The interstellar e − and e + background is approximated by [24] where E and Φ are in units of GeV and GeV − $^{1}$cm − $^{2}$s − $^{1}$sr − $^{1}$, respectively. The interstellar (IS) positron fraction is given by where we have introduced two real parameters k$_{+}$ and k$_{−}$ to normalize the positron and electron background, which reflects the background uncertainties [29]. In the numerical calculations we take k$_{+}$ ( k$_{−}$ ) = 0 . 9(0 . 7). For the solar modulation effects we adopt the Gleeson and Axford analytical force-field approximation in which the final flux Φ( E$_{⊕}$ ) at the top of atmosphere is given by where Φ( E$_{IS}$ ) denotes the primordial interstellar flux. E$_{IS}$ and E$_{⊕}$ have the following relation where | Z | = 1 is the e ± charge number. For the solar modulation parameter φ$_{F}$ , we take a typical value of φ$_{F}$ = 500 MV. We assume that the solar modulation effect is the same for electron and positron, which leads to PF ( E$_{⊕}$ ) = PF ( E$_{IS}$ ). Note that even in the case of charge-independent modulation, the solar modulation effect leads to a translation for the positron factor PF , thus must be considered. Making use of the above mentioned formulas we calculate the positron fraction and the total electron and positron flux for the eight cases SH(LH)-I ∼ IV. The injection spectra dn k $_{e}$/dE is obtained by using the package Pythia 8.1 [30]. In each case we consider some typical values of the parameters m$_{D}$ and τ$_{D}$ which can reproduce the experimental data well. The favored values for the parameters are listed in Tab. III and the corresponding numerical results are given in Fig. 3. From the case SH(LH)-I to SH(LH)-III, the required mass of S$_{D}$ increases from 2 to 8 TeV while the lifetime decreases from τ$_{D}$ = 1 . 5 × 10 26 s to 12 E$_{IS}$ = E$_{⊕}$ + | Ze | φ$_{F}$ , (26) Φ( E$_{⊕}$ ) = p 2 ⊕ p 2 IS Φ( E$_{IS}$ ) , (25) PF ( E$_{IS}$ ) = Φ DM e + + k$_{+}$ Φ sec e + Φ DM e + + Φ DM e − + k$_{+}$ Φ sec e + + k$_{−}$ (Φ prim e − + Φ sec e − ) . (24) Φ prim e − ( E ) = 0 . 16 E − 1 . 1 1 + 11 E 0 . 9 + 3 . 2 E 2 . 15 , Φ sec e − ( E ) = 0 . 7 E 0 . 7 1 + 110 E 1 . 5 + 600 E 2 . 9 + 580 E 4 . 2 , Φ sec e + ( E ) = 4 . 5 E 0 . 7 1 + 650 E 2 . 3 + 1500 E 4 . 2 , (23)