eqs={D[Ns[t],t]==-(1/(τc))Ns[t]Ne[t]+Gs,D[Ne[t],t]==(1/(τc))Ns[t]Ne[t]-(1/(τl))Ne[t]-(1/(τr))Ne[t]+Ge,D[Ni[t],t]==(1/(τr))Ne[t],D[Nm[t],t]==(1/(τl))Ne[t],Ns[0]==N0,Ne[0]==0,Nm[0]==0,Ni[0]==0};

Gsv=0.0;
Gev=(1/(2(t2-t1)))(Tanh[((t-t1)/s)]-Tanh[((t-t2)/s)])

conds={N0->200.0,τc->1.0,τr->20.0,τl->500.0,t1->1.0,t2->20.0,s->0.1,Gs->Gsv,Ge->Gev};

eqn={D[Ae[t],t]==Ge,Ae[0]==0}/.conds
sol=DSolve[eqn,{Ae[t]},t][[1]]
Plot[{(Ge/.conds),Ae[t]/.sol/.conds},{t,0,30},PlotRange->All]

gencons={Gs->Gsv,Ge->Gev};
conds={N0->20000.0,τc->64000.0,τr->20.0,τl->500.0,t1->1.0,t2->20.0,s->0.1};
difeqs=eqs/.gencons/.conds
tf=160;
sol=NDSolve[difeqs,{Ns[t],Ni[t],Ne[t],Nm[t]},{t,0,tf}][[1]];
Plot[{(Ns[t]+Ni[t]/.sol),(Ne[t]/.sol),(Nm[t]/.sol)},{t,0,tf},PlotRange->All,Filling->Bottom]
Plot[{(Ne[t]/.sol),(Nm[t]/.sol)},{t,0,tf},PlotRange->All,Filling->Bottom]
Plot[{(Ni[t]/.sol),(Ns[t]/.sol),(Ni[t]+Ns[t]/.sol)},{t,0,tf},PlotRange->All,Filling->Bottom]

Gsv=(10000/(2(t4-t3)))(Tanh[((t-t3)/s)]-Tanh[((t-t4)/s)])
Gev=(1/(2(t2-t1)))(Tanh[((t-t1)/s)]-Tanh[((t-t2)/s)])
gencons={Gs->Gsv,Ge->Gev};
conds={N0->20000.0,τc->64000.0,τr->20.0,τl->500.0,t1->1.0,t2->20.0,t3->180.0,t4->185.0,s->0.1};
difeqs=eqs/.gencons/.conds;
tf=360;
sol=NDSolve[difeqs,{Ns[t],Ni[t],Ne[t],Nm[t]},{t,0,tf}][[1]];
Plot[{(Ns[t]+Ni[t]/.sol),(Ne[t]/.sol),(Nm[t]/.sol)},{t,0,tf},PlotRange->All,Filling->Bottom]
Plot[{(Ne[t]/.sol),(Nm[t]/.sol)},{t,0,tf},PlotRange->All,Filling->Bottom]
Plot[{(Ni[t]/.sol),(Ns[t]/.sol),(Ni[t]+Ns[t]/.sol)},{t,0,tf},PlotRange->All,Filling->Bottom]