LOCAL AND GLOBAL STABILITY ANALYSIS OF MEASLES EPIDEMIC MODEL AT DISEASE-FREE EQUILIBRIUM

Philip Onyema Ochi; Apeh Andrew Agada; Ifeoma B. Nworah; Damascus Arinze Nworah; Achi Nathan Goni

doi:10.33003/fjs-2024-0801-2219

Philip Onyema Ochi AMERICAN UNIVERSITY OF NIGERIA ACADEMY, YOLA
Apeh Andrew Agada SKYLINE UNIVERSITY, NIGERIA
Ifeoma B. Nworah 3Federal Institute of Industrial Research (FIIRO), North-East centre, Yola, Adamawa State
Damascus Arinze Nworah Bioresources Development Centre, National Biotechnology Development Agency, Jalingo, Taraba State, Nigeria.
Achi Nathan Goni Bethel Baptist High School, Kaduna, Nigeria

DOI: https://doi.org/10.33003/fjs-2024-0801-2219

Keywords: Measles, SVEIR Model, Basic reproduction number, Local stability, Numerical simulation

Abstract

In this study, a continuous mathematical model for the dynamics of Measles (rubeola) outbreak at constant recruitment rate was formulated. In the model, we partitioned the population into Susceptible (S), Vaccinated (V), exposed (E), Infected (I) and recovered (R) individuals. We analyzed a SVEIR compartmental nonlinear deterministic mathematical model of measles epidemic in a community with constant population. Analytical studies were carried out on the model using the method of linearized stability. The basic reproductive number R₀ that governs the disease transmission is obtained from the largest eigenvalue of the next-generation matrix. The disease-free equilibrium is computed and proved to be locally and globally asymptotically stable if R₀<1 and unstable if R₀>1 respectively. Finally, we simulate the model system in MATLAB and obtained the graphical behavior of each compartment. From the simulation, we observed that the measles infection was eradicated in the environment when R₀<1.

Author Biographies

Philip Onyema Ochi, AMERICAN UNIVERSITY OF NIGERIA ACADEMY, YOLA

DEPARTMENT OF MATHEMATICS AND MATHEMATICS TEACHER

Apeh Andrew Agada, SKYLINE UNIVERSITY, NIGERIA

MATHEMATICS DEPARTMENT AND LECTURER II

Ifeoma B. Nworah, 3Federal Institute of Industrial Research (FIIRO), North-East centre, Yola, Adamawa State

NIL

Achi Nathan Goni, Bethel Baptist High School, Kaduna, Nigeria

NIL

References

Guerra, Fiona M.; Bolotin, Shelly; Lim, Gillian; Heffernan, Jane; Deeks, Shelley L.; Li, Ye; Crowcroft, Natasha S. (December 2017). "The basic reproduction number (R0) of measles: a systematic review". The Lancet. Infectious Diseases. 17 (12): e420–e428. doi:10.1016/ S1473-3099(17)30307-9. ISSN 1474-4457. PMID 28757186.

Rota PA, Moss WJ, Takeda M, de Swart RL, Thompson KM, Goodson JL (July 2016). "Measles". Nature Reviews. Disease Primers. 2: 16049. doi:10.1038/nrdp. 2016 .49. PMID 27411684. DOI: https://doi.org/10.1038/nrdp.2016.49

Bope, Edward T.; Kellerman, Rick D. (2014). Conn's Current Therapy 2015. Elsevier Health Sciences. p. 153. ISBN 9780323319560. Archived from the original on 8 September 2017.

Guerra FM, Bolotin S, Lim G, Heffernan J, Deeks SL, Li Y, Crowcroft NS (2017). "The basic reproduction number (R0) of measles: a systematic review". The Lancet Infectious Diseases. 17 (12): e420–e428. doi:10. 1016 /S1473-3099(17)30307-9. PMID 28757186. DOI: https://doi.org/10.1016/S1473-3099(17)30307-9

Dept of Health, Saskatchewan (DHS) (2015) "Measles (Red Measles, Rubeola)". . Archived from the original on 10 February 2015. Retrieved 10 February 2015.

Milner DA (2015). Diagnostic Pathology: Infectious Diseases E-Book. Elsevier Health Sciences.p. 24. ISBN 9780323400374. Archived from the original on 8 September 2017.

Stanley J (2002). Essentials of Immunology & Serology. Cengage Learning. p. 323. ISBN 978-0766810648. Archived from the original on 8 September 2017.

Caserta, MT, ed. (2013). "Measles". Merck Manual Professional. Merck Sharp & DohmeCorp. Archived from the original on 23 March 2014. Retrieved 23 March 2014.

Atkinson W (2011). Epidemiology and Prevention of Vaccine-Preventable Diseases (12 ed.). Public Health Foundation. pp. 301–23. ISBN 9780983263135. Archived from the original on 7 February 2015. Retrieved 5 February 2015.

Marx JA (2010). Rosen's emergency medicine: concepts and clinical practice (7th ed.).Philadelphia: Mosby/Elsevier. p. 1541. ISBN 9780323054720. Archived from the original on 8 September 2017.

Delamater, P. L.; Street, E. J.; Leslie, T. F.; Yang, Y. T.; Jacobsen, K. H. (2019). "Complexity of the Basic Reproduction Number (R0)". Emerging Infectious Diseases. NIH website. 25 (1): 1–4. doi: 10.3201/eid2501.171901. PMC 6302597. PMID 30560777. DOI: https://doi.org/10.3201/eid2501.171901

Chen (2018). Measles (Report). Medscape. Archived from the original on 25 September 2011. Retrieved 13 May 2020.

Russell, SJ; Babovic-Vuksanovic, D; Bexon, A; Cattaneo, R; Dingli, D; Dispenzieri, A; Deyle, DR; Federspiel, MJ; Fielding, A; Galanis, E (September 2019). "Oncolytic Measles Virotherapy and Opposition to Measles Vaccination". Mayo Clinic Proceedings. 94(9): 1834–39. doi: 10. 1016 /j.mayocp.2019.05.006. PMC 6800178. PMID 31235278. DOI: https://doi.org/10.1016/j.mayocp.2019.05.006

Kabra SK, Lodha R (August 2013). "Antibiotics for preventing complications in children with measles". The Cochrane Database of Systematic Reviews. 8(8): CD 001477. doi:10. 1002/14651858. CD 0014 77.pub4. PMC 7055587. PMID 23943263.

GBD (2015) Mortality and Causes of Death Collaborators (October 2016). "Global, regional, and national life expectancy, all-cause mortality, and cause-specific mortality for 249 causes of death, 1980-2015: a systematic analysis for the Global Burden of Disease Study 2015". Lancet. 388(10053): 1459–1544. doi:10.1016/S0140-6736(16) 31012-1. PMC 5388903. PMID 27733281.

GBD 2013 Mortality Causes of Death Collaborators (January 2015). "Global, regional, and national age-sex specific all-cause and cause-specific mortality for 240 causes of death,1990-2013: a systematic analysis for the Global Burden of Disease Study 2013". Lancet. 385 (9963): 117–71. doi:10.1016/S0140-6736(14)61682-2. PMC 4340604. PMID 25530 442. DOI: https://doi.org/10.1016/S0140-6736(14)61682-2

WHO (2014) "Measles Fact sheet N°286". . Archived from the original on 3 February 2015. Retrieved 4 February 2015.

WHO (2018) "Measles cases spike globally due to gaps in vaccination coverage". World Health Organization . Retrieved 21 December 2018.

WHO (2019). "Measles – European Region". Archived from the original on 8 May 2019.

WHO (2019) "Measles fact sheet". Archived from the original on 1 June 2019. Retrieved 20 May 2019.

CDC (2013). Prevention of Measles, Rubella, Congenital Rubella Syndrome, and Mumps, 2013: Summary Recommendations of the Advisory Committee on Immunization Practices (ACIP) . MMWR 2013; 62 (RR04);1-34.

CDC (2014) "Measles (Rubeola) Signs and Symptoms". Archived from the original on 2 February 2015. Retrieved 5 February 2015.

CDC (2019) "U.S. measles cases surge nearly 20 percent in early April, 2019. Retrieved 16 April 2019.

Heffernan, J., Smith, R. & Wahl, L. (2005). Perspectives on the basic reproductive ratio, Journal of the Royal Society Interface 2 (4) 281–293. 240.DOI:10.1098/rsif.2005.0042 DOI: https://doi.org/10.1098/rsif.2005.0042

Van den Driessche, P. & Watmough, J. (2002) Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Mathematical biosciences 180 (1), 29–48. https://doi.org/10.1016/S0025-5564(02)00108-6 DOI: https://doi.org/10.1016/S0025-5564(02)00108-6

Castillo-Chavez, C. and Song, B. (2004). Dynamical models of tuberculosis and their applications. Math. Biosci. Eng, 1(2):361–404.http://dx.doi.org/10.3934/mbe.2004.1.361 DOI: https://doi.org/10.3934/mbe.2004.1.361

LOCAL AND GLOBAL STABILITY ANALYSIS OF MEASLES EPIDEMIC MODEL AT DISEASE-FREE EQUILIBRIUM

Abstract

Author Biographies

References

Announcements

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