A mathematical model on the transmission dynamics of typhoid fever with treatment and booster vaccination
Abdulai Kailan Suhuyini, Baba Seidu
Abstract
Typhoid fever is a potentially fatal illness that is caused by the bacteria Salmonella typhi . In this study, a deterministic mathematical model was formulated to look into transmission dynamics of typhoid fever with treatment and booster vaccination. The reproduction number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> is calculated using the next-generation matrix approach. Then, a stability analysis on the equilibrium points was performed using Routh–Hurwitz criteria. It was revealed that the disease-free equilibrium point is locally asymptotically stable whenever <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> is less than 1 together with other conditions. We also showed that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>≤</mml:mo><mml:mn>1</mml:mn></mml:math> does not guarantee global stability of the typhoid-free equilibrium point and corroborated the result by showing the possible existence of backward bifurcation at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math> . The model parameters in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> were also subjected to sensitivity analysis, which revealed that the transmission rate, infection through an exposed person, and bacteria are the most influential parameters of the reproduction number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> . Numerical simulations were run to determine the impact of various parameters on the dynamics of typhoid.