# Statistical Properties of SIS Processes with Heterogeneous Nodal Recovery Rates in Networks
## 5 Discussion  
Figure 3 demonstrates the time evolution of the fraction of infected nodes for one instance of the simulation as well as a comparison between the simulation and the theory. For the setup of the simulation, the SIS process with heterogeneous recovery rates had a metastable state. Also, the theoretical framework, i.e., HNIMFA with heterogeneous recovery rates, can provide an approximation of the metastable-state mean value of the fraction of infected nodes of the SIS process with promising accuracy. Although the framework introduces an overestimation of \(dv_{i}/dt\), the dense graph rather than the sparse graph is more resilient to the impact of the independence assumption [28].  
On the other hand, it appears that the HNIMFA can provide approximations to the SIS process with heterogeneous recovery rates in finite-size complete graphs for various setups,  
Figure 6: The steady-state variance \(\mathrm{Var}[I]\) of the number of infected nodes for mixture portion \(p=0.5\). Markers and curves represent the simulation results and the theoretical results calculated by (17).  
Figure 7: The steady-state variance \(\mathrm{Var}[I]\) of the number of infected nodes for a mixture portion \(p=0.8\). Markers and curves represent the simulation results and theoretical results calculated with (17), respectively.  
as demonstrated in Figure 4. For the theory employing the independence assumption (4), there are observable deviations near the phase transition point, which is also referred to as the epidemic threshold.  
According to the simulation results shown in Figures 5-7, the variability in the number of infected nodes is tiny, while the infection rate is low and followed by a sharp increase to a maximum. Then, the variability decreases to a non-zero value. The maximum fluctuation of the variability occurs near the phase transition point, where the discrepancy between the theoretical mean infection fraction (14) and the corresponding simulation results also reaches its maximum.  
The proposed variance expression (17) can provide rough approximations of the simulation results for different parameter setups with varying accuracy. For all the setups, the simulations show that the underlying trend for the regime of a high infection rate is stable. This trend is also predicted by the proposed theory. The proposed theory underestimates the simulations for \(p=0.2\), as shown in Figure 5, while it overestimates the simulations for \(p=0.8\), as shown in Figure 7. Notably, the approximation accuracy is sound for the regime with a relatively high infection rate for the case where \(p=0.5\).  
The reasons that there are discrepancies between the proposed variance expression and the simulations are two-fold. Firstly, the approximation (17) omits the co-variance part of the exact expression (12). This kind of manipulation makes it possible to estimate the variance easily, but it introduces theoretical deviations. Secondly, the network topology plays an important role in determining the range of deviations if deviations are inevitable. For a typical topology, such as a complete graph where each pair of nodes is connected, the nodal-pair impact is large such that the co-variance may become large.