Mathematical modelling of the dynamics of typhoid fever and two modes of treatment in a Health District in Cameroon

In this paper, we propose a novel mathematical model for indirectly transmitted typhoid fever disease that incorporates the use of modern and traditional medicines as modes of treatment. Theoretically, we provide two Lyapunov functions to prove the global asymptotic stability of the disease-free equilibrium (DFE) and the endemic equilibrium (EE) when the basic reproduction number $ (\mathcal{R}_0) $ is less than one and greater than one, respectively. The model is calibrated using the number of cumulative cases reported in the Penka-Michel health district in Cameroon. The parameter estimates thus obtained give a value of $ \mathcal{R}_0 $ = 1.2058 > 1, which indicates that the disease is endemic in the region. The forecast of the outbreak up to November 2026 suggests that the number of cases will be 21,270, which calls for urgent attention on this endemic disease. A sensitivity analysis with respect to the basic reproduction number is conducted, and the main parameters that impact the widespread of the disease are determined. The analysis highlights that the environmental transmission rate $ \beta $ and the decay rate $ \mu_b $ of the bacteria in the environment are the most influential parameters for $ \mathcal{R}_0 $. This underscores the urgent need for potable water and adequate sanitation within this area to reduce the spread of the disease. Numerically, we illustrate the usefulness of recourse to any mode of treatment to lessen the number of infected cases and the necessity of switching from modern treatment to the traditional treatment, a useful adjuvant therapy. Conversely, we show that the relapse phenomenon increases the burden of the disease. Hence adopting a synergistic therapy approach will significantly mitigate typhoid disease cases and overcome the cycle of poverty within the afflicted communities.