# Statistical Properties of SIS Processes with Heterogeneous Nodal Recovery Rates in Networks
## 6 Conclusions  
The heterogeneity of epidemics' dynamics poses challenges for their theoretical modeling and analysis and makes it hard to exactly predict the statistical properties of the topological impact on the dynamics. In this work, we proposed employing the HNIMFA framework to analyze the SIS process with heterogeneous recovery rates in finite-size networks. Specifically, we focused on the scenario where the SIS process in a finite-size complete graph shows two nodal recovery rates. We first proposed several lemmas to prove the equality of nodes with the same recovery rate. We also proposed expressions for approximating the metastable-state mean value and variance of the number of infected nodes for the SIS process with heterogeneous recovery rates. Then, we employed the \(\varepsilon\)-SIS model as a simulation method and conducted numerous simulations to verify the proposed theories. We finally analyzed the reasons for the presence of deviations.  
In our future work, we will employ methods employing high-order approximations to improve the accuracy of the approximate computations [13; 29; 30]. The basic NIMFA framework, as a node-based approximation, employs low-order mean-field approximation and neglects co-variance in the expression of the approximation of the variance, which leads to estimation inaccuracy for the variance. Employing high-order approximations may help to propose theories with higher accuracy. In addition to NIMFA, other theory frameworks [13; 29; 30] which could predict the mean value may also be capable of predicting variance more accurately. On the other hand, it would be best to maintain a balance between the order of approximation and the simplicity of the model [10; 13]. Markov theory is accurate but computationally complex for relatively large finite-size networks. The NIMFA framework has a simple expression but decreases the prediction accuracy. High-order pairwise approximations or improved master equations [10; 13] have the potential to predict the variance accurately while having simple expressions. Additionally, we will extend HNIMFA to accommodate the discrete-time SIS process, which shows statistical differences from a continuous-time type of model [11].  
**Author Contributions:** Conceptualization, [PERSON] and L.J.; methodology, [PERSON]; software, [PERSON] and L.J.; validation, [PERSON], L.J. and W.F.; formal analysis, [PERSON]; investigation, [PERSON]; resources, [PERSON] and L.J.; data curation, [PERSON] and L.J.; writing--original draft preparation, [PERSON], L.J. and W.F.; writing--review and editing, [PERSON], L.J. and W.F.; visualization, [PERSON] and L.J.; supervision, [PERSON] and L.J.; project administration, [PERSON]; funding acquisition, [PERSON] and [PERSON]. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This study was supported by the Beijing Advanced Innovation Center for Future Blockchain and Privacy Computing (No: GJJ-23).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The raw data supporting the conclusions of this article will be made available by the authors on request.  
**Acknowledgments:** We would like to thank the anonymous reviewers for their insightful comments and suggestions.  
**Conflicts of Interest:** The authors declare no conflicts of interest.  
**Abbreviations**  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} SIS & Susceptible-infected-susceptible \\ NIMFA & N-intertwined mean-field approximation \\ HNIMFA & Heterogeneous N-intertwined mean-field approximation \\ NDN & Named data networking \\ \end{tabular}