Degenerate Lotka-Volterra Mappings and their Corresponding Bigraphs as a Discrete Model of the Evolution of the Interaction of Two Viruses
DOI:
https://doi.org/10.71310/pcam.3_67.2025.02Keywords:
Lotka-Volterra mapping, simplex, skew-symmetric matrix, convex hull, trajectory, partially oriented graph, biographAbstract
This paper proposes a discrete model of the interaction between two airborne viruses, based on an operator acting in a four-dimensional simplex. The model describes the progression of an epidemic in a closed population, divided into five compartments: susceptible individuals, individuals in the latent stage of the first virus, those infected with the first virus, those infected with the second virus, and individuals who have recovered from the first virus. The mathematical structure of the model captures complex transitions between states and interactions between strains, including cases of co-infection. Special attention is given to the analysis of the sets of initial and final states of the disease, defined by systems of inequalities. Depending on the model parameters, these sets may lie on different faces of the simplex, representing various scenarios of epidemic onset and resolution. Two main epidemiological scenarios are considered: one involving complete recovery after infection with the first virus, and another involving progression to co-infection without full recovery. The model is applicable to the analysis of tuberculosis co-infection with viral hepatitis B and C and allows assessment of the influence of various parameters on patient survival during multi-drug therapy. Finally, a numerical experiment is conducted, presenting trajectories, phase portraits, and 30-day dynamics of disease spread, illustrating system behavior under different initial conditions and parameter settings.
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