Mathematical Analysis of the Role of Detection Rate on Dynamical Spread of Ebola Virus Disease

Akanni John Olajide

doi:10.21467/jmsm.3.1.37-52

Authors

Akanni John Olajide Ladoke Akintola University of Technology, Ogbomoso

DOI:

https://doi.org/10.21467/jmsm.3.1.37-52

Abstract

In this paper, a non-linear mathematical model of the Ebola virus disease with detection rate is proposed and analyzed. The whole population under consideration is divided into five compartments e.g. susceptible, latently infected, infected undetected, infected detected, and recovered to study the transmission dynamics of the Ebola virus disease. Based on the immunity level, susceptible individuals move to exposed class or directly to infected detected class once they come into contact with an infective. This has been incorporated through the progression rate which is slow. The equilibria of the model and the basic reproduction number R₀ are computed. It is observed that the disease-free equilibrium of the model is locally asymptotically stable when R₀<1. The model exhibits forward bifurcation under certain restrictions on parameters, which indicate that the model has a single endemic equilibrium for R₀<1. This suggests that an accurate estimation of parameters and the level of control measures are required to reduce the infection prevalence of the Ebola virus in the endemic region and just R₀<1 is enough to eliminate the disease from the population. R₀needs to be lowered much below one to confirm the global stability of the disease-free equilibrium. Numerical simulation is performed to demonstrate the analytical results. It is found that the increase in the rate of detection rate leads to a decrease in the threshold value of R₀. Numerical simulations have been carried out to support the analytic results.

Keywords:

Nonlinear system, Reproduction number, Sensitivity and Bifurcation analysis

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References

Bishop B.M. Potential and emerging treatment options for Ebola virus disease. Ann Pharmacother(2014) 143 -161.

Marsh G. A., Haining J. and Robinson R. Ebola:Reston virus infection of pigs:clinical signifcance and transmission potential. J Infect Dis., 204(Suppl 3):(2011)S804–S809.

Amira Rachah. A mathematical model with isolation for the dynamics of Ebola virus, IOP Conf. Series: Journal of Physics: Conf. Series 1132 (2018) 012058, doi:10.1088/1742-6596/1132/1/012058.

J. Astacio, D. Briere, M. Guillen, J. Martinez, F. Rodriguez and N. Valenzuela-Campos. Mathematical models to study the outbreaks of Ebola, Biometrics Unit Technical Report, Numebr BU-1365-M, Cornell University (1996) 231 pages.

G. Chowell, N. W. Hengartner, C. Castillo-Chavez, P. W. Fenimore, and J. M. Hyman. The basic reproductive number of Ebola and the effects of public health measures: the cases of Congo and Uganda, J.Theor. Biol., vol 229, no. 1, (2004) pp. 119-126.

C. Rizkalla, F. Blanco-Silva, and S. Gruver. Modeling the impact of Ebola and bushmeathunting on western lowland gorillas, EcoHealth J. Consortium, vol 4, (2007) pp. 151-155.

Mhlanga, A. Dynamical analysis and control strategies in modeling Ebola virus disease. Advance differential equation 2019, 458 (2019).

Qiu X., Wong G. and Audet J. Reversion of advanced Ebola virus disease in nonhuman primates with ZMapp. Nature. 514: (2014). 47–53

Abdulrahman N., Sirajo A., and Abdulrazaq A. A mathematical model for the controlling the spread of Ebola virus disease in Nigeria. International Journal of Humanities and Management Sciences (IJHMS) Volume 3, Issue 3, (2015). ISSN 2320–4044.

J. A. Lewnard, M. L. NdeffoMbah, J. A. and Alfaro-Murillo. Dynamics and control of Ebola virus transmission inMontserrado, Liberia: a mathematical modelling analysis, The Lancet Infectious Diseases, vol. 14, no. 12, (2015) pp. 1189–1195.

Kelly, J., et al. Projection of Ebola outbreak size and duration with and without vaccine use in Equateur, Democratic Repulic of Congo, as of May 27, 2018. PLoS ONE14, e0213190 (2019).

A. Rachah and D. F.M. Torres. Mathematical Modelling, Simulation, and Optimal Control of The 2014 Ebola Outbreak in West Africa. Hindawi Publishing Corporation, Discrete Dynamics in Nature and Society, Volume 2015, Article ID 842792, (2015)9 pages.

Akanni J.O. and Akinpelu F.O. An HIV/AIDs model with vertical transmission, treatment and progression rate. Asian Research Journal of Mathematics, 1(4):(2016)1-17, Article no.ARJOM.28549.

Lineberry T. W. and Bostwick J. M. Methamphetamine abuse: A perfect storm of complications Mayo Clin Proc. 2006; 81:(2006)77–84.

Plüddemann A., Dada S. and Parry C. Monitoring alcohol and substance abuse trends in South Africa. SACENDU Research brief,13 (2):(2010)1–16.

A. Plüddemann and C. D. H. Parry. Methamphetamine use and associated problems among adolescents in the Western Cape province of South Africa. MRC South Africa.(2012) 211 pages.

LaSalle JP (1976). The stability of dynamical systems. In: Regional conference series in applied mathematics. SIAM, Philadelphia, Pa.

Castillo-Chavez C. and Song B. Dynamical models of tuberculosis and theirapplications. Math Biosci Eng.,1:(2004) 361–404.

Chitnis, N.; Cushing, J. M. and Hyman, J. M. Bifurcation Analysis of a Mathematical model for malaria transmission. SIAM J. Appl. Math. 67 (1), (2006) 24-45.

J. Arino, C. C. McCluskey, and P. van den Driessche. Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM Journal on Applied Mathematics, vol. 64, no. 1,(2003) pp. 260–276.

S. Olaniyi and O.S. Obabiyi. Qualitative analysis of malaria dynamics with nonlinear incidence function, Applied Mathematical Sciences, 8, No 78, (2014)3889–3904.

Garba S. andGumel A., Bakar M. Backward bifurcation in dengue transmissiondynamics. Math Biosci., 215:(2008) 11–25.

A. D. Adediipo, J. O. Akanni and O. M. Shangodare. Bifurcation and Stability Analysis of the Dynamics of Gonorrhea Disease in the Population, World Scientific News 143 (2020) 139-154.

Mupere E., Kaducu O.F. andYoti Z. Ebola haemorrhagic fever among hospitalised children and adolescents in northern Uganda: epidemiologic and clinical observations. Afr Health Sci.1:(2001) 60–65.

Dowell S. F. Ebola hemorrhagic fever: why were children spared? Pediatr Infect Dis J. 15:(1996)189–191.