Mathematical Modelling of Listeriosis Epidemics in Animal and Human Population with Optimal Control.

Article Sidebar

PDF
Published: Nov 1, 2020
DOI: https://doi.org/10.5556/j.tkjm.51.2020.2860
Keywords:
Listeriosis Backward bifurcation endemic equilibrium reproductive number Pontryagin Maximum principle.

Main Article Content

Shaibu Osman
Department of Mathematics, Pan African University, Institute for Basic Sciences, Technology and Innovations, Box 62000-00200, Nairobi-Kenya.
https://orcid.org/0000-0003-3692-3846
Oluwole Daniel Makinde
Faculty of Military Science, Stellenbosch University, Private Bag X2, Saldanha 7395.
David Mwangi Theuri
Department of Mathematics, Jomo Kenyatta University of Agriculture and Technology, Nairo-Kenya, Box 62000-00200, Nairobi-Kenya.

Abstract

Listeriosis is a serious disease caused by the germ Listeria monocytogenes. People usually become ill with listeriosis after eating contaminated food including meat. The disease primarily affects pregnant women, newborns, older adults, and people with weakened immune systems. In this paper, we propose and scrutinize a model problem describing the transmission dynamics of Listeriosis epidemic in animal and human population using the stability theory of differential equations. The model is qualitatively analysed for the basic reproduction number as well as possibility of forward and backward bifurcation with respect to the stability of disease free and endemic equilibria. The impact of the model parameters on the disease was evaluated via sensitivity analysis. An extension of the model to include time dependent control variables such as treatment, vaccination and education of susceptible (human) is carried out. Using Pontryagin’s Maximum Principle, we obtain the optimal control strategies needed for combating Listeriosis disease. Numerical simulation of the model is performed and pertinent results are displayed graphically and discussed quantitatively.

Article Details

How to Cite
Osman, S., Makinde, O. D., & Theuri, D. M. (2020). Mathematical Modelling of Listeriosis Epidemics in Animal and Human Population with Optimal Control. Tamkang Journal of Mathematics, 51(4), 261–287. https://doi.org/10.5556/j.tkjm.51.2020.2860
Issue
Vol. 51 No. 4 (2020)
Section
Papers
Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

References

F. Allerberger and M. Wagner, Listeriosis: a resurgent foodborne infection, Clinical Microbiology and Infec- tion, 16 (2010), 16–23.

R. M. Anderson and R. M. Mayetal., Population biology of infectious diseases, Report of the Dahlem Work- shop, Berlin, 14–19 March 1982. Berlin, German Federal Republic; Springer-Verlag.

M. K. Borucki, J. Reynolds, C. C. Gay, K. L. McElwain, S. H. Kim, D. P. Knowles and J. Hu, Dairy farm reservoir of Listeria monocytogenes sporadic and epidemic strains, Journal of Food Protection®, 67 (2004), 2496–2499.

C. W. Donnelly, Listeria monocytogenes : acontinuing challenge, Nutrition Reviews, 59(2001),183–194.

W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Volume1, SpringerScience & Business Media, 2012.

N. C. Grassly and C. Fraser, Mathematical models of infectious disease transmission, Nature Reviews Microbiology, 6 (2008), No.6.

K. Hattaf, A. A. Lashari, Y. Louartassi and N. Yousfi, A delayed SIR epidemic model with general incidence rate, Electronic Journal of Qualitative Theory of Differential Equations, 3 (2013), 1–9.

H. W. Hethcote, Qualitative analyses of communicable disease models, Mathematical Biosciences, 28(1976), 335–356.

H. Hof, Listeria monocytogenes : acausative agent of gastroenteritis? European Journal of Clinical Microbiology and Infectious Diseases, 20 (2001), 369–373.

T. Jemmi and R. Stephan, R., Listeria monocytogenes: food-borne pathogen and hygiene indicator, Rev Sci Tech, 25 (2006), 571–80.

H. R. Joshi, S. Lenhart, M. Y. Li and L. Wang, Optimal control methods applied to disease models, Contemporary Mathematics, 410 (2006), 187–208.

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, In Proceedings of the Royal Society of London A: mathematical, physical and engineering sciences, 115 (1927), 700–721.

J. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Appl, SIAM, Philadelphia,1976.

A. A. Lashari, Optimal Control of an SIR Epidemic Model with a Saturated Treatment, Appl. Math, 10(2016), 185–191.

A. Lianou and J. N. Sofos, A review of the incidence and transmission of Listeria monocytogenes in ready-to-eat products in retail and food service environments, Journal of Food Protection®, , 70 (2007), 2172–2198.

C. L. Little, S. M. Pires, I. A. Gillespie, K. Grant and G. L. Nichols, Attribution of human Listeriamono cytogenes infections in England and Wales to ready-to-eat food sources placed on the market: adaptation of the Hald Salmonella source attribution model, Foodborne Pathogens and Disease, 7(2010), 749–756.

E. G. D. Murray, R. A. Webb and M. B. R. Swann, A disease of rabbits characterised by a large mononuclear leucocytosis, caused by a hitherto undescribed bacillus Bacterium monocytogenes (n. sp.), The Journal of Pathol-

ogy and Bacteriology, 29 (1926), 407–439.

K. O. Okosun, M. Mukamuri and D. O. Makinde, Global stability analysis and control of leptospirosis, Open Mathematics, 14 (2016), 567–585.

S. Osman, O. D. Makinde and D. M. Theuri, Stability Analysis and Modelling of Listeriosis Dynamics in Human and Animal Populations, Global Journal of Pure and Applied Mathematics, 14 (2018), 115–137.

L. S. Pontryagin, Mathematical Theory of Optimal Processes, CRC Press, 1987.

V. Rebagliati, R. Philippi, M. Rossi, A. Troncosoetal.,(2009). Prevention of foodborne listeriosis, Indian Journal of Pathology and Microbiology, 52 (2009), 145.

M. Rossi, A. Paiva, M. Tornese, S. Chianelli and A. Troncoso,[Listeria monocytogenes outbreaks : a review of the routes that favor bacterial presence], Revista chilena de infectologia: organo oficial de la Sociedad Chilena de Infectologia, 25 (2008), 328–335.

S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, Journal of Differential Equations, 188 (2003), 135–163.

M. A. Safi and S. M. Garba, Global stability analysis of SEIR model with holling type II incidence function, Computational and Mathematical Methods in Medicine, 2012.

W. F. Schlech and D. Acheson, Foodborne listeriosis, Clinical Infectious Diseases, 31(2000), 770–775.

A. Schuchat, B. Swaminathan and C. V. Broome, Epidemiology of human listeriosis, Clinical Microbiology Reviews, 4 (1991a), 169–183.

A. Schuchat, B. Swaminathan and C. V. Broome, Epidemiology of human listeriosis, Clinical Microbiology Reviews, 4 (1991b), 169–183.

H. L. Smith and X.-Q. Zhao, Global asymptoticstability of traveling waves in delayed reaction-diffusion equations, SIAM Journal on Mathematical Analysis, 31 (2000), 514–534.

B. Swaminathan and P. Gerner-Smidt, The epidemiology of human listeriosis, Microbes and Infection, 9 (2007), 1236–1243.

P. D. Tamma, S. E. Cosgrove and L. L. Maragakis, Combination therapy for treatment of infections with gram-negative bacteria, Clinical Microbiology Reviews, 25 (2012), 450–470.

P. Van den Driessche, and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical biosciences, 180 (2002), 29–48.

H. M. Wexler, Bacteroides: the good, the bad, and the nitty-gritty, Clinical Microbiology Reviews, 20 (2007), 593–621.