Abstract

During the recent COVID-19 pandemic, quarantine and testing policies have been of vital importance since the causative agent has been a novel virus and no vaccine was developed at the time. In this work, a new epidemiological deterministic model is proposed, analyzed, and discussed. Such a model includes quarantine periods of people with symptoms that have been tested positive, and it will trigger a trace of their close contact, who will be also tested and put in quarantine if the result is positive. Moreover, how the model parameters affect its stability is analyzed with the basic reproduction number R₀. Since the COVID-19 outbreak in Spain (approximately 13/03/2020) until 25/04/2021, different restrictions have been applied. For discussion of a real case study, data have been gathered and used from the Spanish Autonomous Community of Cantabria to estimate the parameters and to see how the restrictions have affected their values. In the parameter estimation process, it has been assumed that the constructed model follows the structure of an ARX model. Finally, by considering that the gathered data are subject to certain errors, the paper discusses how to adequate the model usefulness for its use in fitting and processing data through an estimation mechanism involving the provided daily total and positive performed tests.

1. Introduction

In late December of 2019, China reported to the World Health Organization (WHO) several cases of pneumonia apparently linked to the Hunan Wholesale Seafood Market in Wuhan City, and they pointed out that this disease might be caused by a new virus. Chinese scientists analyzed the virus from a hospitalized person, and the 11th of January, China confirmed that the abnormal arising of pneumonia cases was due to a novel coronavirus, similar to SARS and MERS, so they named it SARS-CoV-2 (afterwards it will be renamed to COVID-19). It was also found out that fever and sore cough were the common symptoms, and that pneumonia was a less common consequence of the disease causing virus [1]. An abundant background literature on many medical aspects and transmission mechanisms of COVID-19 is available, some of them being linked with general description of epidemic models. See, for instance, [2–23] and references there in and also [24–28] for some general results on epidemic model parameterizations of interest. Some specific related details are now described. By late January of 2020, some cities of China locked down so as to slow down the increase in hospitalized people, and some airlines decided to suspend flights to (or from) China. On the 30th of January, the WHO (World Health Organization) reported that there were 7,818 confirmed cases all over the world, from which 7,736 cases were in China, 1,370 were severe and 170 deaths [21]. Moreover, the Centers for Disease Control and Prevention (CDC) wrote a press release which outlined the first case of person-to-person spread in U.S, and consequently world countries geared up [8]. The applied prevention measures were not enough to avoid the virus spreading and on March 11 determined that COVID-19 was a global pandemic [19] since there were 132,758 confirmed cases in the world and 4,955 deaths [18]. On March 17, Europe and Russia closed the borders, and by the time, many countries announced lock downs (for instance, Spain, France, and so on) [1].

When Spain imposed a quarantine period on the 13th of March of 2020, its epidemiological situation was critical: there were 4,209 confirmed cases from which 575 were new ones, and 42% of the total cases were hospitalized [5]. Even if the rules were very restrictive (only on necessary situations it was allowed going outside and wearing the face mask was mandatory), the cases continued raising until 25/03/2020 with a pick value of 720 new cases and 825 deaths [6], and it was not until 25/05/2020, when the number of new cases was approximately 2,000 and 68 the number of deaths [6], that more permission was given to the inhabitants (i.e., it was possible working out outside at certain time ranges). On the 21th of June 2020, the Spanish Government announced the so-called “new normality,” the mask face and social distancing (1.5 m) were still mandatory although at this point each autonomous community could impose more constrains but their capability was limited [7]. During summer vacation, the situation maintained stable; however, when people started working (approximately after 09/09/2020), the epidemiological situation worsen. To tackle this problem, each Autonomous Community (AC) combined and applied differently restrictions (as, for instance, curfew or bars close down), and considering the later results, inhabitants cast doubt on whether the imposed rules were worth or not. Therefore, a mathematical model which could represent the COVID-19 behavior over certain conditions could be very useful.

The aim of epidemiological models is to understand and predict the dynamics of the propagation of a disease, and it must exhibit an equilibrium between three elements [11], namely, accuracy, transparency, and flexibility. Accuracy is the capacity to reproduce the real life behavior and a necessary feature to establish good control actions, perhaps the complexity of the model is proportional to the mentioned accuracy. On the other hand, transparency considers how the model components interact in the dynamics, and it has to be noticed that it is a property that is opposed to complexity and consequently to accuracy. Finally, flexibility is the ability to adapt the model to other circumstances.

If the model fits well with the real disease propagation dynamics, it can be used to infer optimal control actions, so it makes possible implementing more efficient measures and optimizing the use of resources. In 2001, the epidemic foot-and-mouth outbreak in the UK and the result of three different models applied to this case predicted a large-scale spreading and a better control of the epidemic if the infected and exposed animals were culled [26]. In many research studies, it has been considered a spreading of a very high death rate virus such as smallpox [11] to verify whether a massive vaccination, which could generate health problems, is advantageous or not. However, the obtained results differ from each other since there are many epidemiological uncertainties. As far as COVID-19 is a pandemic, there are many data sources that provide information about the epidemiological situation, so it is feasible to create a parametric model from real data and reduce epidemiological uncertainties. Thus, clear control strategies could be built up and implemented.

In this work, a deterministic mathematical model is built up, which spans the well-known SEIR epidemic model by adding people in quarantine and those being asymptomatic. To reduce the model dimension, the infected people with symptoms and asymptomatic will be gathered, so a SEIQR epidemic model is obtained which is integrated by the following subpopulations: Susceptible (S), Exposed (E), Infectious (I), Quarantined (Q), and Recovered or Immune (R). In this model, it is assumed that only exposed and infectious people will cause new infections since people in Q, which represent the individuals who stay in hospital and at home during some period due to the disease, are concerned about their illness, and they try not being in contact with others. On the other hand, it will be considered that some percentage of the individuals in the infectious subpopulation I, more precisely the symptomatic ones, will be reported and carried to Q, and it will trigger a tracing method as it is shown in [3, 23]. Thus, we can say that there are the subpopulations including infected individuals which are the exposed, the infectious, and the quarantined subpopulations. In particular, the asymptomatic infected subpopulation is included in the exposed one.

After constructing the SEIQR model schematic, defining how individuals interact, and defining the necessary parameters, the respective nonlinear system of differential equations will be obtained and analyzed. The theoretical expression for the basic reproduction number, which is a feature dependent on the system’s parameters and a threshold that determines whether the disease is out of control or not, will be obtained through the next generation matrix [27]. For stability analysis purposes, the equilibrium points, that is, the disease-free one and the endemic equilibrium one, which determine the states in which the disease is extinguished from the society and when the disease prevail (i.e., the usual flu disease), respectively, are calculated. The system is linearized around the equilibrium points, and the eigenvalue problem is solved to evaluate when their real parts become positive/negative (condition instability/stability of the system).

Some of the model parameters (for instance, the virus incubation period and the probability to get infected by symptomatic/asymptomatic people) are a disease characteristic, so their values have been collected via other studies [2, 3, 12, 13, 27, 29]. On the other hand, in [30], a model with susceptible, exposed, infectious, quarantined, and removed is discussed with parametrical values fitted to Saudi Arabia data which test the model and analyze the effects of hospital beds availability, quarantines, and media effects on the foreseen dynamics. In [31], a sliding-mode vaccination controller is designed to fight against COVID-19 and testing simulations are performed on data of Iran and Russia. In [32], machine learning designs are used in a susceptible-infected-recovered model to discuss the influence of vaccination in the COVID-19 pandemic evolution in Brazil as well as the impacts of the immunization speed and vaccines efficacy.

However, parameters that depend on the society behavior (i.e., the average number of contacts per day) must be determined with the respective real data. Since the autonomous community of Cantabria gathers wide data about the epidemiological situation, it has been used to estimate the unknown parameters, such as the average contact rate and the tracing effectiveness. It has been assumed that the discrete differential expressions of variables such as Q and I follow an ARX model structure. Thus, the ARX model parameters have been estimated with the least square method and their values have been used to infer the unknown epidemiological values of the parameters. Finally, the differential equations have been solved with the obtained parameters, and the results have been compared with real data.

The paper is organized as follows. Section 2 describes the new proposed epidemic SEQIR (Susceptible-Exposed-Quarantined-Infectious-Recovered) model. Such a model has several specific novel characteristics, whose effects are integrated and analyzed together in a coordinated way, namely,(a)The proportions of asymptomatic and symptomatic individuals in the quarantined group and in the infectious group have, in general, different average transmission rates of contagion to susceptible individuals as it is also potentially distinct from the above mentioned ones the average transmission rate of the exposed individuals to the susceptible ones.(b)Various average transmission rates are parameterized by the infective contacts between individuals. In particular, the values of the average transmission rates are directly proportional to the average potentially infectious contacts. Those contacts occur among the various infected types of individuals (that is, those being currently allocated in the exposed subpopulation, including the asymptomatic infective, in the symptomatic infectious subpopulation, and in the quarantined subpopulation) with susceptible individuals.(c)Various average transmission rates are also parameterized in a direct proportion by the corresponding probabilities of infecting susceptible individuals.(d)The model is also parameterized by the average rate of attendance to the doctor for disease positivity testing or evaluation of symptoms checking and by the effectiveness in tracing the positive cases.(e)The increase in the attendance rate of symptomatic suspect individuals contributes positively to the incorporation to the quarantined subpopulation of those being positively tested.(f)The total day-to-day quarantined subpopulation incorporates the individuals who have been tested positive. This includes the hospitalized individuals including those under intensive care and also those quarantined at their homes having no serious symptoms.

The positivity of the model, in terms of the nonnegativity of all the components of the state trajectory solution under any given finite nonnegative initial conditions, is also proved and discussed in that section. Section 2 also includes a brief discussion on other approaches given for the pandemic study including the potential vaccination and treatment control relevance in the damping of the infection as well as the influences of the intervention measures in the pandemic evolution. On the other hand, Section 3 deals with the disease-free and the endemic equilibrium points which are proved to be unique. The disease-free equilibrium point is proved to be locally asymptotically stable if the basic reproduction number is less than unity, while the endemic one is locally asymptotically stable if such a value exceeds unity. In addition, the endemic equilibrium point is proved to be nonreachable in the first case since its infective components are negative so that the eventual convergence of the state trajectory solution to it would contradict the previously proved positivity of the model performed in the former section. Furthermore, it is proved that only one of the two equilibrium points is a global asymptotic attractor depending again on the value of the basic reproduction number. Section 4 gives some simulations with the proposed model being evaluated for the COVID-19 pandemic with demographic data from Spain. Section 5 develops in more detail a case study against COVID-19 which is performed with official data acquisition taken from the Cantabria Autonomous Community (a region in North Spain). In particular, the estimations of the average periods lasted by people to get to a Health Center or to a doctor for eventual positivity inspection, the efficiency of the positivity tracing, and the average number of contagion contacts are estimated from real data along the initial confinement period and also for a subsequent period. In this subsequent period, the formerly applied strong public intervention measures, of almost total general confinement of the whole population, were left while still public intervention rules were maintained, for instance, a duty of general use of masks, social distance keeping, partial restrictions of mobility, closing of nocturnal amusement places or very significant limitation rules on the number of attendees to spectacles and restaurants, and so on. Finally, conclusions end the paper. A subsection with a glossary of the main symbols is listed below, and the proofs of the stated mathematical results are given in appendixes.

1.1. Nomenclature

 S, E, Q, I, R: subpopulations of susceptible, exposed, quarantined, infectious, and recovered individuals. It is assumed that the exposed subpopulation includes the asymptomatic infective individuals, while the symptomatic infective (or infectious) integrated in the subpopulation . N: total population : average recruitment parameter , , : exposed to susceptible, asymptomatic to susceptible, and symptomatic to susceptible average transmission rates : average incubation rate : average recovery rate : average immunity loss rate : average quarantine rate : average natural death rate : average disease mortality rate : average probability of evolution from exposed to symptomatic : average probability of evolution from exposed to asymptomatic : average rate of symptomatic visits to doctor for testing , , : average numbers of close contacts per day of a member from a group of (either exposed, asymptomatic or symptomatic infectious) to susceptible; and , , and are the respective probabilities to infect a susceptible individual : daily total and positive tests  average number of recovered quarantined individuals who were confined because positive testing

2. Model Setting and Its Description and Positivity

Epidemic models can operate in either deterministic or in stochastic frameworks depending on how the population of one block transforms into the others; that is, deterministic models consider that the rates (namely, the chance to be infected) maintain constant with respect to time, and therefore there is a unique solution for each initial condition. In contrast, stochastic models add random or probabilistic variables into the rates, so the model will give a set of probable solutions. Even if human behavior and infections have stochastic components, when the real system (stochastic) is composed by a large group of people, deterministic models could be used to represent the system. Then, the ratios are constant and could be determined by statistical results [11].

The graphical representation of the system is made by a group of blocks (or compartments), each one represents a certain part of the population, and the arrows make the connection between two blocks (A ⟶ B) and specify the way in which part of the population from A transforms into the inhabitants of the block B. To build up the model flow chart, it is important to know the features of the tackling disease which are the stages of the disease, incubation period, immunity period, and so on.

Regarding COVID-19, the WHO reported that there are two main ways in which the disease is transmitted: presymptomatic and asymptomatic transmissions [20]. Presymptomatic transmission, which is possible during the incubation period (time between exposure and the symptoms onset) which is 5-6 days [17, 20], is more likely to happen 1–3 days before symptoms appear [26]. The preliminary way of transmission is via symptomatic cases, but even if the percentage of asymptomatic cases is 16%-17%, [2, 13], it has been seen that they show similar viral loads [10, 13, 26, 27, 29]. Some research studies have spotlighted the importance of presymptomatic and asymptomatic cases since they contribute to the virus high spread [27, 29]. Therefore, many countries were forced to include quarantine periods to reduce the spreading. The flow chart depicted in Figure 1 shows a deterministic model of COVID-19 which includes the characteristics mentioned before. Note that Figure 1(a) shows five different compartments, namely,(i)Susceptible (S): this class is constructed by a group of individuals who can get the disease and are not yet infected.(ii)Exposed (E): those individuals of a group S who have been in contact with infected people and therefore are infected too, but they do not present symptoms because they are still incubating the virus. They move freely; thus, they will maintain an average number of close contacts per day .(iii)Symptomatic (): this group consists of people who got the disease and present symptoms. Soon after the incubation time, an individual of S_y will probably move freely (the average number of close contacts per day will be equal to ) since the host manifests presymptoms, which commonly are confused with tiredness or symptoms of a common cold/flu. After the list of symptoms will become larger and their intensity will increase, therefore, he/she will go to the doctor and put in quarantine. It is assumed that the average number of close contacts of people from S_y is lower than .(iv)Quarantined (Q): this group is integrated by subgroups coming from S and . People from who presented clear symptoms will have tested positive and put in quarantine. Their positive results will trigger a tracing through their close contacts, which is assumed that are individuals from S. Traced contacts will be tested, and those with positive result will be put in quarantine. It is assumed that the average number of contacts per day of people in Q is nearly zero because they are aware of their disease and they will interact less with others. Note that Q covers people who have tested positive and are staying at home or hospital; therefore, only people from Q might die due to the disease.(v)Asymptomatic (A): a group of infected people who does not present symptoms. They move freely with an average number of close contacts equal to .(vi)Recovered (R): when infected, people recover from the disease and have some immunity to it during a period of time.

(a)

(b)

(a)
(b)

Figure 1 

(a) Flow chart of the epidemic model and (b) reduced flow chart of the epidemic model.

The newborns are introduced in the class S with a recruitment Λ, and from the classes, it removed a portion with a rate μ (daily natural death rate). It is assumed that the population is homogeneously mixed, so it is possible to determine the transmission rates , , and , which indicate the probability to infect susceptible people when they are in contact with exposed, asymptomatic, and symptomatic individuals, respectively. The corresponding expressions are , , and where is the average number of close contacts per day of a member from exposed, asymptomatic, and symptomatic, respectively, to susceptible, while , , and are the respective probabilities to infect a susceptible individual. As it was mentioned before, it will be assumed that , , and . These are necessary parameters to specify how susceptible people get infected; one individual from S has a probability , , and of contacting a person from E, I, and A, respectively, so the chance to get infected is . Therefore, all susceptible people who will get disease per day is

Exposed people after the incubation time ( days) will evolve to asymptomatic people with a probability . Those ones evolving to symptomatic individuals may not consider, at the early beginning, that they are still sick. However, after days (λ indicates the rate at which people from group visit the doctor, and it is supposed that its value will be larger as the test stock, sanitary resources, and the disease impact on society are increased), their symptoms will increase and they will go to the doctor, test positive, and put in quarantine. This will start a tracing, which effectiveness is determined by , of the positive tested symptomatic people with close contacts , and those who test positive will go to Q. Therefore, the number of new exposed per day will be equal to the susceptible people who get the disease per day (see expression (1) minus traced people who have tested positive):

People in Q include those who have been tested positive, which means hospitalized ones, people in Intensive Care Unit (ICU), and those who are in their homes. Some people from Q will remain there during the quarantine period ( days), and others will die from COVID-19 with a rate α. After days, both asymptomatic and symptomatic individuals will recover from the disease, and they will be part of the recovered population R during a period . Now, six blocks are shown in the whole scheme from Figure 1(a), so that six differential equations are needed to represent the system. Managing with six equations might be a difficult task; therefore, a new compartment I involving together A and is defined to reduce the number of equations from six to five, that is,and in Figure 1(b), the block I (infectious) substitutes the blocks A and . In the following, the dependence on time is omitted in the equations for the sake of exposition and simplicity when no confusion is expected. The differential system representing the deterministic model is given by the following set of first-order differential equations:subject to nonnegative initial conditions , and . The above model is being considered in the sequel for analysis. All the parameters are nonnegative and, furthermore, . The total population and its time derivative become

The following result holds on nonnegativity of the trajectory solution for any given finite nonnegative initial conditions. The proof is given in Appendix A.

Theorem 1. Assume that . Then, the following properties hold:(i) for all so that the five subpopulations are nonnegative for all time(ii)The total population is nonnegative and bounded for all time and then the subpopulations are also bounded for all time.

Close techniques for evaluating the nonnegativity of the solution have been used in some background literatures for other epidemic models. See, for instance, [27–32]. Also, it can be pointed out that other related alternative mathematical models for infective disease have been proposed in the recent literature with applications to parameterizations related to COVID-19. For instance, in [33], mixed contagions of susceptible individuals from asymptomatic and symptomatic infectious ones are studied based on provided official data taken from some European countries. In [34], SARS-CoV-2 (COVID-19) transmission data from Spain and Italy were analyzed and processed to estimate the transmission rate levels through SIR models with undelayed and delayed propagation infection related to the periods of different public intervention measures. In [35], confinements and quarantines have been considered as impulsive actions which decrease at certain time instants, and along certain duration time periods, the amount of individuals is able to produce contagions. In [36], an analysis of the contagion propagation levels from small amounts of exposed and infected outsiders coming in a certain habitat was studied under eventual delayed resusceptibility. In [37], a model for COVID-19 including a potential herd immunity has been developed for Austria, Luxembourg, and Sweden. Also, the effect of delaying the vaccination while assuming unlimited vaccine units supply is studied in [38]. A very complete recent work on epidemic modeling is given in [39]. Such a well-organized work discusses different deterministic, stochastic, and optimization models and includes also a discussion on the pandemic evolution in France. On the other hand, some mathematical techniques of interest for its application in mathematical epidemic models related to positivity and stability as well as numerical methods, estimation methods, or probability and statistics tools can be found in [39–47].

3. Equilibrium Points and Stability Analysis

Now, the disease-free and the endemic equilibrium points of the above model are seen to be unique, and their components are characterized. It is also seen that the endemic equilibrium is not reachable (or, roughly speaking, it does not exist in view of Theorem 1 (i)) for small values of the transmission rate since the infective components are negative.

3.1. Equilibrium Points

The equilibrium points are found by zeroing the time derivatives of the subpopulations in (4)–(8). The disease-free equilibrium point satisfies . Looking at (8), it is seen that those constraints imply that and, from the first one, so that is indeed unique. The interpretation is that, in the absence of disease, the whole population is susceptible. Note that, in the absence of recruitment, i.e., if , then the disease-free equilibrium point implies the population extinction.

The existence, uniqueness, and characterization of the endemic equilibrium point are discussed in the two following results which are, respectively, proved in Appendixes B and C.

Lemma 1. Assume that

Then, the following properties hold:(i)We have(ii)We have Let us define the relative transmission rates related to some transmission rate reference value leading to , , and . Then, a necessary condition for the endemic equilibrium point to exist is that .(iii)The quarantined value at the endemic equilibrium point , if it exists (i.e., if ), satisfies the subsequent constraint: and is unique if for some . Furthermore, and .(iv), where(v)We have where

Note that the reference transmission rate is fixed for each given triple of transmission rates , , and with their relative values being , , and .

The following brief discussion concludes that a unique feasible endemic equilibrium point can also exist irrespective of the parameter which describes the attendance rate to the doctor or Health Service for checking or treatment.

Lemma 1 leads directly to the next result whose proof is given in Appendix C.

Theorem 2. Assume that and . Then, the endemic equilibrium point is unique subject to an endemic equilibrium population given by

And, as a result, there is a unique reachable endemic equilibrium point given by (11), (12), (14), (15), and (17). If , then the endemic equilibrium point coincides with the disease-free one.

Remark 1. Theorem 2 establishes that the endemic equilibrium point does not exist for sufficiently small transmission rates so that, in this case, the disease-free equilibrium point is the only feasible equilibrium point. For instance, note that if the three transmission rates are zero, then the susceptible individuals at the endemic equilibrium would be infinity what contradicts Theorem 1(ii) so that, for zero transmission rates, the endemic equilibrium does not exist, as expected.

Remark 2. Note that it becomes attractive to refer the three defined transmission rates to a prescribed reference one by using the relative values of the defined one with respect to such a reference transmission rate. In particular, it is easy in that way to characterize the existence of the endemic equilibrium point referring to such a reference transmission rate exceeding a threshold which depends on the remaining parameters including the three relative transmission rates. It will be seen later on in Section 3.2 that the necessary condition for the reachability of the endemic equilibrium point (equivalent to its existence from its necessary positivity for that concern) is also equivalent to the basic reproduction number to fulfill , which implies in turn the instability of the disease-free equilibrium point. Thus, one will conclude that if the disease-free equilibrium point is locally asymptotically stable, then the endemic equilibrium one is not reachable. Simulations corroborate that if , then the disease-free equilibrium point is locally asymptotically stable. As a result, the disease-free equilibrium point is a global attractor if and equivalently if .

Remark 3. Note that the sufficient conditions and of Theorem 2 for the uniqueness of the endemic equilibrium point are independent of the equilibrium subpopulations amount since it is based on the fact that .
For the particular case, which reflects that people, in average, do not go to the doctor under symptoms since the delayed time to go to the doctor is . For such a case, it can be deduced as well that the endemic equilibrium is unique. However, the reasoning has to be specific for the discussion of the particular model (4)–(8) obtained by zeroing the terms affected by a factor in the differential equations. Since yields to division by zero in some relevant equations in the proof of Lemma 1, the next related result is stated and proved separately from Lemma 1 and Theorem 2. Its proof is given in Appendix D.

Theorem 3. If , then there are a unique disease-free equilibrium point and a unique endemic equilibrium. The first one is ; similarly for , and the endemic one satisfies the subsequent relations:whereFurthermore, the endemic equilibrium point is reachable if the reference transmission rate exceeds a minimum threshold according to

3.2. Local Asymptotic Stability Analysis

The next technical result, proved in Appendix E, relies on the evaluation of the Jacobian matrix with respect to any equilibrium point. It will be then used to characterize the local asymptotic stability of both equilibrium points.

Lemma 2. The Jacobian matrix of the linearized trajectory around any equilibrium point is given by with entries as follows:The above Jacobian matrix can be decomposed as with the transmission and transition matrices, respectively, (being independent of the transmission rates and of the considered equilibrium point) and , whereby using the relative transmission rates , , and with respect to the reference transmission rate .

The above result is useful for the local stability analysis around the disease-free and endemic equilibrium points, respectively, and , via the respective Jacobian matrices and . In the calculation of the entries of the Jacobian matrix around , it has been taken into account that the total population is not constant with respect to time due to the mortality associated to the disease. For instance,generates one of the additive terms in and so on for the remaining entries of the Jacobain matrix around the considered equilibrium point.

The following remark gives some considerations of conceptual interest related to the Jacobain matrix around the equilibrium points and its partition into the sum of the transition and transmission matrices.

Remark 4. Note that the transmission matrix is a Metzler stability matrix, i.e., its diagonal entries are negative and its off-diagonal entries are nonnegative so that all its eigenvalues are real negative and its associated fundamental (or transition) matrix is nonsingular and has nonnegative entries, for all time. Note also that the transmission matrix is zero if the transmission rates are zero (i.e., it is zero in the absence of disease). The fact that has nonnegative entries for all time implies that any solution trajectory is nonnegative for all time under any given nonnegative initial conditions, in the absence of disease which is a necessary condition for the nonnegativity of the whole model to have a nonnegative solution trajectory under nonnegative initial conditions (Theorem 1).
On the other hand, is stable since is stable if and only if the spectral radius of , identical to that of is small enough. According to equation (23), this fact happens for the disease-free equilibrium point if is small enough. It can be proved that this is equivalent also to and it will be seen later on that this is also further equivalent to the basic reproduction number to be less than one. Finally, according to (10), this also implies that the endemic equilibrium point is not reachable.
Local asymptotic stability and instability results around the equilibrium points are stated in the subsequent result, whose proof is given in Appendix F.

Theorem 4. The following properties hold:(i)The Jacobian matrix around the disease-free equilibrium point is stable if and only if the basic reproduction number with the average critical reference transmission rate being defined in (10) and then the disease-free equilibrium point is locally asymptotically stable. Moreover, the spectral radius of equalizes the basic reproduction number where, such that for the disease-free equilibrium point, the transmission and transition matrices are as follows:(ii)The disease- free equilibrium point is unstable if .(iii)Assume that . Then, the endemic equilibrium point is locally asymptotically stable if and only if .(iv)The above conditions in Properties (ii) and (iii) can be weakened in the sense that the disease-free equilibrium point is locally asymptotically stable if and only if and the endemic one is locally asymptotically stable if and only if .

Remark 5. It is now discussed how the calculation of the basic reproduction number can be simplified by manipulating the infective variables, that is, the exposed, infectious, and quarantine subpopulations, only. Note from equations (27)–(28) that the Jacobian matrix at the disease-free equilibrium point is as follows:which makes obvious by its direct inspection that the linearized subsystem of the infective components , , and around the disease-free equilibrium points is not coupled to the linearized subsystem of noninfective components and . Note that the Jacobian matrix of the infective linearized subsystem around the disease-free equilibrium point is given bywhere the corresponding transition and transition matrices are as follows:which concludes that the local stability around the disease-free equilibrium point holds if which guarantees that is a stability matrix since is a stability matrix of eigenvalues , , and . Therefore, the local stability around the disease-free equilibrium point can be analyzed more easily from the Jacobian matrix of the infective linearized subsystem since if as for any initial conditions sufficiently close to the disease-free equilibrium point, then the disease-free equilibrium point is locally asymptotically stable and conversely this happens if the eigenvalues of have negative real parts.