional calculus in bioengineering. Begell House; 2006.
Mainardi F. Fractional calculus and waves in linear viscoelasticity. Imperial College Press; 2010.
Miller KS, Ross B. An introduction to the fractional calculus and fractional differential equations. Wiley; 1993.
Baleanu D, Diethelm K, Scalas E, Trujillo JJ. Fractional calculus: Models and numerical methods. World Scientific; 2012.
Baleanu D, Tenreiro Machado JA, Luo ACJ. Fractional dynamics and control. Springer; 2012.
Diethelm K. The analysis of fractional differential equations. Springer; 2010.
Atangana A. Fractional operators with constant and variable order with application to geo-hydrology. Academic Press; 2018.
Gorenflo R, Loutchko J, Luchko Y. Mittag-Leffler functions, related topics and applications. Springer; 2020.
Lakshmikantham V, Leela S, Devi JV. Theory of fractional dynamic systems. Cambridge Scientific Publishers; 2009.
Group B: Fractional Derivatives, Operators & Integral Transforms
Agrawal OP. Formulation of Euler–Lagrange equations for fractional variational problems. J Math Anal Appl. 2002;272(1):368–379.Almeida R. A Caputo fractional derivative of a function with respect to another function. Springer; 2017.
Atangana A, Baleanu D. New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model. Thermal Science. 2016;20(2):763–769.
Fractional integral operator associated with the I-functions. J Math Model Fract Calc. 2025;2(1):55–62.
Integral transform and generalized M-series fractional integral operators. J Math Model Fract Calc. 2025;2(1):1–18.
Group C: Numerical Methods & Computational Techniques
Diethelm K, Ford NJ, Freed AD. A predictor–corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 2002;29:3–22.
Ford NJ. Numerical solution of delay differential equations. Springer; 2010.
Li C, Deng W, Liu F. Numerical approaches to fractional calculus and fractional ordinary differential equations. J Comput Phys. 2011;230(9):3352–3368.
Lin Y, Xu C. Finite difference/spectral approximations for the time-fractional diffusion equation. J Comput Phys. 2007;225(2):1533–1552.
Garrappa R. Numerical solution of fractional differential equations: A survey and a software tutorial. Mathematics. 2018;6(2):16.
Cermak J, Korytowski A, Kocan M. Numerical methods for solving delay fractional differential equations. Mathematics. 2020;8(6):987.
Group D: Fractional-Order Epidemic & Infectious Disease Models
Ahmed EH, Elsonbaty A, Elsonbaty MA. A fractional order model of COVID-19 with vaccination strategy. Chaos Solitons Fractals. 2021;146:110885.
Alzahrani E, Shaikh AS, Khan A. Mathematical model of a fractional-order COVID-19 system with quarantine, treatment, and vaccination. Alexandria Eng J. 2022;61(1):507–520.
El-Saka HAA. On the solutions of fractional differential equations and their applications to epidemic models. Adv Differ Equ. 2018;2018:90.
El-Saka HAA. Fractional order model of hepatitis B infection: Stability analysis and optimal control. Math Methods Appl Sci. 2020;43(18):10085–10100.
Ezzinbi K, Lakhel EM, Moussaoui A. Global stability and optimal control of a fractional order COVID-19 model. Adv Differ Equ. 2020;2020:296.
Kumar S, Singh J. Fractional-order modeling of tuberculosis with Caputo–Fabrizio derivative. Chaos Solitons Fractals. 2019;125:44–54.
Nisar KS, Abdeljawad T, Hammouch Z. A fractional-order dengue fever model with memory effects. Adv Differ Equ. 2020;2020:115.
Ullah S, Khan A, Baleanu D. Optimal control of COVID-19 dynamics using fractional-order models. Alexandria Eng J. 2021;60(1):557–568.
Group E: Optimal Control Theory
Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF. The mathematical theory of optimal processes. New York: Interscience; 1962.
Hattaf K, Yousfi N. Optimal control of fractional-order HIV infection model including immune response and therapy. Comput Math Appl. 2017;73(5):891–902.
Dwivedi A, Dwivedi V, Verma S. Fractional-order approaches to optimal control in HIV-AIDS dynamics. Numer Algebra Control Optim. 2025;15(4):1286–1320.
Group F: Cardiovascular & Biomedical Fractional Models
DOvidio M, Garra R. A fractional model for cardiac tissue. Physica A. 2018;502:434–446.
Girejko E, Kocan M, Smolarkiewicz M. A model of cardiovascular diseases with fractional order and delays. Fractals. 2020;28(5):2050116.
Gutierrez JB, Lee J, Sriram R. An age-structured model of cardiovascular disease. Theor Biol Med Model. 2006;3(1):1–12.
Da Silva RMFL. Effects of radiotherapy in coronary artery disease. Curr Atheroscler Rep. 2019;21(12):50.
Kim L, Loccoh EC, Sanchez R, et al. Cardiovascular disease after cancer radiotherapy. Curr Cardiol Rep. 2020;22(11):151.
Liu F, Zlotnik VA, Liu Q. Modeling of fractional diffusion in tissues. J Math Biol. 2012;65:1235–1251.
Group G: Epidemic Dynamics, Reinfection & Seasonal Models
Chen L, Zhao Y, Sun J. Seasonal dynamics of an SEIR epidemic model with periodic transmission. J Appl Math Comput. 2021;67(3):1123–1142.
Li Q, Wang R. Impact of waning immunity on recurrent epidemics in a multi-strain model. Appl Math Model. 2020;77:911–928.
Tang B, Xiao Y, Wu J. Impacts of hospitalization and isolation on influenza dynamics. Math Biosci. 2015;264:43–53.
Zhang X, Wang K, Liu H. Global dynamics of a non-autonomous epidemic model with seasonal forcing. Math Biosci Eng. 2019;16(5):4305–4326.
Wang P, Liu H, Zhou Y. Mutation-driven immune escape in respiratory virus epidemics. Bull Math Biol. 2020;82(5):105.
Romero-Severson E, Del Valle S, Klinkenberg D. Quantifying reinfection risk in paramyxovirus transmission models. Epidemics. 2021;36:100468.
Liu X, Zhang J, Li R. Modeling reinfection dynamics of human metapneumovirus with immune evasion. J Theor Biol. 2022;547:111190.
Jin Y, Zhao H, Chen Q. Hospitalization and severity patterns of human metapneumovirus in China. Infect Dis Model. 2023;8(1):55–69.
Group H: Recent Advanced Fractional Dynamic Models
Dwivedi A, Verma S. Dynamical study of a fear-influenced fractional predator–prey model with disease spread. Eur Phys J Plus. 2025;140(4):306.
