# The Multi-Compartment SI(RD) Model with Regime Switching: An Application to COVID-19 Pandemic

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**rough**type—considered here, but for a SEIR type model in which the coefficients are supposed to take two (or three values) on two (or three) complementary intervals of the time domain. A study of a regime switching models for the evolution of an infected population that influenced this work is [17]. A recent study of a SI(RD) model with regime switching with a logistic function is presented in [18]; the authors use a neural network to detect the regime change points, and a grid search for the estimation of the usual parameters. A more recently published study of a SIR model with deaths and with changing parameters applied to Spanish data is given in [19]; the SI(RD) model studied in this article was, independently, being discussed by us at least since November 2020 (see [20,21]). In this work, we consider data from the first wave of the COVID-19 pandemic in Portugal. Other very recent works that also consider data from Portugal are [22,23].

- We study the model proposed—a susceptible-infected-(recovered-dead) model with regime switching—proving that there are two ways to achieve the regime switching (the rough way and the smooth way), and that these are equivalent for all practical purposes;
- Given a bivariate set of data—(daily cases-lethality)—we propose a new simple method to estimate the parameters of the model specially suited to the jagged character of the observations using a binomial model; this amounts to hypothesise that there is a mechanism that corrupts, by noise, the observed trajectories of the ordinary differential equations of the model.
- We show that, taking into account important qualitative information, there are secondary parameters—that modify the primary parameters—that can be set according to reasonable scenarios, such that in one of these scenarios, the lethality rate can be recovered with a very small error.

## 2. The SI(RD) Model

**susceptible**to being infected; I, those who have been

**infected**and are able to spread the disease to susceptible individuals; R, those who have

**recovered**from the disease and are immune to subsequent re-infection and D, those who

**died**while infected. The movement of individuals is one-way only according to the scheme: $S\to I\to (R,D)$. The parameters of the model are:

- $\alpha $ the daily infection rate of the susceptible population;
- $\beta $ the recovery rate is a rate which controls how fast members progress into the I and R groups, respectively;
- $\mu $ is the death rate among those infected;
- $\gamma =\alpha /\beta $ is a composite parameter, which—in case $\mu =0$—is often used, and is referred to as the contact number.

**proportions**of the population in each group, that is, s in S, i in I, r in R and d in D; as so, $s,i,r,d$ refer, respectively, to the

**fraction**of the population in the susceptible, infected, recovered and dead subpopulation groups. The ODE system in the Cauchy problem of Formula (1) can also be found in [19] (p. 2).

**numerically**the system of four ordinary differential equations which, for constant or regular time varying parameters, has a unique regular solution.

- If the fraction of the population in the infected group is initially increasing (i.e., ${i}^{\prime}\left(0\right)={i}_{0}>0$), an epidemic has begun. For any $t>0$ in a neighbourhood of zero—such that $i\left(t\right)>0$—and if almost all population is susceptible—that is, ${s}_{0}\approx 1$—we have that:$${i}^{\prime}\left(t\right)>0\iff i\left(t\right)(-\beta -\mu +\alpha s\left(t\right))>0\iff \frac{\alpha}{\beta +\mu}>\frac{1}{s\left(t\right)}\approx \frac{1}{{s}_{0}}\approx 1\phantom{\rule{0.277778em}{0ex}}.$$As a consequence, for the SI(RD) model, we define the composite parameter $\tilde{\gamma}$, corresponding to $\gamma :=\alpha /\beta $ in the SIR model—since in SIR model, there is no $\mu $ parameter—by:$$\tilde{\gamma}=\frac{\alpha}{\beta +\mu}\phantom{\rule{0.277778em}{0ex}}.$$
- At the peak point of an epidemic—with $i\left(t\right)\ne 0$—we should have that ${i}^{\prime}\left(t\right)=0$, by Formula (2), and so, the peak point of an epidemic occurs at a time ${t}_{\pi}$ such that:$$s\left(\right)open="("\; close=")">{t}_{\pi}$$

**Remark**

**1**

**(Basic reproduction number).**The basic reproduction number ${\mathcal{R}}_{0}$ is the expected number of secondary cases produced by a single infected individual in a completely susceptible population. Usually, it is defined as,

- T—the transmissibility which is the probability an individual infecting another given there was contact between them;
- ${C}_{a}$—the average rate of contact between susceptible and infected individuals;
- ${D}_{u}$—the duration of the infection in individuals; in the SI(RD) model, the infection ends with two possible outcomes: either recovered or dead.

**Remark**

**2**

**Remark**

**3**

**(On the positive and bounded solutions of ODE system in Formula (1)).**We observe that the sum of the righthand terms of the four equations in Formula (1) is zero. As a consequence of this observation, we have that, for all t and by integration, the sum $s\left(t\right)+i\left(t\right)+r\left(t\right)+d\left(t\right)$ is constant; since at time ${t}_{1}$, we want to deal with proportions we have that ${s}_{0}+{i}_{0}+{r}_{0}+{d}_{0}=1$ and so, we may conclude that for all t, we have that $s\left(t\right)+i\left(t\right)+r\left(t\right)+d\left(t\right)=1$. Furthermore, if the solutions are all non-negative, the relation $s\left(t\right)+i\left(t\right)+r\left(t\right)+d\left(t\right)=1$, valid for all t, this implies that all the solutions are bounded. The proof to show that the solutions are bounded for the SIR type models may be done in several ways; the most efective proceeds by considering one equation at a time, by expressing the equation in integral form in such a way that an exponential function appears, guaranteeing the positivity of solutions (see for instance [28]). By dealing with one equation after another, it is possible to take into account relations such as the one observed in Formula (4) above.

## 3. SI(RD) Models with Regime Switching (Two Different Values for the Parameter ${\mathcal{R}}_{\mathbf{0}}$)

**pure functional**regime switching. Another basic rendition amounts to just change the parameter, that is considering ${y}^{\prime}\left(t\right)=F(t,y\left(t\right),{\theta}_{2})$ for $t\in [{T}_{a},{T}_{b}]\subset [0,T]$ and ${\theta}_{2}\in \Theta \in {\mathbb{R}}^{d}$ another chosen fixed parameter different from ${\theta}_{1}$; this type of regime switching may be called a

**pure parametric**regime switching. A more general

**mixed**regime switching can be considered by changing both the righthand side of the ODE and the parameter in some subintervals of the original time domain $[0,T]$. In this work we consider, and compare, two different forms of regime switching with the same purpose, to wit, the

**rough (R)**regime switching and the

**smooth (S)**regime switching. Let us consider a first example of

**pure parametric**regime switching, the

**rough**regime switching in the context of a standard existence and unicity theorem for ODE for instance, the global Cauchy-Lipschitz theorem (see [29] (pp. 152–154) and [30] (pp. 119–121)) or with the denomination Picard-Lindelöf theorem [31] (p. 8)).

**Definition**

**1**

- (j)
- The function $F(t,y,\theta ):[0,T]\times {\mathbb{R}}^{4}\u27f6{\mathbb{R}}^{4}$ is a continuous function.
- (jj)
- For some constant $M=M\left(\theta \right)>0$, the uniform Lipschitz condition in the variable t given by,$$\left(\right)open="\parallel "\; close="\parallel ">F(t,{x}_{1},\theta )-F(t,{x}_{2},\theta )\phantom{\rule{0.277778em}{0ex}},$$

**glueing**the solution ${y}_{1}\left(t\right)$ for $t\in [0,{T}_{a}]$ and the solution ${y}_{2}\left(t\right)$ for $t\in [{T}_{a},T]$ and so ${y}_{1,2}\left(t\right)$ is the continuous function in $[0,T]$ that coincides with ${y}_{1}\left(t\right)$ for $t\in [0,{T}_{a}]$ and with ${y}_{2}\left(t\right)$ for $t\in [{T}_{a},T]$.

**Remark**

**4**

**rough**regime switching here described by ${y}_{1,2}\left(t\right)$ is not differentiable at the point ${T}_{a}$ in general. In fact, supposing that $F({T}_{a},{y}_{1}\left({T}_{a}\right),{\theta}_{1})\ne F({T}_{a},{y}_{1}\left({T}_{a}\right),{\theta}_{2})$ and observing that by construction, under the hypothesis of the continuity of the function $F(t,y,\theta )$, we have that:

**rough**regime switching function described above can, nevertheless, be considered as a solution of an ODE, although only in an extended sense.

**smooth (S)**regime switching.

**Definition**

**2**

**Remark**

**5**

**smooth (S)**regime switching, provides a solution for the problem of the change in the dynamics of the phenomena in our ODE model representation, given a choice of function G a smooth solution in $[0,T]$, and given a choice of fixed times ${t}_{0},{t}_{1}\in [0,T]$.

**smooth**regime switching model, as just described with a function such as the one in Figure 1, that is:

**smooth**regime switching case—results presented in Section 5—the system of ODE in Formula (10) is to be integrated numerically. The software used was Mathematica ™ (see [32]). The existence and uniqueness of solutions for such a model pose no problem, as pointed out in Remark 5.

## 4. Data and Parameter Estimation

#### 4.1. On the Assumed Value for $\delta $ the Duration of the Disease

#### 4.2. Determination of the Initial Date of the Decreasing Trend Period

**used for estimation of the contact number**.

#### 4.3. Estimation of $\tilde{\gamma}$ the Contact Number

- (1)
- The length of the period $\tau $ is determined and we have ${\left({i}_{n}\right)}_{0\le n\le {n}_{\tau}}$ a sequence of daily observations of the number of infected. There is a sequence of random variables ${\left({I}_{n}\right)}_{1\le n\le {n}_{\tau}}$ such that the initial ${i}_{0}$ is arbitrary but the remaining observed data ${\left({i}_{n}\right)}_{1\le n\le {n}_{\tau}}$ is a realisation of this sample.
- (2)
- There exists a random variable ${Z}^{\tau}$, taking two values $u,d$—with u representing the magnitude of the
**upward**jump and d representing the magnitude of a**downward**jump—such that $\mathbb{P}[{Z}^{\tau}=u]={p}_{u}$ and $\mathbb{P}[{Z}^{\tau}=d]={p}_{d}$ with ${p}_{u}+{p}_{d}=1$ and ${p}_{u},{p}_{d}>0$. For ${\left({Z}_{n}^{\tau}\right)}_{n\ge 1}$ a sample of ${Z}^{\tau}$ we have, for $n\ge 1$, that:$${I}_{n+1}={Z}_{n+1}^{\tau}{I}_{n}\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{0.277778em}{0ex}}{I}_{0}\equiv {i}_{0}\phantom{\rule{0.277778em}{0ex}}.$$The values $u,{p}_{u},d,{p}_{d}$ are the parameters to be estimated and they depend on the period considered.

- (1)
- The
**maximum likelihood estimator**of ${p}_{u}$, as a consequence of the distribution of ${Z}^{\tau}$, is given by:$$\widehat{{p}_{u}}\left(N\right):=\frac{\#\left(\right)open="\{"\; close="\}">1\le n\le N:{I}_{n+1}{I}_{n}}{}N$$ - (2)
- By the law of large numbers we have that for $N\ge 1$ large enough,$${\widehat{s}}_{N}:=\frac{1}{N}\sum _{n=1}^{N}\frac{{i}_{n+1}}{{i}_{n}}\approx {\mathbb{E}}^{\mathbb{P}}\left[{Z}^{\tau}\right]\phantom{\rule{1.em}{0ex}},\phantom{\rule{1.em}{0ex}}\widehat{{s}_{N}^{2}}:=\frac{1}{N-1}\sum _{n=1}^{N}{\left(\right)}^{\frac{{i}_{n+1}}{{i}_{n}}}2$$
- (3)
- By the
**method of moments**, we can determine the couple of estimators $(\widehat{u},\widehat{d})$ for the parameter couple $(u,d)$ as the solutions of the equations derived from the law of ${Z}^{\tau}$ given by,$$\left(\right)$$$$\left(\right)$$$$\left(\right)$$

**Remark**

**6**

#### 4.4. Estimation of $\mu $ the Lethality Rate

**delay of three days**in the first period and a

**delay of eleven days**in the second period. That is,

**Remark**

**7**

**Remark**

**8**

#### 4.5. Further Hypothesis on the Results of the Estimation Procedure

**more than 20 times**the number of confirmed cases” (see https://www.cnbc.com/2020/10/05/who-10percent-of-worlds-people-may-have-been-infected-with-virus-.html (accessed on 6 October 2020)). As a consequence, since our model applies to the whole population, we will further assume the following:

- We will consider that the real number of infected is
**20 times higher**than the number reported. As a consequence, the model parameters $\tilde{\gamma}$ and $\mu $ are to be modified to take this assumption in consideration. - The estimated value for the parameter $\tilde{\gamma}$ has to be replaced by the value ${\tilde{\gamma}}_{c}$ given by,$${\tilde{\gamma}}_{c}=\tilde{\gamma}\frac{1+19\zeta}{20}\phantom{\rule{0.277778em}{0ex}},$$
- In the same way, the estimated value for the parameter $\mu $ should be replaced by the value ${\mu}_{c}$ given by,$${\mu}_{c}=\mu \frac{1+19\theta}{20}\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.277778em}{0ex}},$$
- For the initially immune ${I}_{i}$ we consider that in the first period these are supposed to be 2.5% of the population. In the second period the initially immune are those who either recovered or died at the end of the preceding period.
- For the
**smooth**regime switching model drive function G, shown in Figure 1, the parameter choice ${\mathbf{\lambda}}_{0}:=(0,\eta ,\theta ,\tau )=(0,0.85,1.39,0.994)$ ensures that for ${t}_{0}=16$ we have $G({t}_{0},{\mathbf{\lambda}}_{0})=0.00238659\approx 0$, for ${t}_{1}=22$ we have $G({t}_{1},{\mathbf{\lambda}}_{0})=0.181604$, for ${t}_{2}=26$ we gave $G({t}_{2},{\mathbf{\lambda}}_{0})=0.477708\approx 0.5$ and that for ${t}_{3}=40$ we gave $G({t}_{3},{\mathbf{\lambda}}_{0})=0.960981\approx 1$ and so, the smooth regime switching unfolds, essentially, between day 16, the declaration of starting time of the lockdown, and day 40.

## 5. Results of the Model

**rough**regime switching ODE we take day 20 that is, 23 March the following day to the emergency state declaration, in Portugal, to be the beginning of the second sub-period with a different regime.

#### 5.1. Results of the Model in the Basis Scenario

**smooth**and the

**rough**case in the case of the base scenario (see Table 5 for the values of the secondary parameters $\zeta $, $\theta $ and $\delta $ in the basis scenario). The yellow constant line at height one—explained in Remark 3—is just the sum of proportions and is drawn for control purposes. The terminal values, at date 67 corresponding to May 9th, for the proportions in each subpopulations—susceptible, infected, recovered and dead—are given in Table 6.

**Remark**

**9**

**smooth**regime switching; on the right-hand side, if they differ, they differ slightly (see Table 6).

**Remark**

**10**

**rough**regime switching, as expected.

**Remark**

**11**

**rough**regime switching—given by the model in Table 6, at day 67, is 2430.14 deaths per million; by reversing the procedure we applied to the estimated lethality rate in order to have a rate applied to the whole infected population in Formula (17) we have that the number of deaths of infected in regard to the diagnosed population is 2430.14/20 = 121.507 deaths per million. The data number of deaths caused by infection corresponding to the pandemic status at day 67, the end of the second period under analysis, should be taken at date 78 in Table 2, corresponding to 20 May 2020, due to the 11-day time lag between the deaths and the infected, and this value is 1263; in [44], we get, for 20 May, 1247 deaths because they did not count the number of deaths which occurred on 20 May, which was 37. For coherence with our computations, we will consider the death toll as 1263. Taking into account the estimate of the Portuguese population of 10,188,013 (see https://www.worldometers.info/world-population/portugal-population/ (accessed on 15 October 2020), we have the number of deaths of infected per million equal to 123.969. So, the ratio of the data number/model number is 123.969/121.507 = 1.0203, that is,

**an excess of 2% of the data over the model**. This can be considered as a good agreement between model and data given that we used an estimate for the total population number in Portugal. We nevertheless observe that the true death toll of the epidemic may be larger than the accounted (see [45], for an early suggestion of this possibility) and that there is also the possibility that the model studied does not cover some important aspects of the epidemic evolution. This problem may be a feature common to the compartmental models; for instance in [5] (p. 857a) the authors predict a 0.06% percentage of deaths in a time horizon of 350 days—a very high value for a death of infected estimate of the whole population—which is a result comparable to the

**rough**regime switching evolution for the extreme scenario I in Table 7 ahead.

#### 5.2. Results of the Model in the Extreme Scenarios I and II

**Remark**

**12**

**For the asymptomatic and/or undiagnosed the most plausible value of the contact number is 50% of the contact number of the diagnosed. The most plausible mortality rate, caused by the disease, for the asymptomatic and/or undiagnosed is zero and the most plausible value for the duration of the disease is 7.5 days.**

## 6. On the Regime Switching SI(RD) ODE Models: Existence and Unicity Results

**rough**regime switching considered in this work—that it seems being used by some authors (see [16], for instance)—is to consider that the coefficients of the SIR type model—for instance, SEIR or SI(RD) models—are not constant but take two (or three) different values in two (or three) complementary intervals of the time domain. In fact, we will observe in Remark 14 that the two approaches to the

**rough**regime switching are

**essentially**the same. For the alternative

**rough**regime switching, the usual Lipschitz type existence theorem or even the Peano existence theorem for ODE do not apply as the function in the righthand term of the ODE—with such discontinuous time varying parameters—is not continuous. Existence and unicity of solutions for such a model is a crucial question. A different approach is needed; following [46] (pp. 41–44), we consider the definition of an

**extended solution**of a differential equation,

**Definition**

**3**

**non**necessarily continuous function, with $I\subset [0,+\infty [$ and $\mathcal{D}\subset {\mathbb{R}}^{4}$. An extended solution $\mathit{{\rm Y}}\left(t\right)$ of the ODE in Formula (18) is an

**absolutely continuous**function $\mathit{{\rm Y}}\left(t\right)$ (see for a precise definition [47] (pp. 144–150)) such that $f(t,\mathit{{\rm Y}}(t\left)\right)\in \mathcal{D}$ for $t\in I$ and Formula (18), or equivalently, Formula (19), is verified for all $t\in I$, possibly with the exception of a set of null Lebesgue measure.

**extended solution**under general conditions.

**Theorem**

**1**

- (i)
- $f(t,\mathit{y})$ is measurable in the variable t, for fixed $\mathit{y}$, and continuous in the variable $\mathit{y}$, for fixed t.
- (ii)
- there exists a Lebesgue integrable function $m\left(t\right)$, defined on a neighbourhood of the initial time, let us say I, such that $\left|f(t,\mathit{y})\right|\le m\left(t\right)$ for $(t,\mathit{y})\in I\times \mathcal{D}$.

**extended solution**according to Definition 3.

**Theorem**

**2**

**Proof.**

**extended solutions**for the

**rough**regime switching models, we observe that the ODE in Formulas (5)—with time changing, discontinuous, parameters $\tilde{\gamma}\left(t\right)$, $\delta \left(t\right)$ and $\mu \left(t\right)$—may be represented as:

**extended solution**for equation given in Formula (18)—for a given initial condition and in the case of the

**rough**regime switching models here considered—exists as a consequence of the estimate in Formula (22) by observing that the components of $\mathit{y}$ are proportions, and so $L\left(\u2225\mathit{y}\u2225\right)$ is bounded, and that $N\left(t\right)$ in Formula (23) is bounded and measurable, and so it is Lebesgue-integrable in any compact interval of the time variable.

**extended solution**, that we know to exist, is unique, in the sense that two solutions may only differ on a set of Lebesgue measures equal to zero. □

**Remark**

**13**

**piecewise**continuous, positive and L nondecreasing, such that for some $c>0$

**extended solutions**instead of usual solutions, which have a continuous derivative, and as $L\left(t\right)$ and $N\left(t\right)$ both satisfy the hypotheses of Wintner’s theorem for parameters $\tilde{\gamma}\left(t\right)$, $\delta \left(t\right)$ and $\mu \left(t\right)$, taking each two (or three) distinct values in two (or three) complementary intervals of the time domain.

**Remark**

**14**

**essentially**only one

**rough**regime switching model). We also observe that our definition of a

**rough**regime switching as the continuous function obtained by

**glueing**the usual solutions of ODE in complementary time intervals—as described in Section 3—is equal almost everywhere in the time variable, with respect to the Lebesgue measure, to any

**extended solution**of the alternative

**rough**regime switching obtained by solving an ODE with discontinuous parameters, which is the best we can expect for

**extended solutions**.

**Remark**

**15**

**smooth**regime switching). It is clear that the existence and unicity of a solution for the

**smooth**regime switching model in Section 3 may be dealt either by the usual Cauchy-Lipschitz existence and unicity theorem or by the results of this section, with the necessary adaptations of the definitions of the matrices $\mathit{M}(\Theta (t),\mathit{{\rm Y}}(t\left)\right)$ and ${\mathit{M}}_{\Delta}(\Theta \left(t\right),{\mathit{y}}_{1},{\mathit{y}}_{2})$. An example of

**smooth**regime switching was studied in [17] with two coupled ODE, having in each equation two evolution regimes in such a way that the transition in time, between these regimes, is achieved by a coupling function similar to the function in Formula (9).

**Remark**

**16**

## 7. Discussion, Conclusions and Further Work

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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$2,3,4,8,9,9,2,18,19,34,57,76,86,117,194,143,235,260,320,460,302,633,549,724,$ |
---|

$902,792,446,1035,808,783,852,638,754,452,712,699,815,1516,515,598,349,514,$ |

$643,750,181,663,521,657,516,603,371,444,595,472,163,295,183,551,295,161,92,$ |

$242,178,480,533,553,138,175,98,234,219,187,264,227,226,173,223,228,252,288,$ |

$271,152,165,219,285,304,350,257,297,200,195,366,331,377,382,342,192,421,294,$ |

$310,270,283,227,346,300,336,417,375,377,292,259,345,367,311,451,323,457,266,$ |

$229,313,328,374,413,328,232,287,443,418,602,342,291,306,233,375,339,312,313,$ |

$246,135,127,252,229,313,263,209,135,111,203,255,204,238,153,106,112,167,213,$ |

$290,186,131,157,120,278,325,235,198,121,132,214,253,291,219,241,145,123,192,$ |

$362,399,401,374,320,244,231,390,418,406,486,315,249,388,646,585,687,497,673,$ |

$613,425,605,770,780,849,552,623,463,793,700,919,864,665.$ |

$0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,2,3,6,3,8,10,10,17,16,24,19,21,20,27,22,37,20,29,16,34,35,$ |
---|

$29,26,35,34,31,32,32,30,28,30,27,21,27,23,35,34,26,23,25,20,25,16,18,16,20,20,11,15,16,$ |

$9,12,9,9,19,12,9,6,13,15,13,16,16,14,12,13,14,14,12,14,13,14,13,14,14,12,11,8,10,9,5,6,$ |

$7,5,7,1,7,5,3,2,1,1,3,1,2,4,6,3,6,6,6,3,4,8,3,8,11,7,9,6,9,2,13,2,8,6,2,6,8,3,3,2,5,2,6,$ |

$5,3,7,4,1,2,3,3,2,8,2,1,0,1,1,3,3,4,6,3,2,3,6,2,3,3,1,5,2,2,4,2,2,5,4,2,2,6,3,1,3,2,3,2,$ |

$4,5,2,3,3,3,3,3,5,7,4,4,3,10,6,5,13,8,5,3,3,5,8,9$ |

**Table 3.**Estimates of ${p}_{u},{p}_{d}$ and of ${\mathcal{R}}_{0}$ for the two successive sub-periods.

Periods | $\widehat{{\mathit{p}}_{\mathit{u}}}\left(\mathit{N}\right)$ | u | $\widehat{{\mathit{p}}_{\mathit{d}}}\left(\mathit{N}\right)$ | d | $\widehat{\tilde{\mathit{\gamma}}}=\widehat{{\mathcal{R}}_{0}}$ |
---|---|---|---|---|---|

$[1,25]$ | 0.833333 | 2.33474 | 0.166667 | −2.04242 | 2.33474 |

$[26,67]$ | 0.487805 | 1.94555 | 0.512195 | 0.411811 | 0.411811 |

Periods | $\mathbf{\Delta}$ | $\widehat{\mathit{\mu}}$ | St. Error | t-Stat | p-Value | Adj. ${\mathit{R}}^{2}$ | BIC | AIC |
---|---|---|---|---|---|---|---|---|

$[1,25]$ | 3 | 0.0306118 | 0.00155902 | 19.6353 | $2.71833\times {10}^{-16}$ | 0.938957 | 123.008 | 120.57 |

$[26,67]$ | 11 | 0.0350414 | 0.00210826 | 16.621 | $8.02743\times {10}^{-20}$ | 0.867616 | 302.68 | 299.21 |

Parameters | Extreme Scenario I | Basis Scenario | Extreme Scenario II |
---|---|---|---|

$\zeta $ | 0.3 | 0.5 | 0.7 |

$\theta $ | 0.25 | 0 | 0.5 |

$\delta $ | 5 | 7.5 | 10 |

Regime | $\mathit{\zeta}$ | $\mathit{\theta}$ | $\mathit{\delta}$ | Susceptible | Infected | Recovered | Deaths |
---|---|---|---|---|---|---|---|

Constant | 0.5 | 0 | 7.5 | 0.588081 | 0.0147572 | 0.368176 | 0.00398521 |

Smooth RS | 0.5 | 0 | 7.5 | 0.71313 | 0.000537805 | 0.258525 | 0.00280666 |

Rough RS | 0.5 | 0 | 7.5 | 0.746942 | 0.000271394 | 0.225357 | 0.00243014 |

Regime | $\mathit{\zeta}$ | $\mathit{\theta}$ | $\mathit{\delta}$ | Susceptible | Infected | Recovered | Deaths |
---|---|---|---|---|---|---|---|

Constant | 0.3 | 0.25 | 5 | 0.801685 | 0.000413846 | 0.107069 | 0.0658329 |

e Smooth RS | 0.3 | 0.25 | 5 | 0.815603 | $6.82517\times {10}^{-6}$ | 0.0981462 | 0.0612436 |

Rough RS | 0.3 | 0.25 | 5 | 0.822341 | $3.1775\times {10}^{-6}$ | 0.0937749 | 0.058881 |

Regime | $\mathit{\zeta}$ | $\mathit{\theta}$ | $\mathit{\delta}$ | Susceptible | Infected | Recovered | Deaths |
---|---|---|---|---|---|---|---|

Constant | 0.7 | 0.5 | 10 | 0.361501 | 0.0420978 | 0.0849807 | 0.48642 |

Smooth RS | 0.7 | 0.5 | 10 | 0.604511 | 0.00413745 | 0.0279379 | 0.338413 |

Rough RS | 0.7 | 0.5 | 10 | 0.67301 | 0.00232243 | 0.0214171 | 0.27825 |

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Esquível, M.L.; Krasii, N.P.; Guerreiro, G.R.; Patrício, P.
The Multi-Compartment SI(RD) Model with Regime Switching: An Application to COVID-19 Pandemic. *Symmetry* **2021**, *13*, 2427.
https://doi.org/10.3390/sym13122427

**AMA Style**

Esquível ML, Krasii NP, Guerreiro GR, Patrício P.
The Multi-Compartment SI(RD) Model with Regime Switching: An Application to COVID-19 Pandemic. *Symmetry*. 2021; 13(12):2427.
https://doi.org/10.3390/sym13122427

**Chicago/Turabian Style**

Esquível, Manuel L., Nadezhda P. Krasii, Gracinda R. Guerreiro, and Paula Patrício.
2021. "The Multi-Compartment SI(RD) Model with Regime Switching: An Application to COVID-19 Pandemic" *Symmetry* 13, no. 12: 2427.
https://doi.org/10.3390/sym13122427