# Statistical Properties of SIS Processes with Heterogeneous Nodal Recovery Rates in Networks
## 2 Literature Review  
The SIS processes with homogeneous and fixed nodal infection rates and recovery rates in a static and single-layer network have been analyzed thoroughly in many previous studies. [PERSON] et al. [1] modified the basic compartmental model and proposed a heterogeneous mean-field approximation model to investigate the impact of the statistical properties of large networks on the dynamical behaviors of the SIS process in a coarse-grained manner. [PERSON] [16] proposed the N-intertwined mean-field approximation (NIMFA), which is a time-continuous individual-based mean-field model for finite-size networks. The metastable state of the homogeneous SIS process in finite-size networks was claimed to be approximated by the steady state of the NIMFA model. Several steady-state statistical properties of the dynamics were also verified under the theoretical framework of NIMFA. For instance, the epidemic threshold can be approximated by the inverse of the maximum eigenvalue of the adjacency matrix of the underlying network. The metastable-state mean fraction of infected nodes can also be calculated via NIMFA. [PERSON] et al. [17] proposed a discrete-time model to study the statistical properties of the SIS process in networks. [PERSON] and [PERSON] [18] applied the NIMFA framework to investigate the metastable-state variance of the number of infected nodes. Several general approximations were derived to calculate the variance for the SIS process with homogeneous infection and recovery rates in complete and bipartite graphs. It is of great interest to investigate the heterogeneous SIS process, which is defined as the SIS process where at least one nodal infection rate or recovery rate is not the same as that of the others. However, the modeling of SIS processes with heterogeneous recovery rates or infection rates has attracted limited attention. [PERSON] and [PERSON] [19] investigated the SIS process with heterogeneous virus-spread behaviors in networks with finite sizes and extended the NIMFA model to a heterogeneous NIMFA (HNIMFA) model to analyze this complex type of scenario in a general manner. Similarly, [PERSON] et al. [20] extended the NIMFA framework to accommodate the SIS process with two different infection rates and the same recovery rates in finite-size bipartite and star graphs. [PERSON] et al. [20] derived the expressions to estimate the metastable-state mean value of the infection fraction and experimentally verified the accuracy of the approximations. [PERSON] et al. [21] proposed a susceptible-infected-water-susceptible model with different infection rates, which describes the disease spread in different mediums. [PERSON] and [PERSON] [22] proposed a susceptible-infected-recovered-infected model where the nodal susceptibility could change after recovery. In other research regarding SIS process modeling, [PERSON] et al. [23] investigated the impact of time-varying infection rates on infection behaviors. [PERSON] et al. [15] applied the HNIMFA framework to characterize the scenario of interest flooding attacks and mitigation methods. The SIS model with various recovery rates was employed to estimate the metastable-state mean value of the infection fraction and assess the impact of a mixture of mitigation methods on the infection fraction. [PERSON] et al. [24] proposed an approximate SIS pairwise model to investigate the variances of system state variables and the temporal variance of the infection. The authors also found that the variances of the pairwise model and the mean-field SIS model are nearly equal for dense networks. Finally, [PERSON] et al. [25] employed the HNIMFA framework to estimate the metastable-state average variance of the number of infected nodes for bipartite and star graphs and verified its accuracy. It is of great interest to investigate the statistical properties of the SIS epidemic dynamics with complex heterogeneous infection and recovery behaviors. The obtained insights into the complex epidemic dynamics will improve understanding and control of real-world epidemics.