A Mathematical Model Analysis for the Transmission Dynamics of Leptospirosis Disease in Human and Rodent Populations
Habtamu Ayalew Engida, David Mwangi Theuri, Duncan Kioi Gathungu, John Gachohi, Haileyesus Tessema Alemneh
Abstract
This work is aimed at formulating and analyzing a compartmental mathematical model to investigate the impact of rodent-born leptospirosis on the human population by considering a load of pathogenic agents of the disease in an environment and the incidence rate of human infection due to the interaction between infected rodents and the environment. Firstly, the basic properties of the model, the equilibria points, and their stability analysis are studied. We also found the basic reproduction number <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mfenced open="(" close=")"> <a:mrow> <a:msub> <a:mrow> <a:mi>R</a:mi> </a:mrow> <a:mrow> <a:mn>0</a:mn> </a:mrow> </a:msub> </a:mrow> </a:mfenced> </a:math> of the model using the next-generation matrix approach. From the stability analysis, we obtained that the disease-free equilibrium (DFE) is globally asymptotically stable if <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M2"> <e:msub> <e:mrow> <e:mi>R</e:mi> </e:mrow> <e:mrow> <e:mn>0</e:mn> </e:mrow> </e:msub> <e:mo><</e:mo> <e:mn>1</e:mn> </e:math> and unstable otherwise. The local stability of endemic equilibrium is performed using the phenomenon of the center manifold theory, and the model exhibits forward bifurcation. The most sensitive parameters on the model outcome are also identified using the normalized forward sensitivity index. Finally, numerical simulations of the model are performed to show the stability behavior of endemic equilibrium and the varying effect of the human transmission rates, human recovery rate, and the mortality rate rodents on the model dynamics. The model is simulated using the forward fourth-order Runge-Kutta method, and the results are presented graphically. From graphical stability analysis, we observed that all trajectories of the model solutions evolve towards the unique endemic equilibrium over time when <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" id="M3"> <g:msub> <g:mrow> <g:mi>R</g:mi> </g:mrow> <g:mrow> <g:mn>0</g:mn> </g:mrow> </g:msub> <g:mo>></g:mo> <g:mn>1</g:mn> </g:math> . Our numerical results revealed that decreasing the transmission rates and increasing the rate of recovery and reduction of the rodent population using appropriate intervention mechanisms have a significant role in reducing the spread of disease infection in the population.