MATHEMATICAL INVESTIGATION OF A MASS ACTION EPIDEMIC MODEL
DOI:
https://doi.org/10.60787/tnamp.v23.630Keywords:
Global stability, Lyapunov stability, Mass action, System of ODEs, Invariant regionAbstract
A mass action epidemic model incorporating vital dynamics was developed and examined. This type of model is particularly applicable to several childhood illnesses such as Mumps, Rubella, as well as highly contagious diseases like influenza. We demonstrated that the biologically meaningful region—where solutions are feasible—is positively invariant, meaning that any solution starting within this region remains there for all time. The model features two equilibrium points: the disease-free equilibrium (DFE) and the endemic equilibrium. By utilizing the basic reproduction number, , we established that the DFE is locally asymptotically stable when . Additionally, we proved the global stability of the DFE through an appropriately chosen Lyapunov function. We also found that the endemic equilibrium exists only when . Numerical simulations highlighted the critical role of the contact rate in influencing the transmission dynamics within mass action models.
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