*3.1. The SEIR Continuous Model with CPT Proportional to Infective Class*

The modification of Equations (1)–(4), by considering the introduction of CPT proportional to the infective class, yields the following equations:

$$\frac{dS}{dt} = \mu - \beta S(t)I(t) - \mu S(t) \tag{5}$$

$$\frac{dE}{dt} = \beta S(t)I(t) - \delta E(t) - \mu E(t) \tag{6}$$

$$\frac{dI}{dt} = \delta E(t) - aI(t) - \mu I(t) - \varepsilon I(t) \tag{7}$$

∗) *with:*

$$\frac{dR}{dt} = \varkappa I(t) - \mu R(t) + \varepsilon I(t) \tag{8}$$

An endemic-free or non-endemic equilibrium always exists for any parameters of the model. However, we show that there is a threshold that determines the existence of an endemic equilibrium, say <sup>T</sup> *<sup>ε</sup>* , so that the endemic equilibrium exists only if <sup>T</sup> *<sup>ε</sup>* is above a certain value; otherwise, an endemic equilibrium does not exist. We sum up this property in the following theorem.

**Theorem 1.** *In the SEIR model (Equations (5)–(8))*, *the following properties hold:*

*(a) A non-endemic equilibrium always exists*, *given by* (*S*<sup>0</sup> <sup>∗</sup>, *E*<sup>0</sup> ∗, *I*<sup>0</sup> ∗, *R*<sup>0</sup> ∗) = (1, 0, 0, 0);

> ∗, *Ie* ∗, *Re*


**Proof of Theorem 1.** By solving Equations (5)–(8) simultaneously under steady-state conditions (i.e., when all LHSs of the equations are equal to zero), the system has two equilibria, i.e., (*S*<sup>0</sup> <sup>∗</sup>, *E*<sup>0</sup> ∗, *I*<sup>0</sup> ∗, *R*<sup>0</sup> <sup>∗</sup>) and (*Se* ∗, *Ee* ∗, *Ie* ∗, *Re* <sup>∗</sup>), with (*S*<sup>0</sup> <sup>∗</sup>, *E*<sup>0</sup> ∗, *I*<sup>0</sup> ∗, *R*<sup>0</sup> ∗) = (1, 0, 0, 0) and:

$$\begin{gathered} S\_{\mathfrak{c}}{}^{\*} = \frac{\delta\mathfrak{a} + \delta\mathfrak{c} + \delta\mu + \mu\mathfrak{a} + \mu\mathfrak{c} + \mu^{2}}{\delta\mathfrak{d}}, \\ E\_{\mathfrak{c}}{}^{\*} = \frac{\left(\mu\delta + \mu\mathfrak{a} + \delta\mathfrak{a} + \delta\mathfrak{c} + \mu\mathfrak{c} + \mu^{2} - \mathfrak{f}\mathfrak{s}\right)\mu}{(\delta + \mu)\mathfrak{f}\mathfrak{d}}, \\ I\_{\mathfrak{c}}{}^{\*} = -\frac{\left(\mu\delta + \mu\mathfrak{a} + \delta\mathfrak{a} + \delta\mathfrak{c} + \mu\mathfrak{c} + \mu\mathfrak{c} + \mu^{2} - \mathfrak{f}\mathfrak{s}\right)\mu}{(\delta\mathfrak{a} + \delta\mathfrak{c} + \delta\mathfrak{a} + \mu\mathfrak{a} + \mu\mathfrak{c} + \mu\mathfrak{c} + \mu^{2})\mathfrak{f}}, \\ R\_{\mathfrak{c}}{}^{\*} = -\frac{\left(\mu\delta + \mu\mathfrak{a} + \delta\mathfrak{a} + \delta\mathfrak{c} + \mu\mathfrak{c} + \mu\mathfrak{c} + \mu^{2} - \mathfrak{f}\mathfrak{s}\right)(\mathfrak{a} + \mathfrak{c})}{(\delta\mathfrak{a} + \delta\mathfrak{c} + \delta\mathfrak{a} + \mu\mathfrak{a} + \mu\mathfrak{c} + \mu^{2})\mathfrak{f}}. \end{gathered}$$


1 − *Se* <sup>∗</sup> − *Ee* <sup>∗</sup> − *Ie* <sup>∗</sup> <sup>=</sup> (<sup>T</sup> *<sup>ε</sup>* <sup>−</sup> <sup>1</sup>) *<sup>α</sup>*+*<sup>ε</sup> <sup>β</sup>* , with <sup>T</sup> *<sup>ε</sup>* <sup>=</sup> *βδ* (*α*+*μ*+*ε*)(*δ*+*μ*). Hence, it is clear that if <sup>T</sup> *<sup>ε</sup>* <sup>=</sup> *βδ* (*α*+*μ*+*ε*)(*δ*+*μ*) > 0, then *Ie* <sup>∗</sup> > 0 and *Ee* ∗ > 0. -

Note that when *<sup>ε</sup>* <sup>=</sup> 0 (i.e., when there is no CPT intervention), then <sup>T</sup> <sup>0</sup> <sup>=</sup> *βδ* (*α*+*μ*)(*δ*+*μ*). Thus, the condition that should be satisfied in order for an endemic equilibrium to exist is <sup>T</sup> <sup>0</sup> <sup>=</sup> *βδ* (*α*+*μ*)(*δ*+*μ*) <sup>&</sup>gt; 1. This can be written as <sup>T</sup> <sup>0</sup> <sup>=</sup> *<sup>β</sup>* <sup>1</sup> (*α*+*μ*) *<sup>δ</sup>* <sup>1</sup> (*δ*+*μ*) > 1 and can be read verbally as the multiplication of four epidemiological factors, namely, (the rate of infection)·(the length of stay within the infectious period)·(the rate of transition from exposed class to infectious class)·(the length of stay within the incubation period). We called <sup>T</sup> <sup>0</sup> the basic threshold number and <sup>T</sup> *<sup>ε</sup>* the effective threshold number. Thus, it is clear that <sup>T</sup> <sup>0</sup> <sup>&</sup>gt; <sup>T</sup> *<sup>ε</sup>* .

To provide a deeper interpretation of this threshold, let us consider a clinical intervention. In the health context, any intentional action designed to obtain an outcome is called a clinical intervention. If, in the absence of clinical intervention, we have <sup>T</sup> <sup>0</sup> <sup>&</sup>gt; <sup>1</sup> (hence, an endemic equilibrium exists), then we could apply a clinical intervention (such as CPT), so that it is possible to reduce the threshold to be less than 1 by changing <sup>T</sup> <sup>0</sup> to <sup>T</sup> *<sup>ε</sup>* for a certain choice of *<sup>ε</sup>* <sup>&</sup>gt; 0, resulting in <sup>T</sup> *<sup>ε</sup>* <sup>&</sup>lt; 1 (removing the endemic equilibrium from the system). This is the basic idea behind controlling/eliminating contagious diseases from a mathematical point of view. Finding this kind of threshold is vital in the study of mathematical epidemiology. In the modern literature, this threshold is usually called the basic reproduction number (sometimes the basic reproduction/reproductive ratio). It is not easy to find this number for more complex transmissions of a disease. There are some good and rigorous literature studies regarding this concept, such as [19,22–24] and [25] (pp. 285–319), that provide a more systematic way of constructing the basic reproduction number. We prove, by standard theory, that the <sup>T</sup> <sup>0</sup> and <sup>T</sup> *<sup>ε</sup>* mentioned above are indeed the basic reproduction number and the effective reproduction number, respectively. We begin by defining the basic reproduction number.

The basic reproduction number of an infection is the expected number of cases produced by one case in a population where all the individuals are susceptible to infection. The authors of [19] (p. 4) defined the basic reproduction number, with the symbol R0, as the expected number of secondary cases per primary case in a "virgin" population. In the same book, they showed that <sup>R</sup>0:<sup>=</sup> lim*n*→<sup>∞</sup> *<sup>K</sup><sup>n</sup>* 1/*<sup>n</sup>* [19] (p. 75), where *<sup>K</sup>* is the nextgeneration matrix defined therein. According to the authors, this is a natural definition of the basic reproduction number from which its value can be computed. However, there is another way to compute the basic reproduction number other than from this definition. In fact, there are some methods that are easier to use to obtain the basic reproduction number. As an example, the following method is suggested in [24]. The authors looked at an epidemic multi-compartment model *dxi dt* = *fi*(*x*) = F*i*(*x*) − V*i*(*x*), *i* = 1, ... , *n* (as in Equations (5)–(8) above). They showed that the function *fi*(*x*) can be decomposed into the rate of appearance of new infections in the *i*th compartment, F*i*(*x*), and the rate of transfer of individuals from/into the *i*th compartment, V*i*(*x*). Furthermore, they defined *F* and *V* to be the Jacobian matrix evaluated at the non-endemic equilibrium and showed that the basic reproduction number can be calculated as the spectral radius <sup>R</sup><sup>0</sup> <sup>=</sup> *<sup>ρ</sup>*(*FV*−1). The following theorem provides the reproduction numbers of the SEIR model (Equations (5)–(8)).

**Theorem 2.** *The SEIR model (Equations (5)–(8)) has the following reproduction numbers:*

