If you have a simple function of t, you can just add the time dependence right there:
function sir_ode!(du,u,p,t)
(S,I,R,C) = u
(a,ω,c,γ) = p
β = a * (1+sin(ω*t))
N = S+I+R
infection = β*c*I/N*S
recovery = γ*I
@inbounds begin
du[1] = -infection
du[2] = infection - recovery
du[3] = recovery
du[4] = infection
end
nothing
end
In a more complicated scenario, you might use events.