In this article, a delay differential equations model is constructed to observe the spread of rabies among human and dog populations by considering two delay effects on incubation period and vaccine efficacy. Other parameters that affect the spread of rabies are also analyzed. Using the basic reproduction number, it is shown that dog populations and the two delays gives a significant effect on the spread of rabies among human and dog populations. The existence of two delays causes the system to experience Transcritical bifurcation instead of Hopf bifurcation. The numerical simulation shows that depending only on one control method is not enough to reduce or eradicate rabies within the dog populations; instead, it requires several combined strategies, such as increasing dog vaccinations, reducing contact with infected dogs, and controlling puppies' birth. The spread within the human population will be reduced if the spread within the dog population is reduced.

This study aims to propose and analyze a mathematical model of the competitive interaction of the pathogen-immune system. Some effects of the existence of natural delays and the addition of therapeutic proteins are considered in the model. A delay arises from the indirect response of the host body when a pathogen invades. The other comes from the maturation of immune cells to produce immune memory cells since the immune system and antigenic substances responsible for provoking the production of immune memory cells. Analytical investigations suggest several sufficient conditions for the existence of a positive steady-state solution. There is a critical pair of delays at which oscillatory behavior appears around the positive steady-state solution. Numerical simulations were carried out to describe the results of the analysis and show that the proposed model can describe the speed of pathogen eradication due to the addition of therapeutic proteins as antigenic substances.