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In this paper, we consider a diffusive SIS epidemic reaction-diffusion model with linear source in a heterogeneous environment in which the frequency-dependent incidence function is SI/(c + S + I) with c a positive constant. We first derive the uniform bounds of solutions, and the uniform persistence property if the basic reproduction number $\mathcal{R}_{0}>1$. Then, in some cases we prove that the global attractivity of the disease-free equilibrium and the endemic equilibrium. Lastly, we investigate the asymptotic profile of the endemic equilibrium (when it exists) as the diffusion rate of the susceptible or infected population is small. Compared to the previous results [1, 2] in the case of c = 0, some new dynamical behaviors appear in the model studied here; in particular, $\mathcal{R}_{0}$ is a decreasing function in c∈[0, ∞) and the disease dies out once c is properly large. In addition, our results indicate that the linear source term can enhance the disease persistence.

In this paper, we investigate a lattice system un'(t) = un+1 (t) - 2un (t) + un-1 (t) + f (un(t)) for n ∈ ℕ with Fisher-KPP type of f (u). With Dirichlet boundary condition at n = 0, we first present the unique positive stationary solution {Vn} and show its stability. Then we construct two types of entire solutions: all of them converge to {Vn} as t → ∞, and to 0 locally uniformly in n ∈ ℕ as t → -∞. The difference is that, in a moving frame, as t → -∞, each of the first type of entire solution converges to a traveling wave solution, while the second type converges to a horizontal line. To show the crucial property u∞(t) = 1 for the second type entire solution, we study the eigenvalues of a corresponding matrix precisely, which is a complete different approach from the Fourier transformation as used in continuous systems. [ABSTRACT FROM AUTHOR]