A method of upper and lower-solutions for nonlinear time-dependent reaction-diffusion systems of the type \[ \begin{cases} {\partial \over {\partial t}} u_i(x,t) - d_i \Delta u_i(x,t)= f_i(u_1,\dots,u_m, x) \quad \text{ for } x\in \Omega \subset \mathbb R^m \\ \alpha_i u_i(x,t) + \beta_i {\partial \over \partial n} u_i(x,t) = \psi_i(u_1,\dots,u_{i-1},u_{i+1}, \dots, u_m, x) \quad \text{ on } \partial\Omega\\ u_i(x,0)=u_i^\circ(x) \end{cases} \] is employed, without assuming the quasi-monotonicity on the reaction term. As an application, a mathematical model of interaction between immune cells and a virus arising from biological and medical sciences is investigated. For this model the authors obtain the existence of positive solutions and the \(\omega\)-limit.