t ) n(t,a)+\partial _n(t,a)=-\mu (a)n(a,t)} in the simplest case of equal fertilities in the three epidemic classes, we have that in order to have demographic equilibrium the following necessary and sufficient condition linking the fertility φ ( . ) with the mortality μ ( a ) must hold: 1 = ∫ 0 a m φ ( a ) exp ⁡ ( − ∫ 0 a μ ( q ) d q ) d a ^\varphi (a)\exp \left(-\int _^\right)\,da} and the demographic equilibrium is n ∗ ( a ) = c exp ⁡ ( − ∫ 0 a μ ( q ) d q ) , (a)=c\exp \left(-\int _^\mu (q)\,dq\right),} automatically ensuring the existence of the disease-free solution: d f s ( a ) = ( n ∗ ( a ) , 0 , 0 ) . (a),0,0).} a basic reproduction number can be calculated as the spectral radius of an appropriate functional operator. other considerations within compartmental epidemic models vertical transmission in the case of some diseases such as aids and hepatitis b, it is possible for the offspring of infected parents to be born infected. this transmission of the disease down from the mother is referred to as vertical transmission. the influx of additional members into the infected category can be considered within the model by including a fraction of the newborn members in the infected compartment. vector transmission diseases transmitted from human to human indirectly, i.e. malaria spread by way of mosquitoes, are transmitted through a vector. in these cases, the infection transfers from human to insect and an epidemic model must include both species, generally requiring many more compartments than a model for direct transmission. others other occurrences which may need to be considered when modeling an epidemic include things such as the following: non-homogeneous mixing variable infectivity distributions that are spatially non-uniform diseases caused by macroparasites deterministic versus stochastic epidemic models it is important to stress that the deterministic models presented here are valid only in case of sufficiently large populations, and as such should be used cautiously. to be more precise, these models are only valid in the thermodynamic limit, where the population is effectively infinite. in stochastic models, the long-time endemic equilibrium derived above, does not hold, as there is a finite probability that the number of infected individuals drops below one in a system. in a true system then, the pathogen may not propagate, as no host will be infected. but, in deterministic mean-field models, the number of infected can take on real, namely, non-integer values of infected hosts, and the number of hosts in the model can be less than one, but more than zero, thereby allowing the pathogen in the model to propagate. the reliability of compartmental models is limited to compartmental applications. one of the possible extensions of mean-field models considers the spreading of epidemics on a network based on percolation theory concepts. stochastic epidemic models have been studied on different networks and more recently applied to the covid-19 pandemic.