## 3 Evolutionary Game Theoretical Model
### Model Calculation and Stability Analysis
: _When the conditions are those in Scenarios 4 to 5, \(D_{1}\) (0, 0) is an ESS, and users and OHCs will choose (non-disclosure, negative protection)._  
In these scenarios, since the sum of the benefits obtained by the users who disclose private personal information is between the sum of the costs of the \"positive protection\" strategy and that of the \"negative protection\" strategy, i.e., \(cL_{u}<\Delta<dL_{u}\), some users will choose to disclose their personal information \((x\to 1)\), while other users will choose non-disclosure \((x\to 0)\). On the other hand, for the OHCs, the additional costs to the OHCs are greater than the reduced expected loss of the lowered privacy risk, i.e., \(C_{2}+(c-d)L_{c}>0\), so the OHCs still tend to implement the \"negative protection\" strategy \((y\to 0)\). Over time, when all online health communities choose to implement the \"negative protection\" strategy, users will gradually shift to not disclosing their private personal information \((x\to 0)\) because the relationship between the sum of benefits and the sum of costs will become \(\Delta<dL_{u}\). Therefore, the evolutionary game process will eventually converge to \(D_{1}\) (0, 0), which can be seen in Figure 3(a). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Equilibrium Points** & \(detJ\) & \(trJ\) \\ \hline \(D_{1}\) (0, 0) & \((dL_{u}-\Delta)\times C_{2}\) & \((\Delta-dL_{u})-C_{2}\) \\ \(D_{2}\) (1, 0) & \((dL_{u}-\Delta)\times[(d-c)L_{c}-C_{2}]\) & \((dL_{u}-\Delta)+[(d-c)L_{c}-C_{2}]\) \\ \(D_{3}\) (0, 1) & \((\Delta-cL_{u})\times C_{2}\) & \((\Delta-cL_{u})+C_{2}\) \\ \(D_{4}\) (1, 1) & \((\Delta-cL_{u})\times[(d-c)L_{c}-C_{2}]\) & \((cL_{u}-\Delta)+[C_{2}-(d-c)L_{c}]\) \\ \(D_{5}\) (\(x^{*},y^{*}\)) & \(\frac{C_{2}\times(\Delta-dL_{u})\times(\Delta-cL_{u})\times[(c-d)L_{c}+C_{2}]}{ (c-d)^{2}L_{u}L_{c}}\) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Determinant and trace of the Jacobian matrix. **Proposition 3**. : _When the conditions are those in Scenario 6, \(D_{1}\) (0, 0) and \(D_{4}\) (1, 1) are ESSs, and users and OHCs will choose (non-disclosure, negative protection) or (disclosure, positive protection)._  
In this scenario, the sum of the benefits obtained by users for disclosing private personal information is between the sum of the costs of the \"positive protection\" strategy and that of the \"negative protection\" strategy. As a result, there are users who are willing to disclose privacy \((x\to 1)\) and users who are unwilling to disclose privacy \((x\to 0)\). When the users disclose personal information, the sum of the benefits obtained by the OHCs must be greater than the sum of its costs under the \"positive protection\" strategy, and the additional cost is less than the reduced expected loss of the lowered privacy risk, so the OHCs will tend to implement the \"positive protection\" strategy \((y\to 1)\). Moreover, when users do not disclose private personal information, the OHCs will tend to implement the \"negative protection\" strategy in order to reduce costs \((y\to 0)\). Therefore, the evolutionary game process will eventually form two ESS points: \(D_{1}\) (0, 0) and \(D_{4}\) (1, 1), see Figure 4c.