Asian Research Journal of Mathematics

The infectiousness of individuals with an acute respiratory infection can vary with time since entry into the infectious stage, motivating epidemic models that retain infection-age structure rather than treating all infectious individuals as epidemiologically identical. This study formulates and analyses an infection-age-structured model for COVID-19 disease dynamics with susceptible, vaccinated, exposed, infection-age-structured infectious, and recovered populations. New infectious individuals enter the structured infectious class only after progression from the exposed class, so that the boundary condition is i(0,t) = gE(t). Transmission, recovery, and disease-induced mortality may depend on infection age. An explicit basic reproduction number is obtained as;  

where Λ = (1-b)p, α = θ+μ, y = μ+g, and π(a) is the probability of remaining infectious to infection age a . The disease-free equilibrium is locally asymptotically stable when  R₀<1 and unstable when R₀>1; a positive endemic equilibrium exists when R₀>1. For the illustrative parameter set used in the numerical analysis, the baseline value is R₀=0.4694. Multiplying the transmission kernel by three gives R₀=1.4081 and produces persistent infection in the corresponding numerical trajectory. The normalised functional sensitivity density, \(β(a)π(a)/∫_0^Tβ (s)π(s) ds,\) attains its maximum at approximately 2.49 days for the specified kernels, indicating that early infection ages make the largest model-specific contribution to . This peak is a property of the assumed mathematical functions and should not be interpreted as a direct empirical estimate of the biological peak of SARS-CoV-2 infectiousness. The results illustrate how infection-age structure can identify temporal contributions to epidemic transmission while retaining a transparent threshold criterion for local disease invasion.