In this research work, we have reviewed and studied two disease models: a disease model with logistic recruitment rate with and without incubation period and a disease model with linear recruitment rate with and without incubation period.  The existence and stability or otherwise of the equilibrium states of the models were studied; also numerical experiments were conducted using a written computer program in MATLAB function ode45; for which table of parameter values were generated using secondary data from the works of Dhar and Sharma (2009) and graphs were plotted, giving pictorial representation of the schemes with the transmission rate from the infected to the susceptible population ‘b’ as the varying parameter. It was observed during the study that in replacing the model with logistic recruitment rate with linear recruitment rate without incubation period, one important critical point (the disease free) was lost. Numerical experiments using published data showed that as the transmission rate from infected to susceptible population increases; the overall population of the infected people for the model with linear recruitment rate goes slightly higher than that of the model with logistic recruitment rate. Our numerical results also showed that as the transmission rate increases, the peak values of the model with logistic recruitment rate decreases and always less than the peak values of the model with linear recruitment rate.  This study provides a comparative analysis of the disease models; the study reveals that the incidence of infection for a disease model with linear recruitment rate is slightly higher than that of a disease model with logistic recruitment rate.