On some epidemic models

Abstract

The qualitative behavior of the solution x x of the equation \[ x ( t ) = k ( p ( t ) − ∫ 0 t A ( t − s ) x ( s ) d s ) ( f ( t ) + ∫ 0 t a ( t − s ) x ( s ) d s ) , t ≥ 0 x\left ( t \right ) = k\left ( {p\left ( t \right ) - \smallint _0^tA\left ( {t - s} \right )x\left ( s \right )ds} \right )\left ( {f\left ( t \right ) + \smallint _0^ta(t - s)x(s)ds} \right ),t \ge 0 \] is studied. This equation arises in the study of the spread of an infectious disease that does not induce permanent immunity.