Stability Analysis of a Cholera Model

International Journal of Research and Innovation in Applied Science (IJRIAS) | Volume V, Issue XII, December 2020 | ISSN 2454–6186

Stability Analysis of a Cholera Model

   Oluwaseun Biodun AKINDUKO
Department of Mathematical Sciences
Adekunle Ajasin University, Akungba-Akoko, Ondo State, Nigeria

Abstract:- In this paper, a cholera model has been formulated. Using Lipchitz condition, the efficacy of the model was ascertained by testing for the existence and uniqueness of the solution of the model. The disease free equilibrium (DFE), the endemic equilibrium (EE) and the local asymptotic stabilityof the DFE were also conducted. Using the next generation matrix approach, the basic reproduction number, R_0was derived. The DFE was proved to be locally asymptotically stable when R_0<1.

Keywords: existence and uniqueness of solution, Lipchitz condition, Cholera disease, local stability

Table 1:Description of parameters for the model
Parameter Description
b Per capital birth rate of humans
μ Per capital natural death rate of humans
a Rate of exposure to contaminated water
K Concentration of vibrio cholera in water
η Natural recovery
τ Recovery due to treatment
c Rate of compliance with water hygiene
β Rate of compliance with environmental sanitation
δ Cholera-induced death rate
m Growth rate of vibrio cholera in the aquatic environment
ω Loss rate of immunity by recovered individuals
e Contribution of each infected person to the population of vibrio cholera in the aquatic environment

Introduction

Cholera is an acute diarrhoeal disease caused by Vibro Cholerae bacterium. It has continued to be a global threat to public health and a key indicator of lack of social development (Adagbada et al., 2012). Based on a report by the Nigeria Centre for Disease Control (NCDC) in 2019, there have been seven pandemics of Cholera worldwide, the last of which began in Indonesia in 1961, with an estimate of between 1.3 to 4.0 million cases and 21,000 to 143,000 deaths globally due to Cholera every year. The World Health Organization (WHO) reported that only 5 – 10% of the actual cases have been reported especially in low and middle income countries probably due to poor surveillance systems and inadequate disease notification system.The transmission of the bacteria is majorly through ingestion of contaminated food or water as a result of inadequate access to clean water and poor sanitation, especially in peri-urban slums where basic infrastructures are not available. Severe cases of Cholera can lead to death within few hours due to dehydration and the fertility ratio can be up to 50% especially in people without access to treatment which drops to about 1% with adequate treatment. People with low immunity such as malnourished children or people living with HIV are at a greater risk of death if infected [NCDC, 2019]. In Nigeria, Cholera occurs annually mostly during the rainy season, especially in areas with poor sanitation with the first series of the outbreak reported between 1970 and 1990. Major epidemics also occurred in 1992, 1995-1996 and 1997. The Federal ministry of health reported 37,289 cases and 1434 deaths between January and October 2010. NCDC reported 42,466 suspected cases including 830 deaths with a fatality rate of 1.95% from 20 out of 36 states from January to October 2018

Read More