Science World Journal Vol. 16(No 2) 2021 www.scienceworldjournal.org ISSN: 1597-6343 (Online), ISSN: 2756-391X (Print) Published by Faculty of Science, Kaduna State University Mathematical Modeling and Analysis of Pneumonia Infection Dynamics MATHEMATICAL MODELING AND ANALYSIS OF PNEUMONIA INFECTION DYNAMICS *Marcus Ifeanyi Ossaiugbo and Newton I. Okposo Department of Mathematics, Delta State University, P.M.B. 001, Abraka, Nigeria *Corresponding Author’s Email Address: marcusossaiugbo@gmail.com ABSTRACT Pneumonia is one of the leading causes of death worldwide, especially among children below 5 years, the elderly above 65 years and people with weaker immune system. It is usually referred to as the “captain of the men of death" because of the great toll it exacted on humanity. In this work, we examined the dynamics of the pneumonia disease from a mathematical perspective via a deterministic SEIR model. This consists of investigating the equilibrium, basic reproduction number, stability analysis, and bifurcation analysis. It is observed that the pneumonia free equilibrium is locally asymptotically stable if the basic reproduction number is less than one, and the pneumonia endemic equilibrium is globally asymptotically stable in the invariant region if the basic reproduction number is greater than one. The sensitivity analysis revealed that the rate of transmission and the rate at which exposed individuals become infectious are the most sensitive parameters, and the bifurcation analysis via the centre manifold theory revealed the presence of forward bifurcation. Keywords: Pneumonia, SEIR, Model, Stability, Equilibrium, Bifurcation INTRODUCTION Pneumonia is a condition of the lungs that affects the alveoli; and dry cough, chest pain, fever, and trouble breathing are common symptoms of pneumonia. Viruses or bacteria usually cause Pneumonia (Angela, 2009). Eddy (2005) explained that the lungs of individuals with pneumonia are filled with fluid and this makes breathing difficult, and pneumonia disproportionately affects the young, the elderly, as well as vulnerable individuals whose immune system have been compromised. It preys on weakness and vulnerability. Tilahun (2017) revealed that pneumonia was described 2,500 years ago by Hippocrates, the father of medicine, and that Dr. William Osler, the founder of modern medicine, who studied pneumonia throughout his career, called pneumonia the “captain of the men of death” because of the great toll it exacted on humanity. Pneumonia is associated with the following risk factors: pulmonary disease, cystic fibrosis, asthma, diabetes, heart failure, poor ability to cough such as following a stroke, and a weak immune system. The disease may be classified by where it was acquired with community, hospital, or health care associated (Angela, 2009). For children under five years, the typical signs and symptoms of pneumonia include fever, cough, fast or difficult breathing, ongoing vomiting, unwillingness to drink, convulsions, extremes of temperature, and a decreased level of consciousness (Varinder & Satinder, 2011). George (2005) revealed that the introduction of vaccines and antibiotics in the 20th century improved the chance of survival of pneumonia patients, but among the very young, the very old, the chronically ill, and in developing countries, pneumonia remains a leading cause of death. Eddy (2005) explained that pneumonia often shortens suffering among those already close to death and has thus been called "the old man's friend"; while Angela (2009) opined that pneumonia, can be classified as one of the air-borne diseases, and It accounts for the death of millions of people through inhalation of pathogenic organism, mainly streptococcus pneumonia. Human beings of all ages can be affected by the pneumonia disease, from children to the elderly. This is even worsened when the immune system is lowered (WHO, 2008). In order to understand the dynamics of infectious diseases, several scholars proposed different mathematical models to describe the dynamics of infectious diseases in the community and these models are used for making quantitative predictions of different intervention strategies and their effectiveness. Tilahun et al proposed a nonlinear deterministic mathematical model for the typhoid fever outbreak and the optimal control problem was also studied for a community with varying population. It was revealed that the model exhibits a forward transcritical bifurcation, and that treatment is the best cost effective strategy to eradicate the disease. Joseph (2012) studied the impact of treatment and vaccination in curtailing the spread of pneumonia disease, and it was revealed that the rate of transmission, vaccine protection, and the waning rate of vaccine are the main factors in fueling the spread of the disease, while the vaccination and treatment control parameters are inhibitors of the disease spread. He therefore posited that if the vaccination and treatment control programs targeted at both adults and children can reduce the effective reproduction number,   , below unity, then a combination of both programs can effectively eliminate the pneumonia infection from the population. Several scholars also proposed a model on pneumonia dynamics. Kizito and Julius (2018) studied a model on the spread and control of bacterial pneumonia under treatment and vaccination and it was revealed that the disease-free equilibrium is stable if and only if the basic reproduction number,  0 , is less than unity, and the disease will be wiped out of the population, while for  0 >1, the endemic equilibrium is globally stable and the disease persists. Jacob et al (2013) developed a mathematical model for pneumonia among children under five years of age, and the analysis revealed that reducing the transfer rates between the carrier and the infected class reduces prevalence of the disease. The analysis also revealed a possibility of forward bifurcation. In order to investigate the dynamics of the co-infection of pneumonia and meningitis, Tilahun (2019) developed a deterministic mathematical model using ordinary differential equations which divides the population into seven compartments. He emphasized that in order to make the endemic equilibrium unstable so that it switches to disease-free equilibrium, intervention strategies like high efficacy treatment and vaccination programs are necessary. The results obtained revealed that decreasing the contact rate of Full Length Research Article 73