## 3 Evolutionary Game Theoretical Model
### Model Calculation and Stability Analysis  
We define the expected benefit of choosing to \"disclose\" their personal information for users as \(P_{1}\), the expected benefit of choosing to \"non-disclose\" personal information as \(P_{2}\), and the average expected benefit as \(\overline{P}\). Therefore,  
\[P_{1}=y[aM+b(E+I+R)-cL_{u}]+(1-y)[aM+b(E+I+R)-dL_{u}] \tag{1}\]  
\[P_{2}=y\times 0+(1-y)\times 0=0 \tag{2}\]  
\[\overline{P}=xP_{1}+(1-x)P_{2}=aM+b(E+I+R)+(d-c)yL_{u}-dL_{u} \tag{3}\]  
From Formulas (1)-(3), we obtain the replication dynamic equation of user game strategy as follows:  
\[F(x)=\frac{dx}{dt}=x\big{(}P_{1}-\overline{P}\big{)}=x(1-x)[aM+b(E+I+R)+(d-c)yL _{u}-dL_{u}] \tag{4}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Notation** & & **Description** \\ \hline \(M\) & Medical service support obtained by users after disclosing their personal information, \(M>0\) \\ \(E\) & Emotional support obtained by users after disclosing their personal information, \(E>0\) \\ \(I\) & Information support obtained by users after disclosing their personal information, \(I>0\) \\ \(L_{u}\) & Loss of users after their privacy leaks, \(L_{u}>0\) \\ \(L_{c}\) & Loss of OHCs after users’ privacy leaks, \(L_{c}>0\) \\ \(R\) & Rewards from the OHCs after users disclose their personal information, \(R>0\) \\ \(C_{1}\) & Costs of implementing the “negative protection” strategy in the OHCs, \(C_{1}>0\) \\ \(C_{2}\) & Extra costs of implementing the “positive protection” strategy in the OHCs, \(C_{2}>0\) \\ \(B\) & Benefits of user disclosure of personal information for OHCs influence, \(B>0\) \\ \(a\) & Probability of users disclosing personal information to medical personnel, \(0\leq a\leq 1\cap a+b\
eq 0\) \\ \(b\) & Probability of users disclosing personal information to other ordinary users, \(0\leq b\leq 1\cap a+b\
eq 0\) \\ \(c\) & Probability of privacy leakage under the “positive protection” strategy, \(0<c<1\) \\ \(d\) & Probability of privacy leakage under the “negative protection” strategy, \(0<c<d<1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notation and description. Figure 3: Relationship between online health communities and users in the game model. We define the expected benefit of choosing to implement the \"positive protection\" strategy for OHCs as \(G_{1}\), the expected benefit of choosing to implement the \"negative protection\" strategy as \(G_{2}\), and the average expected benefit as \(\overline{G}\). Therefore,  
\[G_{1}=x(B-C_{1}-C_{2}-cL_{c})+(1-x)(-C_{1}-C_{2}) \tag{5}\]  
\[G_{2}=x(B-C_{1}-dL_{c})+(1-x)(-C_{1}) \tag{6}\]  
\[\overline{G}=yG_{1}+(1-y)G_{2}=(B-dL_{c})x+(d-c)yxL_{2}-C_{1}-yC_{2} \tag{7}\]  
From Formulas (5)-(7), we obtain the replication dynamic equation of OHC game strategy as follows:  
\[F(y)=\frac{dy}{dt}=y\big{(}G_{1}-\overline{G}\big{)}=y(1-y)[(d-c)xL_{c}-C_{2}] \tag{8}\]  
To simplify the calculation, we let \(\Delta=aM+b(E+I+R)\), and obtain the replication dynamic system equations of users and online health communities as shown in Formula (9):  
\[\left\{ \begin{array}{c}F(x)=x(1-x)[\Delta+(d-c)yL_{u}-dL_{u}]\\ F(y)=y(1-y)[(d-c)xL_{c}-C_{2}]\end{array} \right. \tag{9}\]  
Let \(F(x)=F(y)=0\); then, we obtain the following five potential equilibrium points: \(D_{1}\) (0, 0), \(D_{2}\) (1, 0), \(D_{3}\) (0, 1), \(D_{4}\) (1, 1), and \(D_{5}\) (\(x^{\star},y^{\star}\)). Among them,  
\[x^{\star}=\frac{-C_{2}}{(c-d)L_{c}},y^{\star}=\frac{\Delta-dL_{u}}{(c-d)L_{u}}\]  
Furthermore, the Jacobian matrix can be obtained by solving Formula (9), as shown in Formula (10):  
\[J_{(x,y)}= \begin{bmatrix}\frac{\partial F(x)}{\partial x}&\frac{\partial F(x) }{\partial y}\\ \frac{\partial F(y)}{\partial x}&\frac{\partial F(y)}{\partial y}\end{bmatrix} = \begin{bmatrix}(1-2x)[\Delta+(d-c)yL_{u}-dL_{u}]&x(1-x)(d-c)L_{u}\\ y(1-y)(d-c)L_{c}&(1-2y)[(d-c)xL_{c}-C_{2}]\end{bmatrix}  \tag{10}\]  
We used the research method proposed by [PERSON] (1998) [49] to determine the stability of the above potential equilibrium points, in accordance with the methods of [PERSON] et al.