This thesis is based on paper by Liping Feng [2012]. In this thesis, a novel computer virus propagation model with dual delays and multi-state antivirus measures is considered. Using theories of stability and bifurcation, it is proven that there exists a critical value of delay for the stability of virus prevalence. When the delay exceeds the critical value, the system loses its stability and a Hopf bifurcation occurs. Furthermore, the explicit formulas determining the stability and direction of bifurcating periodic solutions are obtained by applying the center manifold theorem. Finally, some numerical simulations are performed to verify the theoretical analysis. The conclusions of this thesis can contribute to a better theoretical basis for understanding the long-term actions of virus propagation in networks. With growing interconnectivity and interoperatability of computer networks, computer viruses have increasing opportunities to spread and cause tremendous loss to customers and organizations. L.A. Gordon et al [2005] have estimated that the average annual loss due to virus infections are over 0.2 million dollars. According to statistics, in 2009, the infection rate of computer viruses in China reached 70.15 percent. It is thus imperative to understand in depth how computer viruses spread. Sound anti-virus policies must be based on accurate information about computer virus prevalence and intrinsic understanding of the factors which governs it. Toward this goal, Kephart and White first combined traditional biological approaches and topological interconnections to understand and predict virus propagations in computer networks. After that, an increasing effort has been made to adopt biological epidemic theories to study computer virology. For example, P. De et al in 2009, Y. Moreno et al in 2002 and M. Garetto et al in 2003 studied the spread of malicious objects (viruses and worms) in different network topologies. However, the authors did not discuss how human countermeasures impact on the propagation of viruses. In fact, human countermeasures, such as cleaning, patching, and filtering play an important role in defending against computer viruses. Zou et al. [2006] analyzed red worm propagation based on the Susceptible– Infected–Susceptible (SI/SIS) model and the Susceptible–Infected–Recovered (SIR) model. They investigated how the spread of red worms is affected by the worm characteristics and human countermeasures in 2002, 2003 and 2006. In the above mentioned works, the common feature is that explicit mathematical analysis is not required. In fact, qualitative dynamical analysis for models can provide perfect theoretical insight into prediction and elimination of computer viruses. In biological epidemiology, there has been a significant progress in understanding different scenarios for disease transmissions and behaviors of epidemics by constructing and analyzing differential equations with and without time delays. According to similarities between biological epidemiology and computer viruses, we can obtain a perfect theoretic instruction for controlling the prevalence of computer viruses. But notably, in these biological epidemic models, many assumptions are not suitable for computer networks. Obviously, these hypotheses are not acceptable in real computer networks. Hence, to effectively predict the long-term actions of computer virus prevalence, it is necessary to construct more reliable models of virus propagation. For this pursuit, we have constructed a modified SIRS model, which is a differential equation with dual time delays by adding impact of multi-state immunization. Furthermore, stability and Hopf bifurcation of this model are analyzed. The rest of this thesis is organized as follows: in Section 2, we present a computer virus model by using the theory of epidemic models and the basic concepts of dynamical systems. In section 3, we discussed the delay differential equantions and a great deal about delay differential equations can be learned by a study of its simplest representative, the linear delayed negative feedback equation. Also we discussed the stability of the equilibria, the analysis of the Hopf bifurcation, center manifold and also defined autonomous functional delay differential equantions as a functional differential equantions. In section 4, we discussed the stability of endemic-equilibrium of system and the existence of Hopf bifurcation, and the direction and stability of Hopf bifurcation are analyzed by center manifold approach. Finally in the last section we gave some numerical simulations and some concluding remark.