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A dynamical study of diarrhea delayed epidemic model: application of mathematical biology in infectious diseases

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Abstract

This manuscript presents the stability analysis of the diarrhea epidemic model with the effect of time delay. The delayed epidemic model for the disease of diarrhea contains four compartments, including susceptible, infective, treated, and recovered classes. The artificial delay parameter is designed with a saturated incidence rate of the model. The mathematical analysis is carried out by studying the equilibria, positivity, boundedness, and reproduction number for the said model. Furthermore, the sensitivity of the parameters is studied to strengthen the mathematical analysis. The diarrhea epidemic model's local and global stabilities are investigated using the acknowledged results of the Routh Hurwitz criterion and Lyapunov function, respectively. Moreover, the numerical results are obtained to support the analysis.

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References

  1. J. Ardkaew, P. Tongkumchum, Statistical modelling of childhood diarrhea in northeastern Thailand. Southeast Asian J. Trop. Med. Public Health 40(4), 807–815 (2009)

    Google Scholar 

  2. B.R. Cherry, M.J. Reeves, G. Smith, Evaluation of bovine viral diarrhea virus control using a mathematical model of infection dynamics. Prev. Vet. Med. 33(1–4), 91–108 (1998)

    Article  Google Scholar 

  3. P. Collier, G. Conway, T. Venable, Climate change and Africa. Oxf. Rev. Econ. Pol. 24, 337–353 (2008)

    Article  Google Scholar 

  4. H. Auld, D. MacIver, J. Klaassen, Heavy rainfall and waterborne disease outbreaks: the Walkerton example. J. Toxicol. Environ. Health A 67(20–22), 1879–1887 (2004)

    Article  Google Scholar 

  5. R.E. Black, Epidemiology of diarrhoeal disease: implications for control by vaccines. Vaccine 11(2), 100–106 (1993)

    Article  MathSciNet  Google Scholar 

  6. A.T. Fuller, Conditions for a matrix to have only characteristic roots with negative real parts. J. Math. Anal. Appl. 23(1), 71–98 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Gatto, L. Mari, A. Rinaldo, Leading eigenvalues and the spread of cholera. SIAM News, 46(7) (2013)

  8. O. Chaturvedi, M. Jeffrey, E. Lungu, S. Masupe, Epidemic model formulation and analysis for diarrheal infections caused by salmonella. SIMULATION 93(7), 543–552 (2017)

    Article  Google Scholar 

  9. M. Farthing, M.A. Salam, G. Lindberg et al., Acute diarrhea in adults and children: a global perspective. J. Clin. Gastroenterol. 47(1), 12–20 (2013)

    Article  Google Scholar 

  10. S.E. Hrudey, P.M. Huck, P. Payment, R.W. Gillham, E.J. Hrudey, Walkerton: lesson learned in comparison with waterborne outbreaks in the developed world. J. Environ. Eng. Sci. 1, 397–407 (2002)

    Article  Google Scholar 

  11. S.M. Pamela. Communicable disease epidemiological, Profile for horn of Africa (2007)

  12. Gunn GJ, Stott AW, Humphry RW. Modeling and costing BVD outbreaks in beef herds. Veterinary J 167:143–149 (2004)

  13. C. Viboud, L. Simonsen, G. Chowell, A generalized-growth model to characterize the early ascending phase of infectious disease outbreaks. Epidemics 15, 27–37 (2016)

    Article  Google Scholar 

  14. B. Lopman, K. Simmons, M. Gambhir, J. Vinje, U. Parashar, Epidemiologic implications of asymptomatic re-infection: a mathematical modeling study of Norovious. Am. J. Epidemiol 179(4) (2013)

  15. E. Bonyah, G. Twagirumukiza, P.P. Gambrah, Mathematical analysis of diarrhoea model with saturated incidence rate. Open J. Math. Sci. 2019(3), 29–39 (2019)

    Article  Google Scholar 

  16. P.R. Hunter, Climate change and waterborne and vector-borne disease. J. Appl. Microbiol. 94, 37–46 (2003)

    Article  Google Scholar 

  17. S. Lenhart, J.T. Workman, Optimal Control Applied to Biological Models (Chapman and Hall/CRC, London, UK, 2007)

    Book  MATH  Google Scholar 

  18. G. Innocent et al., A computer simulation of the transmission dynamics and the effects of duration of immunity and survival of persistently infected animals on the spread of bovine viral-diarrhoea virus in dairy cattle. Epidemiol. Infect. 119, 91–100 (1997)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Air University Islamabad, Islamabad, Pakistan

    Muhammad Naveed & Atif Hasan Soori

  2. Department of Mathematics, Govt. Maulana Zafar Ali Khan Graduate College, Wazirabad, Higher Education Department (HED), Lahore, 54000, Pakistan

    Ali Raza

  3. Department of Mathematics, Science Faculty, Firat University, 23119, Elizig, Turkey

    Mustafa Inc

  4. Department of Medical Research, China Medical University, 40402, Taichung, Taiwan

    Mustafa Inc

  5. Department of Mathematics, Faculty of Science and Technology, University of Central Punjab, Lahore, 54000, Pakistan

    Muhammad Rafiq

  6. Department of Mathematics, Mathematics Research Center, Near East University, Near East Boulevard, PC: 99138, Nicosia/Mersin 10, Turkey

    Muhammad Rafiq

  7. Department of Mathematics and Statistics, The University of Lahore, Lahore, 54590, Pakistan

    Nauman Ahmed

  8. Department of H&BS, Military College of Signals, NUST, Islamabad, Pakistan

    Muhammad Sajid Iqbal

  9. Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon

    Ali Raza & Nauman Ahmed

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  1. Muhammad Naveed
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  2. Ali Raza
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  3. Atif Hasan Soori
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  4. Mustafa Inc
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  5. Muhammad Rafiq
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  7. Muhammad Sajid Iqbal
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Correspondence to Mustafa Inc or Nauman Ahmed.

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Naveed, M., Raza, A., Soori, A.H. et al. A dynamical study of diarrhea delayed epidemic model: application of mathematical biology in infectious diseases. Eur. Phys. J. Plus 138, 664 (2023). https://doi.org/10.1140/epjp/s13360-023-04287-5

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