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Global stability and optimal control in a single-strain dengue model with fractional-order transmission and recovery process

Published 19 Feb 2024 in math.DS and math.OC | (2402.11974v2)

Abstract: The current manuscript introduce a single-strain dengue model developed from stochastic processes incorporating fractional order transmission and recovery. The fractional derivative has been introduced within the context of transmission and recovery process, displaying characteristics similar to tempered fractional ($TF$) derivatives. It has been established that under certain condition, a function's $TF$ derivatives are proportional to the function itself. Applying the following observation, we examined stability of several steady-state solutions, such as disease-free and endemic states, in light of this newly formulated model, using the reproduction number (R_0). In addition, the precise range of epidemiological parameters for the fractional order model was determined by calibrating weekly registered dengue incidence in the San Juan municipality of Puerto Rico, from April 9, 2010, to April 2, 2011. We performed a global sensitivity analysis method to measure the influence of key model parameters (along with the fractional-order coefficient) on total dengue cases and the basic reproduction number (R_0) using a Monte Carlo-based partial rank correlation coefficient (PRCC). Moreover, we formulated a fractional-order model with fractional control to asses the effectiveness of different interventions, such as reduction the recruitment rate of mosquito breeding, controlling adult vector, and providing individual protection. Also, we established the existence of a solution for the fractional-order optimal control problem. Finally, the numerical experiment illustrates that, policymakers should place importance on the fractional order transmission and recovery parameters that capture the underline mechanisms of disease along with reducing the spread of dengue cases, carried out through the implementation of two vector controls.

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References (50)
  1. W. H. Organization, “Dengue guidelines for diagnosis, treatment, prevention and control : new edition,” 2009.
  2. D. J. Gubler and G. G. Clark, “Dengue/dengue hemorrhagic fever: the emergence of a global health problem.,” Emerging infectious diseases, vol. 1, no. 2, p. 55, 1995.
  3. T. Sardar, S. Rana, and J. Chattopadhyay, “A mathematical model of dengue transmission with memory,” Communications in Nonlinear Science and Numerical Simulation, vol. 22, no. 1-3, pp. 511–525, 2015.
  4. J. Ong, J. Aik, and L. C. Ng, “Adult aedes abundance and risk of dengue transmission,” PLoS Neglected Tropical Diseases, vol. 15, no. 6, p. e0009475, 2021.
  5. S. Bhatt, P. W. Gething, O. J. Brady, J. P. Messina, A. W. Farlow, C. L. Moyes, J. M. Drake, J. S. Brownstein, A. G. Hoen, O. Sankoh, et al., “The global distribution and burden of dengue,” Nature, vol. 496, no. 7446, pp. 504–507, 2013.
  6. T. P. Htun, Z. Xiong, and J. Pang, “Clinical signs and symptoms associated with who severe dengue classification: a systematic review and meta-analysis,” Emerging Microbes & Infections, vol. 10, no. 1, pp. 1116–1128, 2021.
  7. D. J. Gubler, “Epidemic dengue/dengue hemorrhagic fever as a public health, social and economic problem in the 21st century,” Trends in microbiology, vol. 10, no. 2, pp. 100–103, 2002.
  8. F. P. Pinheiro, S. J. Corber, et al., “Global situation of dengue and dengue haemorrhagic fever, and its emergence in the americas,” World health statistics quarterly, vol. 50, pp. 161–169, 1997.
  9. M. Andraud, N. Hens, C. Marais, and P. Beutels, “Dynamic epidemiological models for dengue transmission: a systematic review of structural approaches,” PloS one, vol. 7, no. 11, p. e49085, 2012.
  10. M. Aguiar, V. Anam, K. B. Blyuss, C. D. S. Estadilla, B. V. Guerrero, D. Knopoff, B. W. Kooi, A. K. Srivastav, V. Steindorf, and N. Stollenwerk, “Mathematical models for dengue fever epidemiology: A 10-year systematic review,” Physics of Life Reviews, vol. 40, pp. 65–92, 2022.
  11. S. T. R. d. Pinho, C. P. Ferreira, L. Esteva, F. R. Barreto, V. Morato e Silva, and M. Teixeira, “Modelling the dynamics of dengue real epidemics,” Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 368, no. 1933, pp. 5679–5693, 2010.
  12. R. P. Agarwal, S. K. Ntouyas, B. Ahmad, and M. S. Alhothuali, “Existence of solutions for integro-differential equations of fractional order with nonlocal three-point fractional boundary conditions,” Advances in Difference Equations, vol. 2013, no. 1, pp. 1–9, 2013.
  13. I. Podlubny, “Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications,” 1999.
  14. M. Du, Z. Wang, and H. Hu, “Measuring memory with the order of fractional derivative,” Scientific reports, vol. 3, no. 1, p. 3431, 2013.
  15. A. Dokoumetzidis and P. Macheras, “Fractional kinetics in drug absorption and disposition processes,” Journal of pharmacokinetics and pharmacodynamics, vol. 36, no. 2, pp. 165–178, 2009.
  16. A. Dokoumetzidis, R. Magin, and P. Macheras, “Fractional kinetics in multi-compartmental systems,” Journal of pharmacokinetics and pharmacodynamics, vol. 37, no. 5, pp. 507–524, 2010.
  17. M. Saeedian, M. Khalighi, N. Azimi-Tafreshi, G. Jafari, and M. Ausloos, “Memory effects on epidemic evolution: The susceptible-infected-recovered epidemic model,” Physical Review E, vol. 95, no. 2, p. 022409, 2017.
  18. M. Starnini, J. P. Gleeson, and M. Boguñá, “Equivalence between non-markovian and markovian dynamics in epidemic spreading processes,” Physical review letters, vol. 118, no. 12, p. 128301, 2017.
  19. C. Angstmann, B. Henry, and A. McGann, “A fractional order recovery sir model from a stochastic process,” Bulletin of mathematical biology, vol. 78, no. 3, pp. 468–499, 2016.
  20. T. Sardar and B. Saha, “Mathematical analysis of a power-law form time dependent vector-borne disease transmission model,” Mathematical biosciences, vol. 288, pp. 109–123, 2017.
  21. C. N. Angstmann, B. I. Henry, and A. V. McGann, “A fractional-order infectivity and recovery sir model,” Fractal and Fractional, vol. 1, no. 1, p. 11, 2017.
  22. C. N. Angstmann, A. M. Erickson, B. I. Henry, A. V. McGann, J. M. Murray, and J. A. Nichols, “A general framework for fractional order compartment models,” SIAM Review, vol. 63, no. 2, pp. 375–392, 2021.
  23. Z. Wu, Y. Cai, Z. Wang, and W. Wang, “Global stability of a fractional order sis epidemic model,” Journal of Differential Equations, vol. 352, pp. 221–248, 2023.
  24. Z.-H. Lin, M. Feng, M. Tang, Z. Liu, C. Xu, P. M. Hui, and Y.-C. Lai, “Non-markovian recovery makes complex networks more resilient against large-scale failures,” Nature communications, vol. 11, no. 1, p. 2490, 2020.
  25. N. Sherborne, J. C. Miller, K. B. Blyuss, and I. Z. Kiss, “Mean-field models for non-markovian epidemics on networks,” Journal of mathematical biology, vol. 76, pp. 755–778, 2018.
  26. C. N. Angstmann, B. I. Henry, and A. V. McGann, “A fractional-order infectivity sir model,” Physica A: Statistical Mechanics and its Applications, vol. 452, pp. 86–93, 2016.
  27. R. Hilfer and L. Anton, “Fractional master equations and fractal time random walks,” Physical Review E, vol. 51, no. 2, p. R848, 1995.
  28. X. Yang, L. Chen, and J. Chen, “Permanence and positive periodic solution for the single-species nonautonomous delay diffusive models,” Comput. Math. Appl., vol. 32, no. 4, pp. 109–116, 1996.
  29. Springer, 2003.
  30. J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Mathematical biosciences, vol. 180, no. 1-2, pp. 29–48, 2002.
  31. C. Castillo-Chavez, “On the computation of r. and its role on global stability carlos castillo-chavez*, zhilan feng, and wenzhang huang,” Mathematical approaches for emerging and reemerging infectious diseases: an introduction, vol. 1, p. 229, 2002.
  32. M. Y. Li and J. S. Muldowney, “A geometric approach to global-stability problems,” SIAM Journal on Mathematical Analysis, vol. 27, no. 4, pp. 1070–1083, 1996.
  33. M. Y. Li, J. R. Graef, L. Wang, and J. Karsai, “Global dynamics of a seir model with varying total population size,” Mathematical biosciences, vol. 160, no. 2, pp. 191–213, 1999.
  34. M. Y. Li and L. Wang, “Global stability in some seir epidemic models,” in Mathematical approaches for emerging and reemerging infectious diseases: models, methods, and theory, pp. 295–311, Springer, 2002.
  35. H. I. Freedman, S. Ruan, and M. Tang, “Uniform persistence and flows near a closed positively invariant set,” Journal of Dynamics and Differential Equations, vol. 6, pp. 583–600, 1994.
  36. H. Smith, “Monotone dynamical systems,” Handbook of differential equations: ordinary differential equations, vol. 2, pp. 239–357, 2006.
  37. H. Abboubakar, A. K. Guidzavai, J. Yangla, I. Damakoa, and R. Mouangue, “Mathematical modeling and projections of a vector-borne disease with optimal control strategies: A case study of the chikungunya in chad,” Chaos, Solitons & Fractals, vol. 150, p. 111197, 2021.
  38. T. Sardar, S. Mukhopadhyay, A. R. Bhowmick, and J. Chattopadhyay, “An optimal cost effectiveness study on zimbabwe cholera seasonal data from 2008–2011,” PLoS One, vol. 8, no. 12, p. e81231, 2013.
  39. Springer Science & Business Media, 2012.
  40. H. Gaff and E. Schaefer, “Optimal control applied to vaccination and treatment strategies for various epidemiological models,” Math. Biosci. Eng, vol. 6, no. 3, pp. 469–492, 2009.
  41. Springer, 1989.
  42. E. A. Coddington, An introduction to ordinary differential equations. Courier Corporation, 2012.
  43. E. Coddington and N. Levinson, “Theory of ordinary differential equations (mcgraw-hill, new york, 1955).,” MR16: 1022b.
  44. D. E. Kirk, Optimal control theory: an introduction. Courier Corporation, 2004.
  45. S. Lenhart and J. T. Workman, Optimal control applied to biological models. CRC press, 2007.
  46. F. Sabzikar, M. M. Meerschaert, and J. Chen, “Tempered fractional calculus,” Journal of Computational Physics, vol. 293, pp. 14–28, 2015.
  47. V. Duong, L. Lambrechts, R. E. Paul, S. Ly, R. S. Lay, K. C. Long, R. Huy, A. Tarantola, T. W. Scott, A. Sakuntabhai, et al., “Asymptomatic humans transmit dengue virus to mosquitoes,” Proceedings of the National Academy of Sciences, vol. 112, no. 47, pp. 14688–14693, 2015.
  48. A. B. Gumel, “Causes of backward bifurcations in some epidemiological models,” Journal of Mathematical Analysis and Applications, vol. 395, no. 1, pp. 355–365, 2012.
  49. Y. Eshita, T. Takasaki, I. Takashima, N. Komalamisra, H. Ushijima, and I. Kurane, “Vector competence of japanese mosquitoes for dengue and west nile viruses,” Pesticide Chemistry: Crop Protection, Public Health, Environmental Safety, pp. 217–225, 2007.
  50. A. Wilder-Smith, W. Foo, A. Earnest, S. Sremulanathan, and N. I. Paton, “Seroepidemiology of dengue in the adult population of singapore,” Tropical Medicine & International Health, vol. 9, no. 2, pp. 305–308, 2004.

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