Is it possible to simply the combination of two sigmoid functions
$$\sigma\big(k(t_1-t)\big) - \sigma\big(k(t_2-t)\big)$$
I'm not sure what you are asking. Simply is not a verb. If you are asking if it is possible to simplify, it depends on which sigmoid function you are using and what you mean by simplify. I'm gonna go with probably not.
An example sigmoid function is: $\sigma(z) = \dfrac{1}{1+e^z}$. If we were to plug in:
$\dfrac{1}{1+e^{k(t_1-t)}} - \dfrac{1}{1+e^{k(t_2-t)}} = \dfrac{e^{k(t_2-t)}-e^{k(t_1-t)}}{\left( 1+e^{k(t_1-t)}\right) \left( 1+e^{k(t_2-t)} \right)}$
I do not see that as simpler. So, it depends on which sigmoid function you are using and what you mean by "simpler".