Emerging diseases in animals and plants have led to much research on questions of evolution and persistence of pathogens. In particular, there have been numerous investigations on the evolution of virulence and the dynamics of epidemic models with multiple pathogens. Multiple pathogens are involved in the spread of many human diseases including influenza, HIV-AIDS, malaria, dengue fever, and hantavirus pulmonary syndrome [9, 15, 16, 23, 24, 27]. Understanding the impact of these various pathogens on a population is particularly important for their prevention and control. W summarize some of the results that have appeared in the literature on multiple pathogen models. Then we study the dynamics of a deterministic and a stochastic susceptible-infected epidemic model with two pathogens, where the population is subdivided into susceptible individuals and individuals infected with pathogen $j$ for $j = 1, 2$. The deterministic model is a system of ordinary differential equations, whereas the stochastic model is a system of stochastic differential equations. The models assume total cross immunity and vertical transmission. The conditions on the parameters for coexistence of two pathogens are summarized for the deterministic model. Then we compare the coexistence dynamics of the two models through numerical simulations. We show that the deterministic and stochastic epidemic models differ considerably in predicting coexistence of the two pathogens. The probability of coexistence in the stochastic epidemic model is very small. Stochastic variability results in extinction of at least one of the strains. Our results demonstrate the importance of understanding the dynamics of both the deterministic and stochastic epidemic models.