RESUMEN
We use the epidemic threshold parameter, R 0 , and invariant rectangles to investigate the global asymptotic behavior of solutions of the density-dependent discrete-time SI epidemic model where the variables S n and I n represent the populations of susceptibles and infectives at time n = 0 , 1 , , respectively. The model features constant survival "probabilities" of susceptible and infective individuals and the constant recruitment per the unit time interval [ n , n + 1 ] into the susceptible class. We compute the basic reproductive number, R 0 , and use it to prove that independent of positive initial population sizes, R 0 < 1 implies the unique disease-free equilibrium is globally stable and the infective population goes extinct. However, the unique endemic equilibrium is globally stable and the infective population persists whenever R 0 > 1 and the constant survival probability of susceptible is either less than or equal than 1/3 or the constant recruitment is large enough.
RESUMEN
We investigate the basins of attraction of equilibrium points and minimal period-two solutions of the difference equation of the form x(n+1) = x²(n-1)/(ax²(n) + bx(n)x(n-1) + cx²(n-1)), n = 0,1, 2, , where the parameters a, b, and c are positive numbers and the initial conditions xâ1 and x0 are arbitrary nonnegative numbers. The unique feature of this equation is the coexistence of an equilibrium solution and the minimal period-two solution both of which are locally asymptotically stable.