A Model for Childhood Pneumonia Dynamics
Cyrus G Ngari
Kenyatta University, Kenya
David M Malonza
Kenyatta University, Kenya
Grace G Muthuri
Kenyatta University, Kenya
Keywords: Basic reproduction number, Control reproduction number, Infection free point, Endemic equilibrium point, Local and global stability of equilibrium points.
Abstract
This paper presents a deterministic model for pneumonia transmission and uses the model to assess the potential impact of therapy. The model is based on the Susceptible-Infected-Treatment-Susceptible compartmental structure with the possibility of infected individual recovering from natural immunity. Important epidemiological thresholds such as the basic and control reproduction numbers ( R_oand R_c respectively) and a measure of treatment impact are derived. Infection free point was found to be locally stable but globally unstable. We found that if the control reproduction number is greater than unity, then there is a unique endemic equilibrium point and it is less than unity, the endemic equilibrium point is globally asymptotically stable, and pneumonia will be eliminated. Numerical simulations using Matlab software suggest that, besides the parameters that determine the basic reproduction number, natural immunity plays an important role in pneumonia transmissions and magnitude of the public health impact of therapy. Further, treatment regimens with better efficacy holds great promise for lowering the public health burden of pneumonia disease.