## SEIR Epidemic Model with Eventual Vaccination of Newborns  
### The Epidemic Model  
An SEIR epidemic model is proposed where the host population is split into four categories, namely:  
- Susceptible: individuals who have not contracted the disease but are at risk for it, since they do not have the necessary antibodies to cope with it. - Exposed: this is the group of individuals who are already infected, but who are not yet infectious, so they cannot transmit the infection. This state lasts for a limited period of time that varies according to the disease, and is known as the latent period [1] (Chap. Two). - Infectious: those individuals that are infected and can transmit the disease. - Recovered: individuals who have managed to defeat the disease. The body generates immunity as an adaptive response to the infectious agent [15], so that a recovered individual is no longer susceptible. It is considered that the population is homogeneously mixed, that its spatial distribution is uniform, and that all susceptible individuals have the same probability of being infected, which simplifies the problem considerably. The mathematical representation of the SEIR model considered in this case [3] is given by the system of differential equations:  
\[ \begin{array}{l}\dot{S}=(1-q)A-(\beta\,I+\mu)S\\ \\ \dot{E}=\beta\,S\,I-(\mu+\sigma)E\\ \\ \dot{I}=\sigma E-(\mu+\gamma+\alpha)I\\ \\ \dot{R}=qA+\gamma I+\mu R\end{array}  \tag{1}\]subject to an initial condition given by: \(S(0)\geq 0\), \(E(0)\geq 0\), \(I(0)\geq 0\) and \(R(0)\geq 0\). The variables \(S(t)\), \(E(t)\), \(I(t)\), and \(R(t)\) indicate the proportion of susceptible, exposed, infectious, and recovered individuals of the subpopulations, with respect to the total population, found in each compartment at a specific time. The sum of the four subpopulations at the initial time instant is assumed to be unity, in general, so that the subpopulations of the model are fractions of the total initial population. However, this is not a real constraint since the initial conditions can be also fixed to the real values of the various subpopulations. The parameters are constant with positive values related to either demographic characteristics or to the infection type. The symbols used for each variable or parameter are as follows:  
\(S\): Proportion of susceptible individuals  
\(E\): Proportion of exposed individuals  
\(I\): Proportion of infectious individuals  
\(R\): Proportion of recovered individuals  
\(A\): Proportion of new individuals per unit of time  
\(\beta\): Transmission rate  
\(\mu\): Natural mortality rate  
\(\sigma\): Inverse of the latent period  
\(\gamma\): Recovery rate  
\(\alpha\): Mortality rate caused by infection  
\(q\): Fraction of vaccinated newborns  
Note the following facts on the above model:  
* New individuals enter the susceptible compartment per unit of time, either by birth or by immigration. \(\beta\)SI indicates the proportion of infections [4], that is, the proportion of individuals who are no longer susceptible and who become exposed; and finally, \(\mu\)S is the proportion of individuals who die from causes not related to the infection, also per unit of time. * The infected susceptible \(\beta\)SI pass into the exposed compartment and are removed from it once the latent period has passed, that is, \(\sigma\)E individuals are removed per unit of time. Naturally, the fraction \(\mu\)E of those exposed individuals who die from causes unrelated to infection must also be considered. * At the end of the latent period, the exposed \(\sigma\)E become infectious. Moreover, \(\gamma\)I is the proportion of individuals who overcome the disease and who, therefore, pass to the recovered compartment; instead, \(\sigma\)I represents the individuals who die from the infection, and who are removed from the population. * \(\gamma\)I individuals enter the recovered compartment per unit of time, and \(\mu\)R are withdrawn due to mortality from causes not related to the infection.