# Statistical Properties of SIS Processes with Heterogeneous Nodal Recovery Rates in Networks
## 3 Methodology
### Simulation Method  
The simulation method employed in this work is called the \(\varepsilon\)-SIS model. Before introducing the \(\varepsilon\)-SIS model, we first depict the simulation of the SIS process in a fixed network. One can denote by a vector \(X=[X_{1},\cdots,X_{N}]\) the state of the system of the SIS process in a complete graph. For a fixed \(N\), the number \(||X||\) of possible states is equal to \(2^{N}\), which is large but still limited. It is possible to characterize the state change using a Markov chain. Notably, one absorbing state exists and can be represented by \(X=[0,\cdots,0]\). Once trapped in the absorbing state, the state of the system does not change to any other state. For a finite-size state space, the probability that the system traverses from any other state to the absorbing state is non-zero. Thus, the steady state of the SIS process in a finite-size network is the absorbing state.  
On the other hand, the steady state of the NIMFA model actually corresponds to the metastable state of the SIS process in a finite-size network [16]. It is hard to precisely define and detect the metastable state. [PERSON] et al. [27] came up with the so-called \(\varepsilon\)-SIS model, which can be obtained by assigning each node an independent self-infection process such that the absorbing state of the pure SIS process is eliminated. The \(\varepsilon\)-SIS model with a properly chosen self-infection rate provides an approximation to the pure SIS process, and it can avoid the detection of the metastable state of the SIS process. Thus, it was previously employed as the simulation method to obtain empirical results with reliable accuracy for several types of networks [28].  
The self-infection process in the \(\varepsilon\)-SIS model is an independent Poisson process assigned to each node \(i\) with a non-negative rate \(\varepsilon_{i}\). Even if the state of any node becomes susceptible and no infected node exists, each node may be infected by itself with a non-zero probability. In this way, the absorbing state is eliminated by merely introducing the self-infection process into the exact and pure SIS process. As a result, the steady state of the \(\varepsilon\)-SIS model approximately corresponds to the metastable state of the SIS process. It is better to set the critical parameter \(\varepsilon\) as \(\varepsilon<1/N\)[27].  
We employed the \(\varepsilon\)-SIS model as the simulation method and implemented a discrete-time \(\varepsilon\)-SIS simulator to produce empirical results in this work. We initiated a simulation instance by choosing a small number of the nodes as the infected nodes, with the others being susceptible. The simulation instance continued to run for a relatively long time. The changes in the system state were recorded and stored during the simulation. At the end of the simulation instance, we computed several statistical properties of the system based on the records. We set the simulation time step and the end time to \(10^{-3}\) and 200 time units, respectively. We also set the self-infection rate to \(\varepsilon=10^{-3}\) for each node. For the scenario of interest, we set the infection and recovery rates to \(\beta,\delta_{a}=0.5,\delta_{b}=20,\varepsilon=10^{-3}\). For a fixed infection setup (\(\beta\), \(\delta_{a}\), \(\delta_{b}\), \(\epsilon\)), we only needed to run one simulation instance, record the state changes, and finally, calculate the steady-state statistics, such as the average number of infected nodes.  
We show the time evolution of a single run of the simulation, e.g., the time-varying change in the number of infected nodes recorded in a simulation instance. We changed the setup of the infection rate \(\beta\) for several different mixed proportions, \(p=(0.2,0.5,0.8)\), of nodes with a recovery rate of \(\delta_{a}\). Then, we calculated and demonstrated the metastable-state mean number of infected nodes as well as its corresponding variance.