Delay epidemic model with spatial distribution

We develop an epidemiological model integrating time delays (infectious/immune periods) and spatial structure via a symmetric finite-support kernel ϕ ( x − y ) governing location-dependent transmission. The basic reproduction number R 0 dependent on infection transmission rate, disease duration, and probability spatial distribution ϕ is derived. By reducing the delay system to a single integral equation, we establish: (i) Disease-free equilibrium stability loss at R 0 > 1 , leading to the emergence of a positive equilibrium; (ii) Its destabilization (inducing oscillations) for R 0 beyond a higher threshold ( > 1). Theorems proving the existence, non-negativity, and uniqueness of solutions for the system have been established. The model approximation presented in this work replaces the complex spatial integral with a simple expression involving the local infection density I ( x, t ) and its diffusion I xx ( x, t ), significantly simplifying analysis and numerical simulation while preserving the key dynamics of spatial spread for symmetric kernels. An analytical expression for the minimal wave speed of spatial propagation has been investigated. This expression shows that under high transmission, the wave speed depends solely on the infection transmission rate ( β ) and becomes independent of the disease duration ( τ 1 ). This implies that population contact rate is the primary determinant of wave speed across all regimes, while disease duration influences speed only when contact rates are low.