Stability of an equilibrium state in a multiinfectious-type SIS model on a truncated network

Abstract

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...

The dynamics of spreading and immune strategies of sexually transmitted diseases on scale-free network

“…So we think the constant infectivity A is not suitable at least for STDs spread model. From a mathematical aspect, such S I 1 I 2 R S-type models on networks can be viewed as multi-type SIRS models [19,20] if the networks possess the bounded degree property. Ours simulations have proved the necessary to build a multi-type SIRS model for the STDs spread, which can be embodied by the non-usage of condom p. In fact, p is a controllable parameter compare with β 1 , β 2 , η and γ .…”

“…The absence of an intrinsic epidemic threshold has been found in both the susceptible-infected-susceptible (SIS) model [17] and the susceptible-infected-removed (SIR) model [18] in infinite scale-free networks. Papers above normally only prove the existence of epidemic equilibrium, but from a mathematical aspect, such SIS models on a bounded network can be viewed as multiple SIS models [19,20]. In this way, the stabilities of equilibria also can be proved [20].…”

“…Papers above normally only prove the existence of epidemic equilibrium, but from a mathematical aspect, such SIS models on a bounded network can be viewed as multiple SIS models [19,20]. In this way, the stabilities of equilibria also can be proved [20].…”

Mixing and diffusion in a two-type population

“…Their framework includes the SIS model as a particular case, so their analysis of the local stability of the no-infection state applies to the two-type SIS model too. Epidemic thresholds and endemic equilibria in a two-type SIS model considering a bipartite network (which corresponds to the highest possible level of mixing in our setting) are studied in [14,15], while the analysis in [16] deals with random mixing without bias. The models in [13][14][15][16] generalize our approach by considering different interaction frequencies or network degree distributions, but the analysis of the endemic state in those papers corresponds to one particular mixing value (either a bipartite network [14,15] or random mixing [16]), while here we consider arbitrary mixing levels and explore the sensitivity to mixing of the endemic state.…”

“…Epidemic thresholds and endemic equilibria in a two-type SIS model considering a bipartite network (which corresponds to the highest possible level of mixing in our setting) are studied in [14,15], while the analysis in [16] deals with random mixing without bias. The models in [13][14][15][16] generalize our approach by considering different interaction frequencies or network degree distributions, but the analysis of the endemic state in those papers corresponds to one particular mixing value (either a bipartite network [14,15] or random mixing [16]), while here we consider arbitrary mixing levels and explore the sensitivity to mixing of the endemic state. 2 As a first illustration of our main results, consider the case of an infectious disease spreading in a population composed of two distinct groups of individuals (figure 1): one group is more sensitive to contagion (the sensitive group, in red) and the other group is less sensitive (the resistant group, in blue).…”