Stability of an equilibrium state in a multiinfectious-type SIS model on a truncated network modeling to clarify epidemiologically infectious diseases in order to control the spread of infection. To give a little help to this problem, we consider a simple deterministic epidemic model given by ordinary differential equations. For this, expressions of the effects of networks in differential equations are essential, since several well-studied simple epidemic models, such as the SIS, SIR, and SEIR models, have been used to study infectious diseases for a single homogeneous group. 3,[10][11][12] Here, S, I, R, and E mean the susceptible, the infectious, the removal, and the exposed, respectively. Furthermore, the difference in the threshold forms for epidemics caused by the effects of networks is also important.In this direction, Pastor-Satorras and Vespignani 5 fi rst succeeded in studying SIS models on scale-free networks by large-scale simulations and analytical methods based on the mean-fi eld approximations. The scale-free property is one common aspect that real networks possess quite often, as determined by studies of complex networks over the last decade. Scale-free networks can be generated by random procedures with the preference attachment. 6,13 Power-law distributions of vertex degrees make possible a nonnegligible density of vertices with very high degrees in such networks. These factors could encourage the mean-fi eld approximation on scale-free networks. This would be one reasons for Pastor-Satorras and Vespignani's success. They considered a version of SIS models that refl ect the scale-free property, and showed that the epidemic threshold of the infection rate is zero if vertex degree distributions with fi nite means do not have quadratic moments. In other words, an epidemic equilibrium solution exists on such networks unless the infection rate is zero. Thus, an epidemic equilibrium state always exists on scale-free networks with indices between 2 and 3, which are typically observed in all real networks investigated with power-law vertex degree distributions. After this fi rst success, more details of these models were investigated. 6 Several authors [7][8][9] showed that epidemics always occur in various SIS-type models on the same networks as described by Pastor-Satorras and Vespignani. 5 From a mathematical aspect, such SIS-type models on networks are viewed as multi-type SIS models 1,2 (see Abstract In a homogeneous constant population, the basic SIS model potentially has an epidemic equilibrium state with global asymptotic stability since it can be reduced to the logistic equation. On the basic SIS model with a nonhomogeneous constant population, viewed as a multitype SIS model, the global or local asymptotic stability of an epidemic equilibrium state has also been studied. [1][2][3][4] However, this kind of analysis in other models with nonhomogeneous populations has rarely been developed, even though the corresponding models with homogeneous populations are well known. In addition, recent studie...