  In this paper, we first propose a diffusive pathogen infection model with
general incidence rate which incorporates cell-to-cell transmission. By
applying the theory of monotone dynamical systems, we prove that the model
admits the global threshold dynamics in terms of the basic reproduction number
($\mathcal{R}_{0}$), which is defined by the spectral radius of the next
generation operator. Then, we derive a discrete counterpart of the continuous
model by nonstandard finite difference scheme. The results show that the
discrete model preserves the positivity and boundedness of solutions in order
to ensure the well-posedness of the problem. Moreover, this method preserves
all equilibria of the original continuous model. By constructing appropriate
Lyapunov functionals for both models, we show that the global threshold
dynamics is completely determined by the basic reproduction number. Further,
with the help of sensitivity analysis we also have identified the most
sensitive parameters which effectively contribute to change the disease
dynamics. Finally, we conclude the paper by an example and numerical
simulations to improve and generalize some known results.
