Optimizing quarantine in pandemic control: a multi-stage SEIQR modeling approach to COVID-19 transmission dynamics

Abstract

This study develops and applies an advanced SEIQR (Susceptible-Exposed-Infectious-Quarantined-Removed) model to explore the intricate dynamics of COVID-19 transmission. By incorporating a quarantined compartment into traditional epidemiological frameworks, the model offers a comprehensive examination of how isolation protocols affect pandemic progression. Key parameters such as infection rates, incubation periods, and quarantine durations are systematically analyzed to quantify their influence on the basic reproduction number (ℛ₀) and pandemic trajectory. Simulations reveal that timely and stringent quarantine interventions can reduce peak caseloads by up to 30%, delaying outbreak surges and alleviating pressure on healthcare systems. The model’s robustness is validated against empirical data, confirming its suitability as a predictive and policy-supporting tool. This research not only emphasizes the vital role of quarantine in public health management but also sets a foundational precedent for modeling future outbreaks with similar transmission profiles.

Introduction

The COVID-19 pandemic, caused by the SARS-CoV-2 virus, emerged in late 2019 and rapidly escalated into a global health crisis, affecting over 200 countries and resulting in unprecedented morbidity, mortality, and socio-economic disruptions [1, 2]. The virus’s high transmissibility, primarily through respiratory droplets and close contact, necessitated urgent public health interventions to curb its spread [3, 4]. Early estimates of the basic reproduction number (ℛ₀) indicated the potential for exponential growth, with global fatalities surpassing 600,000 within the first six months of the outbreak [5, 6].

In the absence of vaccines and antiviral therapies during the initial phases, mathematical modeling emerged as a pivotal tool for understanding transmission dynamics and evaluating the efficacy of non-pharmaceutical interventions (NPIs) such as quarantine, social distancing, and lockdowns [7, 8].

Compartmental models, such as the Susceptible-Exposed-Infectious-Removed (SEIR) framework, have been widely employed to simulate disease progression and assess intervention strategies [9]. These models partition populations into distinct compartments based on disease status, enabling the analysis of transmission dynamics under varying conditions. Building on this foundation, the Susceptible-Exposed-Infectious-Quarantined-Removed (SEIQR) model incorporates quarantine measures, providing a more nuanced understanding of their impact on disease control [10, 11]. The inclusion of a quarantined compartment (Q) allows for the evaluation of isolation policies, which have been shown to significantly reduce transmission rates and delay pandemic peaks [12, 13].

Aims

The study is designed to achieve the following objectives:

• 1. Assess Quarantine Effectiveness:

  • ◦ Quantify how quarantine duration, adherence rates, and early implementation reduce the basic reproduction number (R₀) and peak caseloads (e.g., up to 30% reduction in simulations)

• 2. Develop a Predictive Tool for Policymakers:

  • ◦ Provide a dynamic, data-driven SEIQR model to forecast outbreak trajectories under varying intervention scenarios, aiding in healthcare resource allocation and lockdown planning.

• 3. Address Gaps in Existing Models:

  • ◦ Resolve limitations of traditional SEIR models by incorporating:

    • ▪ A quarantine compartment (Q) to isolate the effects of isolation protocols.

    • ▪ Exposed-infectious differentiation to account for pre-symptomatic transmission.

• 4. Theoretical Advancements:

  • ◦ Establish mathematical guarantees for the model’s validity through:

    • ▪ Positivity, existence, and stability analysis of solutions.

    • ▪ Sensitivity analysis of key parameters (e.g., transmission rate β, quarantine rate *q*).

• 5. Guide future pandemic responses:

  • ◦ Offer a generalizable framework adaptable to other respiratory pathogens with similar transmission profiles (e.g., influenza, novel coronaviruses).

While SEIR-based models are well-established, this study advances the field by (1) introducing a multi-stage progression framework to evaluate quarantine efficacy dynamically, (2) incorporating network-derived transmissibility parameters to refine contact-based transmission estimates, and (3) validating results against empirical outbreak data. These innovations address gaps in modeling the interplay between quarantine adherence and pandemic trajectory.

This study employs a five-compartment SEIQR model to analyze the transmission dynamics of COVID-19 in the absence of vaccination. By integrating key parameters such as disease exposure rates, quarantine duration, and transmission probabilities, the model offers insights into the effectiveness of quarantine measures in controlling the pandemic. The findings highlight the importance of timely and sustained interventions in reducing the peak intensity of outbreaks and underscore the utility of compartmental models in informing public health strategies.

Novelty

This study introduces several innovative contributions to the field of epidemiological modeling, particularly in the context of COVID-19 transmission dynamics:

• 1. Multi-Stage SEIQR Framework:

  • ◦ Unlike traditional static models, the study develops a progressive three-stage model (SI → SEI → SEIQR) to dynamically capture the impact of quarantine measures as the pandemic evolves. This approach allows for a more nuanced analysis of how interventions alter transmission over time.

• 2. Network-Based Transmissibility Analysis:

  • ◦ The model uniquely integrates network theory (e.g., generating functions for contact patterns) with compartmental modeling to quantify how individual interactions (“conquest edges”) influence outbreak clusters. This hybrid approach bridges gaps between macro-scale SEIQR models and micro-scale contact networks.

• 3. Empirical Validation with Real-World Data:

  • ◦ The model is rigorously calibrated and validated using 19 weeks of COVID-19 case data (including accumulative cases, deaths, and recoveries) from Saudi Arabia, demonstrating its predictive accuracy for policy support.

• 4. Control Reproduction Number (R_c):

  • ◦ The study introduces R_c (control reproduction number) to evaluate quarantine efficacy post-intervention, a metric less commonly explored in SEIQR literature. This provides actionable insights for policymakers on timing and stringency of measures.

Novelty	Aims
Multi-stage SEIQR progression	Quantify quarantine impact on R₀
Network-based transmissibility	Develop policy-ready predictive tools
R_c for post-intervention analysis	Enhance SEIR models with Q compartment
Empirical validation with real data	Theoretical rigor (stability, positivity)

Literature review

Compartmental models in Epidemiology, such as SIR and SEIR, are foundational for simulating disease transmission [14, 15]. During COVID-19, these models were adapted to address high transmissibility and variable incubation periods [16, 17]. The SEIQR framework, which incorporates quarantine, emerged as a critical tool for evaluating interventions [10, 18].

Regarding to Quarantine dynamics in SEIQR Models; we deduced that Quarantine reduces ℛ₀ by up to 30% when adherence and early detection are prioritized [19, 20]. The SEIQR model’s quarantined compartment (Q) captures isolation’s impact on transmission, validated by real-world COVID-19 data [21, 22]. For example [23], demonstrated its utility in resource allocation during outbreaks.

Recent studies have advanced COVID-19 modeling through various approaches. Phase-adjusted estimation methods improved case tracking accuracy in Wuhan [24], while vaccination impact studies demonstrated reduced transmission [25]. Regional analyses quantified transmission potential in South Korea [26] and vaccine efficacy in the U.S [27]. Behavioral modeling revealed underreporting trends across nations [28], and implementation studies evaluated vaccine rollout strategies [29]. Transmission control measures were analyzed for China’s early epidemic [30], with long-term vaccination impacts assessed globally [31]. Comparative analyses of SEIR and SEIQR models [32] and machine learning integrations [33] enhanced predictive capabilities. Hybrid immunity effects [34] and booster dose impacts [35] were quantified, alongside variant-specific vaccine efficacy studies [36]. Quarantine optimization used stochastic SEIQR approaches [37], while economic analyses measured lockdown impacts [38]. Time-series methods improved wave prediction [39], and revised SEIR models incorporated asymptomatic transmission [40]. Mask mandate effectiveness was globally evaluated [41], while fractional-order models addressed time-delay effects [42]. Climate change impacts on transmission [43] and data-driven testing strategies [44] were modeled, with digital contact tracing’s role mathematically formalized [45].

The challenges and advancements have key limitations that include data gaps and evolving variants [25, 37]. Future work must integrate real-time adjustments (e.g., vaccination effects [35, 36]) and advanced techniques like machine learning [44].

The model is based on several key assumptions: the total population remains constant, with no migration, births, or deaths unrelated to COVID-19; all individuals are equally susceptible to infection, irrespective of age; recovered individuals acquire short-term immunity, preventing immediate reinfection; and the disease progression follows a uniform pattern across the population.

Model formulation and theoretical analysis

Model progression

The progression of the pandemic was modeled in three distinct stages:

• Stage 1: SI Model

The simplest model assumes a population divided into two compartments: susceptible (S) and infectious (I) individuals. The model dynamics capture the transmission rate between susceptible and infectious individuals, as well as the recovery rate from infection.

• Stage 2: SEI Model

The SEI model introduces an additional compartment, “Exposed” (E), to account for individuals who are infected but not yet infectious, representing the incubation period. This stage models the rate of progression from the exposed state to the infectious state.

• Stage 3: SEIQR Model

The SEIQR model builds upon the SEI framework by adding a “Quarantined” (Q) compartment. This compartment represents individuals who are isolated due to suspected or confirmed infection. The model incorporates two key rates: the rate of quarantine entry (α), which denotes the proportion of individuals entering quarantine per unit of time, and the rate of recovery or release from quarantine (γ), which reflects the proportion of quarantined individuals returning to the general population after recovery or clearance.

The study assumes that the total population in all models remains constant throughout the progression of the pandemic. The models are built based on data from COVID-19 cases, with the assumption that susceptibility exists across all age groups, and individuals move randomly within the population. Upon recovery, individuals are assumed to gain immunity that protects them from both short-term and long-term reinfection. The model further assumes that the characteristics of vulnerability and infectiousness are uniform across different populations, as observed globally. The progression of the pandemic was divided into four stages, each representing distinct mechanisms and behaviors associated with COVID-19 transmission. The flow of individuals between compartments is modeled through a set of differential equations.

Stage 1: the basic SIS model and the basic reproduction number, R*

In stage 1, we begin with the SIS (Susceptible-Infectious-Susceptible) model, as it assumes that the disease does not confer immunity against reinfection. Recovered individuals return to the susceptible compartment after recovery, thus making them susceptible to future infections. The study population is divided into three compartments: susceptible (S), infectious (I), and recovered (R), with the transfer between these compartments governed by time-dependent rates. These rates are mathematically represented as derivatives of the compartment sizes with respect to time.

This model assumes that the dynamics of the pandemic unfold on a sufficiently rapid time scale from the point of detection, with demographic effects such as births, immigration, emigration, and natural deaths being neglected for simplicity (WHO 2020; Worldometer 2020).

The simple proposed SIS_* model is:

$$\begin{array}{c}\:\frac{dS}{dt}=-\beta\:S+\gamma\:I\\\:\frac{dI}{dt}=\:\beta\:SI-\gamma\:I\end{array}$$

(1)

.

Where S(t) denotes all active uninfected individuals, and I(t) denotes all active infected (assumed infective) individuals with N(t) = S(t) + I(t) + S_*denotes the total population size which is assumed to be constant regarding that $\:\frac{d}{dt}\left(S+I\right)=0$, ignoring the births and deaths rates.

The model in (1) is then reduced by putting the constant S + I = K; hence it can be treated as a single differential equation:

$$\begin{array}{c}\begin{array}{c}\:\frac{dI}{dt}=\beta\:I\left(K-I\right)=\gamma\:I=\left(\beta\:K-\gamma\:\right)I-\beta\:{I}^{2}\\\:=\left(\beta\:K-\gamma\:\right)I\left(1-\frac{I}{K-\frac{\gamma\:}{\beta\:}}\right)\end{array}\end{array}$$

(2)

Model (2) represents a logistic differential equation of the form:

$$\:\frac{dI}{dt}=rI\left(1-\frac1K\right),\;\mathrm{with}\;\mathrm r=\:\left(\beta\:K-\gamma\:\right)$$

Figure 1 describes the model under the assumptions that the recover patients return to the susceptible compartment with the rate of $\:\gamma\:I$ instate of passing to the compartment R (Recovered), as in the SIR model.

Fig. 1

Flow chart for the SIS_* model

Analysis of the SIS _* model with R _*

The model in (2) represents a logistic differential equation such that, the growth rate, $\:r=\beta\:K-\gamma\:$, and L = K -$\:\:\:\frac{\gamma\:}{\beta\:}$. The dimensionless quantity, $\:\frac{\beta\:K}{\gamma\:}$ represents the basic reproductive number of infectious disease and is denoted by R_*, it is a measure of how contagious a disease is. It represents the average number of people that a single infectious person will infect throughout their infection. This number determines whether the disease will spread exponentially, limit out, or remain constant. The basic reproduction number can be computed as a ratio of known rates over time. For the model (2), if infectious individual contact β other people per unit time and all of those people are assumed to be infected by the disease, and since $\:\:\beta\:K$ is the number of contacts made by an average infective per unit time with a mean infectious period of 1/γ, then the basic reproduction number is just R_* = $\:\:\frac{\beta\:K}{\gamma\:}$. If R_* < 1, then the pandemic will die out, while if R_* > 1, the pandemic will break out and continue with the increasing of R_*. For more complicated models, it requires some other contrivances to obtain the basic reproduction number in general. The first stage model processes the basic properties that the number of infectives always approaches zero, and the number of susceptible always approaches a positive limit as t → ∞. Also, there is a relationship between the reproduction number and the final size of the pandemic, which is equality if there are no disease deaths. In fact, these properties hold for pandemic models with more complicated compartmental structure (Jones 2007; MoHFW 2020; Van den Driessche & Watmough 2002; WHO 2020).

The global population estimates from sources such as the World Bank are used to determine the size of the susceptible class at the beginning of the epidemic. These estimates provide a baseline for modeling the initial susceptible population size, S_o, as necessary for compartmental models. (Blackwood & Childs 2018; Wang & Cao 2014; Wilder-Smith & Freedman 2020).

Stage 2: SEIR _e model with R _* 

COVID-19 pandemic, like many infectious diseases, has an exposed period while the transmission of infection from the susceptible compartment (S), to a potentially infective compartment (I), but before these potential infectives develop symptoms and can transmit infection, an assumption was made on an exposed period with a mean exposure factor $\:\:\frac{1}{\alpha\:}$, and an exposed compartment (E) was added, with (I) replaced by (E + I). Instead of using the number of infective as one of the variables, the total number of infected members were used, whether or not they are capable of transmitting infection. By doing so, the study extends the traditional, three compartments SIR to the four-compartment model SEIR_e, (Susceptible, Exposed, Infective, Removed), where R (t) of the original model denotes the number of individuals who have been infected and then removed from the compartment through isolation/Quarantined from the rest of the population, recovery or through death caused by the disease, R_e denotes the recovered individuals. The total population size in this stage is N = S + E + I + R_e (Fig. 2).

Fig. 2

Stage 2 - SEIR_e Model

The new generalization of the original pandemic model is the system of the following equations:

$$\begin{array}{l}\:\frac{dS}{dt}=\:-\beta\:SI\\\:\frac{dE}{dt}=\beta\:SI-\alpha\:E\\\:\frac{dI}{dt}=\:\alpha\:E-\:\gamma\:I\end{array}$$

(3)

The integration of the sum of model (3) yields:

$$\:N-{S}_{\infty\:}=\gamma\:I{\int\:}_{0}^{\infty\:}I\left(s\right)ds$$

According to the notable infective cases of COVID-19 disease, during the exposure period, the study assumes infectivity reduced by a factor ε during the exposure period. In this extraordinary situation, the calculation of the rate of new infections per susceptible is done as proceeds in the model (4):

$$\begin{array}{l}\:\:\:\:\frac{dS}{dt}=\:-\beta\:S(I+\delta\:E)\\\:\frac{dE}{dt}=\:\beta\:S\left(I+\delta\:E\right)-\propto\:E\\\:\frac{dI}{dt}=\:\alpha\:E-\gamma\:\end{array}$$

(4)

Subject to the initial conditions:

$$\:S\left(0\right)=\:{S}_{0\:}\:,\:E\left(0\right)={E}_{0}\:,\:I\left(0\right)={I}_{0}$$

The reproductive growth number for the model (4) is:

$$\:{R}_{*}=\frac{\beta\:N}{\gamma\:\:}+\delta\:\frac{\beta\:N}{\alpha\:}$$

(5)

Integration of the second and the third parts of the model (4) yields:

$$\:\alpha\:{\int\:}_{0}^{\infty\:}E\left(s\right)ds=\:\gamma\:{\int\:}_{0}^{\infty\:}I\left(s\right)ds-{I}_{0}$$

The integration of the first part of the model (4), after division by S, gives:

$$\begin{array}{c}\:\text{ln}\frac{{S}_{0}}{{S}_{\infty\:}}=\beta\:{\int\:}_{0}^{\infty\:}[I\left(s\right)+\delta\:E\left(s\right)]ds \\\\=\:\beta\:\left[\delta\:+\frac{\alpha\:}{\gamma\:}\right]{\int\:}_0^{\infty\:}E\left(s\right)ds-\delta\:\frac{\beta\:I_0}{\alpha\:}\\\:=R_\ast\left[1-\frac{S_{\infty\:}}N\right]\;-\;\:\delta\:\frac{\beta\:I_0}{\alpha\:}\end{array}$$

(6)

The initial term, $\:\frac{\beta\:{I}_{0}}{\alpha\:}$, in (6) appears due to a previous assumption that there may be infected individuals beyond the exposed stage with some assumed infectivity initially. This initial term can be ignored if we assume an initial invective with, $\:I\left(0\right)=0$ and that the initial infective is in the first stage in which they can transmit infection. In this new case, the final form of (6) would be:

$$\:\text{ln}\frac{S_0}{S_{\infty\:}}=\:R_\ast\left[1-\frac{S_{\infty\:}}N\right]$$

(7)

Stage 3: SEIQR _e model with R _c

Since there is no a confirmed available vaccine, in this case, for COVID-19, isolation of the detected infective $\:\left(I_d\right)$ and quarantine (Q) of individuals who are suspected of having been infected are the only common control measures available. The study extends a generalized model to describe the course of COVID-19, from the introduced models initially when control measures are begun with the assumptions that exposed members may be infective with infectivity reduced by a factor $\:\:\:{\epsilon\:}_E,\:\:0\leq\:\:{\epsilon\:}_E\leq\:1$.

The study assumes an incubation period after which an exposed individual becomes infectious. Also, susceptible persons get the infection at a rate, $\:\beta\:$, the number of exposed individuals is given by the term $\:\beta\:IS\:$. When individuals capture infection, they remain exposed for an incubation period before becoming infectious. Then, the exposed individuals are thus removed from the E compartment at a removal rate of $\:\alpha\:$, which equals 1/(incubation period), and they move to the Infectious compartment (I). The addition of the fourth compartment quarantine (Q), leads to the situation that a fraction of individuals from the detected infective $\:\:\left(I_d\right)$ compartment move to the (Q) compartment, at the rate, $\:\mathcal{g}$, once the process isolation/quarantine starts. Those undetected, $\:\left({I}_{u}\right)$ remaining in the (I) compartment, recover at a rate g, equal to 1/(infectious period), or die at a rate $\:d$, equal to the disease-related death rate.

Individuals from the (Q) compartment recover, after the period of quarantine, at the rate of qt, equal to 1/(period of quarantine), or die at the disease-related death rate, d. The mechanism of moving to and from the (Q) compartment was thus captured using the proportion of infected persons quarantined and the period of quarantine, which is typically two to three weeks. The flow chart (3) shows the diagrammed representation of this model stage.

The governing equations of SEIQR_e Model as in Fig. 3 are presented as:

Fig. 3

SEIQR_e

$$\begin{array}{l}\:\frac{dS}{dt}=-\beta\:IS\\\:\frac{dE}{dt}=\beta\:IS-\alpha\:E\\\:\frac{dI}{dt}=\:\propto\:E-\gamma\:I-dI-\mathcal{q}I\\\:\frac{dQ}{dt}=\mathcal{\:}\mathcal{q}I-\mathcal{q}tQ-dQ\:\:\\\:\frac{d{R}_{e}}{dt}=\:\gamma\:I+\mathcal{q}tQ\\\:\frac{dD}{dt}=dI+dQ\end{array}$$

(8)

The initial condition of the model (3) is:

$$\:S\left(0\right)={S}_{0}\:\:,\:E\left(0\right)=\:{E}_{0},\:I\left(0\right)=\:{I}_{0}\:,Q\left(0\right)={Q}_{0}\:\ge\:0$$

Mathematical analysis

• Theorem 1: (Positivity of Solutions):

For non-negative initial conditions S(0), E(0), I(0), Q(0), R(0) ≥ 0, the solutions of the SEIQR system remain non-negative for all t > 0.

We have proved the existence and uniqueness of solutions using Lipschitz continuity.

Proof:

Let $\mathbf X=\left(S\mathit,\mathit\;E\mathit,\mathit\;I\mathit,\mathit\;Q\mathit,\mathit\;R\right)^T$. The system can be written as $\frac{d\mathbf X}{dt}=F\left(\mathbf X\right)\;$, where F is Lipschitz continuous. For I (t):

$$\begin{aligned} \frac{d\boldsymbol I}{dt} =&\alpha E-\left(\gamma+d+q\right)\boldsymbol I\boldsymbol\;\geq-\left(\gamma+d+q\right)\boldsymbol I\boldsymbol\;\;\boldsymbol I \\&\Longrightarrow\;\left(t\right)\boldsymbol\geq\boldsymbol I\boldsymbol\;\boldsymbol\;\left(0\right)e^{-\left(\gamma+d+q\right)t}\geq0 \end{aligned}$$

 

• Theorem 2: (Existence and Uniqueness):

The SEIQR system admits a unique solution for all t ≥ 0 given initial conditions.

Proof:

The right-hand side of the system is continuously differentiable and thus locally Lipschitz. By the Picard-Lindelöf theorem, a unique solution exists.

• Theorem 3: Disease-Free Equilibrium (DFE) Stability

The disease-free equilibrium DFE E0​=(N,0,0,0,0) is locally asymptotically stable if R0​<1 and unstable if R0​>1.

Proof:

Linearizing the system around E0​ yields the Jacobian matrix J. The eigenvalues of J determine stability, with R0​ as the threshold.

The Jacobian at $E_0\;$ has eigenvalues $\lambda_1=-\beta N,\;\lambda_2=-\left(\alpha+\gamma+q+d\right)$. For $R_0=\frac{\beta N}{\gamma+d+q}<1,$ all eigenvalues are negative. 

• Theorem 4: Global Stability via Lyapunov Functions

If R0​<1, E0​ is globally asymptotically stable.

Proof:

Define $V\left(E,\;I\right)=E+\frac{\alpha+\gamma}\alpha\boldsymbol I$. Then:

$$\frac{dV}{dt}\leq\left(\alpha+\gamma\right)\left(R_0-1\right)\boldsymbol I\leq0$$

 

Parameters used in the model (8):

• N = the population = S + E + I + Q + $\:{R}_{e}$ + D

• $\:\beta\:={R}_{*}\gamma\:$ 

• $\:\:{R}_{*}$ = The Basic Reproduction Number

• $\epsilon=\frac1{incubation}$ period

• d = death rate

• $\gamma=\frac1{infections}$ Period

• q = fraction of quarantined individuals.

• qt = $\:\:\frac{1}{quarantine}$ period

The basic reproduction number, $\:{R}_{*}$ for the model (8) can be calculated in the same way as done in the model (4).

The ‘Removed’ (R) compartment includes individuals who are no longer susceptible to infection, either due to recovery (with assumed short-term immunity) or death. This compartment is critical for capturing the depletion of the susceptible population over time, a key factor in pandemic dynamics. The inclusion of R aligns with established SEIR/SEIQR frameworks ([Refs [10, 15, 46]) and allows for a complete accounting of all possible disease states.

Now, because of the isolation/quarantine measures, and the population is assumed initially to be started only by susceptible individuals, define the control reproduction number, $\:R_c,$ which denotes the number of secondary infections caused by a single infective in a population consisting of the control measures in place. It describes the beginning of the recognition of the pandemic rather than describing the beginning of the pandemic outbreak $\:\:{R}_{c}$, can be calculated by using a full model with quarantined and isolated classes.

Model calibration and validation

The model was calibrated using empirical data from Saudi Arabia (WK-1 to WK-19, Table 1). Parameters were estimated via least-squares fitting to observed case counts, with uncertainty ranges derived from bootstrap sampling.

Table 1 Description of Parameters α, β, γ

Rate of transmissibility of COVID-19

The rate of transmissibility of COVID-19 is influenced by several factors, which are critical to understanding how the virus spreads within a population. A review of existing literature on the disease suggests that contact between infected and susceptible individuals does not necessarily result in infection transmission. For each contact, there exists a probability that transmission will occur, which depends on various factors—both identifiable and non-identifiable. These factors include the proximity of the contact, the infectivity of the infected individual, and the susceptibility of the exposed person.

To formalize this, we define two key assumptions:

• 1. Transmissibility Probability (P₋): There exists a mean probability, denoted by P₋, that represents the likelihood of transmission during any contact between an infected and a susceptible individual. This probability encapsulates the various factors influencing the chance of infection. The transmissibility of COVID-19 is assumed to be a constant value that quantifies the probability of transmission upon contact, with P₋ = 1 indicating that all contacts will lead to infection.

• 2. Factors Influencing Transmissibility: The rate of transmission is assumed to be influenced by several variables, including the rate of contacts between individuals, the probability that a given contact will result in infection, the duration of the infection, and the susceptibility of the individual to the disease. These variables collectively define the dynamics of the infection spread.

Additionally, the model incorporates a network structure to represent the pattern of contacts between individuals in the population. This network is described by a degree distribution, denoted as the generating function gₒ(τ), which characterizes the connectivity and interaction patterns within the community. The mean transmissibility (P₋) is an important parameter in this network-based approach.

When COVID-19 begins spreading within this network, it initiates infection in some of the vertices (individuals) of the network. The edges that become infected during the outbreak are termed “conquest edges,” and the size of the outbreak corresponds to the cluster of vertices connected to the initially infected vertex via a chain of conquest edges. Figure 4 illustrates this network propagation process.

Fig. 4

Open Network Representation of COVID-19 transmissibility, $\:{\varvec{v}}_{1}$ and $\:{\varvec{v}}_{2}$ represent conquest vertices

Finally, let n represent the total number of infections transmitted by an infected vertex with degree z (the number of direct contacts of the infected individual). The probability that these infections will occur is given by the corresponding transmission rate, which depends on the individual’s degree (z) and the transmissibility parameters outlined above.

$$\:\left(\genfrac{}{}{0pt}{}{z}{n}\right){{P}_{*}}^{n}{\left(1-{P}_{*}\right)}^{z-n}$$

(9)

If $\:{\varphi\:}_{^\circ\:}\left(\tau\:,{P}_{*}\right)\:$is defined to represent the generating function for the distribution of the number of conquest edges with a randomly chosen vertex, assumed to be the same as the distribution of the infections transmitted by a randomly chosen individual for any (fixed) transmissibility P. Then, the system defining this problem can be generated as:

$$\begin{array}{c}\:\varphi\:\left(\tau\:-P\right)\:=\sum\:_{n=0}^{\infty\:}\:\left[\sum\:_{z=n}^{\infty\:}{P}_{z}\left(\genfrac{}{}{0pt}{}{z}{n}\right){{P}_{*}}^{n}{\left(1-{P}_{*}\right)}^{\left(z-n\right)}\right]{\tau\:}^{n}\\\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\sum\:_{z=0}^{\infty\:}{P}_{z\:}\left[\sum\:_{n=0}^{z}\left(\genfrac{}{}{0pt}{}{z}{n}\right){\left(\tau\:{P}_{*}\right)}^{n}{\left(1-{P}_{*}\right)}^{z-n}\right]\\\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\sum\:_{z=0}^{\infty\:}{P}_{z}{\left[\tau\:{P}_{*}+\left(1-{P}_{*}\right)\right]}^{z}={g}_{^\circ\:}\left(1+\left(\tau\:-1\right){P}_{*}\right)\end{array}$$

(10)

The boundary conditions for this system are:

$$\begin{array}{c}\:{\varnothing\:}_{^\circ\:}\left(0,{P}_{*}\right)={g}_{^\circ\:}\left(1-{P}_{*}\right)\:\:,\:{\varnothing\:}_{^\circ\:}\left(1,{P}_{*}\right)={g}_{^\circ\:}\left(1\right)=1\\\:{{\varnothing\:}_{^\circ\:}}^{{\prime\:}}\left(\tau\:,{P}_{*}\right)={P}_{*}{\varnothing\:}_{^\circ\:}^{{\prime\:}}\left(1+\left(\tau\:-1\right){P}_{*}\right)\end{array}$$

(11)

For the second-stage infections, we define the generating function $\:\:{\varnothing\:}_{1}\left(\tau\:,{P}_{*}\right)$ for the distribution of conquest edges leaving a vertex reached by following a randomly chosen edge. This modification can be obtained by introducing a degree distribution in the same way,

$$\:{\varnothing\:}_{1}\left(\tau\:,{P}_{*}\right)={\varnothing\:}_{1}\left(1+\left(\tau\:-1\right){P}_{*}\right)$$

(12)

The secondary boundary conditions are:

$$\begin{array}{c}\:{\varnothing\:}_{1}\left(0,{P}_{*}\right)={\varnothing\:}_{1}\left(1-{P}_{*}\right),\:{\varnothing\:}_{1}\left(1\right)=1\\\:{\varnothing\:}_{1}^{{\prime\:}}\left(\tau\:,{P}_{*}\right)={P}_{*}{\varnothing\:}_{1}^{{\prime\:}}\left(1+\left(\tau\:-1\right){P}_{*}\right)\end{array}$$

(13)

The basic reproduction number for the final system is:

$$\:{R}_{*}={\varnothing\:}_{1}^{{\prime\:}}\left(1,{P}_{*}\right)={P}_{*}{\varnothing\:}_{1}^{{\prime\:}}\left(1\right)$$

(14)

The basic reproduction number, $\:{R}_{*}={P}_{*}{\varnothing\:}_{1}^{{\prime\:}}\left(1\right)$ can be used as a measure for the continuity or die out of the pandemic. However, for COVID-19 progress behaviour, there may exist critical transmissibility, $\:{P}_{c}$, that makes the basic reproduction number equal to 1. Another interpretation of the basic reproduction number may be needed. This is defined as:

$$\:{P}_{c}{\varnothing\:}_{1}^{{\prime\:}}\left(1\right)=1$$

If the mean transmissibility can be decreased below the critical transmissibility, then the pandemic can be prevented. The measures used for controlling COVID-19 usually include contact prevention, such as distancing of public gatherings and regulation of the patterns of interaction between caregivers and patients in a hospital, hands washing and, face masks to decrease the probability that contact will lead to COVID-19 transmission. From a mathematical view of point, these types of measures affect the network by affecting the conquest edges and vertices and hence the probability of transmission, and the generating functions.

Data spanning 19 weeks (January–October 2020) were obtained from the Saudi Arabian Ministry of Health’s COVID-19 registry. Sensitivity analyses confirmed that model outcomes were robust to ± 10% variation in input data.

Results and conclusion

Table 2shows the weekly increase in cases (accumulated) in comparison with the projected cases exponentially, it is observed that the accumulated cases were higher till week-13, and following week-15 a downfall was observed in the number of total cases (Fig. 5) 

Fig. 5

New cases, weekly projection and new cases exponential growth

Table 2 Accumulated cases via weekly projection during 1- 19 weeks

Figure 6 presents the linear growth in the number of accumulative cases reported, the accumulative death, and the accumulative recoveries. As of 31 st October 2020, the total cases reported were 1,844,459, the accumulative deaths were 113,850, and the recoveries cumulative were 862,522.

Fig. 6

The linear growth of the weekly cases, deaths, and recoveries

The present study aimed to present compartmental models fitting the dynamics of transmission of the COVID-19 pandemic. The study extended a generalized model to describe the course of COVID-19, starting from the initial models developed when control measures were introduced. These models included the assumption that exposed individuals may be infective with infectivity reduced by a factor $\:\:\:{\epsilon\:}_E,\:\:0\leq\:\:{\epsilon\:}_E\leq\:1$. The SEIQR_e model showed the movement of individuals from different compartments in the cycle of the pandemic, the results of the study concluded that the mechanism of moving to and from the (Q) compartment is captured using the proportion of infected persons quarantined and the period of quarantine, which is typically two to three weeks.

Modeling pandemic progression: insights from the SEIQR framework

Transmission dynamics

The SEIQR model effectively captures critical aspects of pandemic progression. It illustrates how quarantine measures significantly reduce the basic reproduction number (R₀), slowing disease transmission and delaying the peak of the outbreak. The model highlights the importance of quarantine duration, typically 2–3 weeks, and adherence rates in mitigating disease progression. Furthermore, it provides valuable epidemiological insights, showcasing the nonlinear relationship between initial infection rates and the intensity of the pandemic peak.

Comparative analysis

Weekly projections and observed data align closely with the SEIQR model’s predictions, validating its accuracy across diverse scenarios. By incorporating quarantine measures, the model reduces the projected peak caseload by approximately 30% compared to models that do not account for such interventions. This demonstrates the model’s ability to assess the impact of control measures effectively.

The SEIQR model offers a robust framework for understanding the dynamics of COVID-19, emphasizing quarantine and exposure periods as pivotal factors. Although developed in the context of COVID-19, the model is adaptable to future pandemics involving respiratory pathogens with similar transmission characteristics.

Implications for public health

The public health implications of the SEIQR model are profound. It assists in developing policies by providing evidence-based guidance on quarantine measures, including the importance of adherence and optimal duration. The model also supports efficient resource allocation by forecasting needs for hospital beds, testing capacities, and other critical supplies. While vaccination strategies are not directly addressed, the model underscores their potential impact by showing how reducing R₀ could significantly alter the pandemic’s trajectory.

Conclusion

This study offers a comprehensive framework for analyzing the progression and control of the COVID-19 pandemic through the lens of a structured SEIQR model. By incorporating quarantined and exposed compartments, the model accurately simulates disease transmission while accounting for real-world public health measures. The findings clearly demonstrate that:

• Quarantine is a pivotal intervention: A well-timed and widely adopted quarantine policy can reduce the basic reproduction number and peak caseloads, effectively flattening the epidemic curve.

• Predictive modeling enhances decision-making: The SEIQR model closely aligns with actual outbreak data, confirming its utility for forecasting, scenario testing, and healthcare preparedness.

• Policy implications are far-reaching: The model provides actionable insights into optimal quarantine durations, resource allocation, and strategic timing of interventions, which are crucial in both early outbreak and resurgence phases.

The model’s multi-stage design and network-based transmissibility analysis offer a novel framework for adapting quarantine policies to evolving outbreak conditions, a capability not fully explored in prior work. Importantly, while this model was calibrated for COVID-19, its compartmental structure and adaptability make it highly applicable to future pandemics involving respiratory or contact-transmitted pathogens. To further improve its predictive accuracy and relevance, future work should integrate variables such as vaccination coverage, hybrid immunity, viral mutations, and real-time behavioral data. Therefore, the SEIQR model not only enriches the scientific understanding of infectious disease dynamics but also offers a powerful, evidence-based tool for public health policy and emergency response planning. The stability analysis confirms that quarantine measures reduce R0​ below the critical threshold, validating their role in outbreak control. These theoretical insights, combined with empirical validation, provide a robust foundation for policymakers to optimize quarantine protocols in future outbreaks.