Editing Compartmental models (epidemiology) (section)

==== The SEIR model with vital dynamics and constant population ====

As described in the example above, so many epidemic processes can be described with a SIR‌ model. However, for many important infections, such as [[COVID-19]], there is a significant latency period during which individuals have been infected but are not yet infectious themselves. During this period the individual is in compartment ''E'' (for exposed). Here, the formation of the next-generation matrix from the [[#The SEIR model|SEIR‌ model]] involves determining two compartments, infected and non-infected, since they are the populations that spread the infection.  So we only need to model the exposed, '''''E''''', and infected, '''''I''''', compartments. Consider a population characterized by a death rate <math>\mu</math> and birth rate <math>\lambda

</math> where a communicable disease is spreading. As in the previous example, we can use the transition rates between the compartments per capita such that <math>\beta</math> be the infection rate, <math>\gamma</math> be the recovery rate, and <math>\kappa</math> be the rate at which a latent individual becomes infectious. Then, we can define the model dynamics using the following equations:<ref name=":3" /><ref name=":5">{{Citation |last1=van den Driessche |first1=P. |title=Further Notes on the Basic Reproduction Number |date=2008 |url=https://doi.org/10.1007/978-3-540-78911-6_6 |work=Mathematical Epidemiology |pages=159–178 |editor-last=Brauer |editor-first=Fred |place=Berlin, Heidelberg |publisher=Springer |language=en |doi=10.1007/978-3-540-78911-6_6 |isbn=978-3-540-78911-6 |access-date=2022-11-08 |last2=Watmough |first2=James |series=Lecture Notes in Mathematics |volume=1945 |editor2-last=van den Driessche |editor2-first=Pauline |editor3-last=Wu |editor3-first=Jianhong|url-access=subscription }}</ref>

<math display="block">\begin{cases} \dot{S} = \lambda - \mu S - \beta SI, \\\\ \dot{E} = \beta SI - (\mu+\kappa)E,  \\\\ \dot{I} = \kappa E - (\mu+\gamma)I, \\\\ \dot{R} = \gamma I - \mu R. \end{cases}
</math>Here we have 4 compartments and we can define vector <math>\mathrm{x} = (S, E, I, R)

</math> where <math>\mathrm{x}_i

</math> denotes the number or proportion of individuals in the ''<math>i

</math>''-th compartment. Let <math>F_i(\mathrm{x})

</math> be the rate of appearance of new infections in compartment ''<math>i

</math>'' such that it includes only infections that are newly arising, but does not include terms which describe the transfer of infectious individuals from one infected compartment to another. Then if <math>V_i^+

</math> is the rate of transfer of individuals into compartment ''<math>i

</math>'' by all other means and <math>V_i^-

</math> is the rate of transfer of individuals out of the ''<math>i

</math>''-th compartment, then the difference <math>F_i(\mathrm{x}) - V_i(\mathrm{x})

</math> gives the rate of change of such that <math>V_i(\mathrm{x}) = V_i^-(\mathrm{x}) - V_i^+
(\mathrm{x})

</math>.

We can now make matrices of partial derivatives of ''<math>F

</math>'' and ''<math>V

</math>'' such that

<math>F_{ij} = {\partial\! \ F_i(\mathrm{x}^*)\over\partial \! \ \mathrm{x}_j}

</math> and <math>V_{ij} = {\partial\! \ V_i(\mathrm{x}^*)\over\partial \! \ \mathrm{x}_j}

</math> , where <math>\mathrm{x}^* = (S^*, E^*, I^*, R^*) = (\lambda/\mu, 0, 0, 0)

</math> is the disease-free equilibrium.

We now can form the next-generation matrix (operator) <math>G = FV^{-1}

</math>.<ref name="Diekmann">{{cite journal | vauthors = Diekmann O, Heesterbeek JA, Metz JA | title = On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations | journal = Journal of Mathematical Biology | volume = 28 | issue = 4 | pages = 365–82 | date = 1990 | pmid = 2117040 | doi = 10.1007/BF00178324 | hdl-access = free | s2cid = 22275430 | hdl = 1874/8051 }}</ref><ref name="vandenDriessche2002" /> Basically, <math>F
</math> is a [[Nonnegative matrix|non-negative matrix]] which represents the infection rates near the equilibrium, and <math>V
</math> is an [[M-matrix]] for linear transition terms making <math>V^{-1}
</math> a matrix which represents the average duration of infectiousness. Therefore, <math>G_{ij}

</math> gives the rate at which infected individuals in ''<math>\mathrm{x}_j

</math>'' produce new infections in ''<math>\mathrm{x}_i

</math>'', times the average length of time an individual spends in a single visit to compartment ''<math>j.

</math>''

Finally, for this SEIR process we can have:

<math>F = \begin{pmatrix} 0 & \beta S^* \\ 0 & 0 \end{pmatrix}

</math> and <math>V = \begin{pmatrix} \mu + \kappa & 0 \\ -\kappa & \gamma + \mu \end{pmatrix}

</math> and so <math>R_0 = \rho( FV^{-1}) = \frac{\kappa\beta S^*}{(\mu+\kappa)(\mu+\gamma)}.
</math>