Modelling, Analysis, and Simulation of Measles Disease Transmission Dynamics
Haileyesus Tessema Alemneh, Asnakew Mesele Belay
Abstract
Measles is one of the top communicable diseases, which is still responsible for 2.6 million deaths every year. Due to this reason, the paper focuses on measles transmission dynamics concerning the impact of indirect contact rate (transmitted from the host of the virus to the healthy individual) and improving the SEVIR model into the SVIRP model. From the model, we first estimated the disease-free equilibrium, calculated the effective reproduction number <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"><a:mfenced open="(" close=")" separators="|"><a:mrow><a:msub><a:mrow><a:mi>R</a:mi></a:mrow><a:mrow><a:mi mathvariant="normal">E</a:mi><a:mi mathvariant="normal">f</a:mi><a:mi mathvariant="normal">f</a:mi></a:mrow></a:msub></a:mrow></a:mfenced></a:math> , and established the stability analysis. The Castillo–Chavez stability criterion is used to demonstrate the global stability of the disease-free equilibrium point, while the linearization method is used to justify its local stability analysis and gives a result <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" id="M2"><i:msub><i:mrow><i:mtext> </i:mtext><i:mi>R</i:mi></i:mrow><i:mrow><i:mi mathvariant="normal">E</i:mi><i:mi mathvariant="normal">f</i:mi><i:mi mathvariant="normal">f</i:mi></i:mrow></i:msub><i:mo><</i:mo><i:mn>1</i:mn></i:math> . The stability analysis of endemic equilibrium point is explained by defining a Lyapunov function, and its global stability exists when <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" id="M3"><n:msub><n:mrow><n:mi>R</n:mi></n:mrow><n:mrow><n:mi mathvariant="normal">E</n:mi><n:mi mathvariant="normal">f</n:mi><n:mi mathvariant="normal">f</n:mi></n:mrow></n:msub><n:mo>></n:mo><n:mtext> </n:mtext><n:mn>1</n:mn></n:math> . To identify the effect of parameters on the transmission dynamics, we performed sensitivity index and numerical simulation. From the result, we obtained that the indirect contact rate has the highest impact in maximizing the transmission dynamics of measles. Also, we found that working on prevention and treatment strategies brings a significant contribution in reducing the disease effect in the community.