1. Introduction
The fundamental model of theoretical epidemiology, Susceptible–Infectious–Recovered (SIR) compartmental model of infectious disease, established by Kermack and Mckendrinck [
1], is utilized by different researchers to understand the propagation and containment of COVID-19. It is worthy to mention here that in epidemiology, the population is split into compartments such as Susceptible (S), Infectious (I), Exposed (E), Recovered (R), Asymptomatic (A), Quarantined (Q), Vaccinated (V), Hospitalized (H), Diseased (D) and so on.
Several studies [
2,
3] on propagation and containment of COVID-19 reveal that a large number of infections are caused by asymptomatic infection. Robinson and Stilianakis [
4] developed the SAIR model to capture asymptomatic infection. Ying and Xiaoqing [
5], Kaushal et al. [
6], Ansumali et al. [
7] and Monteiro [
8] studied the propagation of COVID-19 by using the SAIR model [
4]. The quarantine is an effective measure to control the disease with a large number of asymptomatic groups [
9]. The SIQS and SIQR models were studied by Hethcote et al. [
10] to analyze the effects of quarantine for three different types of incidence functions. Naz and Al-Raeei [
11] developed the SIQD model with a QA incidence to analyze the propagation of COVID-19. Bhadauria et al. [
12] formulated the SIQ model to investigate the lockdown effect of COVID-19. The literature on the propagation and containment of COVID-19 is extensive. For a complete review of past research focusing on COVID-19 using the classical models SIR, SEIR and their extensions, the interested reader is directed to Massonis et al. [
13] and references therein.
The epidemic models are investigated using a variety of methods, including global stability analysis, numerical methods and a data-driven approach. It is worthy to mention here that symmetry approaches have been successfully employed in the recent literature [
14,
15] to analyze models from biomathematics and other fields of applied mathematics [
16,
17,
18,
19,
20]. Naz and Naeem [
21] developed a new technique to construct first integrals and closed-form solutions of dynamical systems from epidemics. Several efficient computer packages [
22,
23,
24,
25,
26] are developed for the computation of Lie symmetries for differential equations. In [
27,
28], the first integrals are utilized to construct the closed-form solutions of the SEI, SIRI and tuberculosis models. A separate strand of the literature focuses on the use of artificial intelligence (AI) and machine learning techniques in the management of COVID-19 disease. Keshavarzi et al. [
29] provided an excellent survey of AI-based models for the development of COVID-19 vaccines and drugs. The information and datasets provided in this review can be utilized to accelerate the exploration of effective viral treatments. A web search engine misinformation notifier extension (SEMiNExt) proposed by Shams et al. [
30] has enabled safer web-based searching on health-related issues by displaying misinformation notifications in real time.
COVID-19, unlike previous viral diseases, has an “asymptomatic” group that has no symptoms but can still spread the disease to others at the same rate as infected symptomatic patients. In this article, we focus only on four compartments: Susceptible , Asymptomatic , Infected and Quarantined . A model with QA incidence is developed to investigate propagation and containment of COVID-19. The SAIQ model with a QA incidence has never been utilized in the literature to examine the propagation and containment of COVID-19 to our knowledge. COVID-19 models are investigated using various methods, but only a few studies focused on closed-form solutions. The closed-form solution facilitates the construction of the diverse epidemic indicators that describe the epidemic phenomenon and enables sensitivity analysis to variations in the data under consideration. We utilized the classical techniques of solving ODEs to establish the closed-form solutions of the systems of four nonlinear ODEs representing the SAIQ model. EP, FOI and ROI are vital epidemic indicators for the control and prevention of disease. We investigated these indicators using closed-form solutions. We provided four different scenarios to analyze COVID-19 19 propagation and containment: (i) lockdown, (ii) quarantine and other preventative measures, (iii) stabilizing the basic reproduction rate to a level where the pandemic can be contained and (iv) containing the epidemic through an appropriate combination of lockdown, quarantine and other preventative measures. It is worth noting that specific parameter values for any country could be used in the closed-form expressions for all the model variables, EP, FOI and ROI, derived here to analyze transmission dynamics and discuss control strategies.
The paper is organized in the following manner. The SAIQ model with QA incidence is established in
Section 2. In
Section 3, the dynamical system of four ODEs is reduced to a single second-order ODE in terms of only one variable
S. The closed-form solutions for the variable
S are obtained from the reduced second-order ODE, and then this is utilized to establish a closed-form solution for other variables of the model. In
Section 4, the closed-form expressions of EP, FOI and ROI are derived by utilizing the closed-form solutions. In
Section 5, specific values of parameters from the literature are used to construct graphs of closed-form expressions for EP, FOI, and ROI.
Section 6 contains the final remarks.
2. The SAIQ Model with QA Incidence
A model to study the propagation and containment of COVID-19 is formulated which has four compartments: Susceptible
, Asymptomatic
, Infected
and Quarantined
, and the total population is
. The rate of the efficient contact between susceptible and asymptomatic is the same as the rate of the efficient contact between susceptible and infected, and we take it as constant
. There is no obvious difference in viral shedding rates between asymptomatic and infected persons, according to the concept of viral shedding [
7,
31]. A QA incidence function [
10,
11] is utilized to analyze the transmission dynamics of COVID-19. The actively mixing population for the SAIQ model is
. The QA incidence for the asymptomatic group is
and for the infected group is
. The FOI for the asymptomatic group is
and the FOI for the infected group is
. Let
be the transfer rate between asymptomatic and infected,
be the segregation rate of asymptomatic and infected individuals, and the quarantined individuals leave the quarantining compartment at a rate
. The parameters of the model are summarized in
Table 1.
The following system of differential equations describes the SAIQ model with infectious force in the asymptomatic and infected groups:
with the initial conditions
here,
, and prime represents the derivative of variable with respect to time. It is worthy to mention here that the population
satisfies
. Ansumali et al. [
7] analyzed the SAIR model with standard incidence. Kaushal et al. [
6] studied the SAIR model with migration effects. The quarantine is an effective measure to control disease with a large number of asymptomatic groups [
9], and thus we replaced the removed/recovered
R compartment by the quarantine compartment
Q.
We utilize Driessche and Watmough’s [
32] next-generation operator technique to obtain the basic reproduction number
. The disease-free equilibrium (DFE) is
The diseased compartments are
For the next-generation operator technique, the related
F and
V matrices of the model, computed at the DFE, are provided as
and
The eigen values of
are
The expression for the basic reproduction number
is
indicates the beginning of the pandemic, implies EP and guarantees the disease’s end.
3. The Closed-Form Solution of SAIQ Model with QA Incidence
In this section, we derive the closed-form solution of the SAIQ model with QA incidence. First, the system of four ODEs is reduced to a second-order ODE in the single variable
S. Then, we establish closed-form solution of the second-order ODE, and thus obtain a closed-form expression for the variable
S. This results in simplification of the original model, and thus it becomes straightforward to derive solutions of other variables of the model. Equations (
1) and () yield
and
Equation (3) with the aid of Equations (
12) and (
13) yields
where
and
, which implies
and
. We assume
, then Equation (
14) simplifies to
and yields
To find the values of arbitrary constants, we employ the initial condition
, which yields
Equations (
16) with the aid of Equation (
17) yields
It is important to note that Equation (
1) has the following alternative form:
We use
S from Equation (
18) in Equation (
19) to obtain the following expression for
:
where
and
Equations (3) and (
20) yield
The solution of first-order liner ODE (
22) is
where
.
Equations (4) and (
20) yield
The solution of first-order liner ODE (
24) is
where
. The summary of the closed-form solution of the SAIQ model (
1)–(4) is as follows:
provided
and
, which implies
and
. The SAIQ model will reach the equilibrium value when
, and thus
.
5. Analysis of EP, FOI and ROI to Analyze Propagation and Containment of COVID-19
In this section, we explore the results obtained in
Section 4 using the specific values of parameters from the existing literature [
5,
6,
7,
8] and [
33,
34]. The total number of infected individuals
, the time span to reach at the epidemic peak
, the epidemic peak
, and the FOI
and ROI
are important indicators to study the propagation and containment of epidemics. It is important to mention here that the closed-form expressions of the total number of infected individuals (
30), the time span to reach at the epidemic peak (
32), the EP (
34), FOI (
37) and ROI (
38) can be applied to any country’s real data to analyze the propagation and containment of epidemics. This can be achieved by inserting the parameters values and initial conditions for a specific country in these expressions.
We provide different scenarios to manage the epidemic through lockdown, quarantine and other preventative measures.
5.1. Scenario I: To Contain the Epidemic through Lockdown
In the first scenario, a lockdown policy is implemented to contain the virus. The segregation rate for asymptomatic and infected people
is set at
. By lowering the rate of efficient contact
between susceptible and asymptomatic/infected, the basic reproduction number
is reduced.
Figure 1 illustrates this scenario graphically. With a reduction in
, the asymptomatic and infected people curve
takes longer to reach its maximum value. The EP gets smaller, and the time it takes to reach there gets longer. With a reduction in
, the FOI and the ROI both slow down. This suggests that lockdown is effective in minimizing FOI and ROI for a limited period of time but at the cost of social and economic consequences. In the long run, this is not a viable method for disease control.
5.2. Scenario II: To Contain the Epidemic through Quarantine and Other Preventative Measures
In the second scenario, the virus is contained through quarantine and other preventative measures. The effective contact rate between susceptible and asymptomatic/infected
is set at
. The segregation rate of asymptomatic and infected groups
is increased to lower the basic reproduction number.
Figure 2 illustrates this scenario visually. With an increase in
, the asymptomatic and infected individuals curve
takes longer to reach its maximum value. The EP gets smaller, and the time it takes to reach there gets longer. The strength of the FOI remains the same as
rises (quarantine and preventive), but the time it takes to get at a certain point rises. With an increase in
, the transfer rate of the susceptible to asymptomatic/infected group is lowered. This indicates that the community is adhering to the quarantine, social distancing and preventive measures in place. This is an excellent technique for preventing disease transmission.
5.3. Scenario III: To Stabilize the Basic Reproduction Rate down to a Level where the Pandemic Can Be Contained
Another effective scenario for controlling disease transmission is to fix
close to the level where transmissibility patterns are still moderate. A reduction in
leads to a decrease in
for a specific value of
.
Figure 3a–c shows the impact of changing
and
with a given reproduction number on EP, FOI and ROI. The EP remains the same since the basic reproduction number is fixed, but the time it takes to reach it increases as
and
drop, as seen in
Figure 3a.
Figure 3b,c shows how decreasing
and
slows down the FOI and ROI. Instead of enforcing a strict lockdown to immediately flatten the curve, it is vital to decelerate disease propagation to prevent a new epidemic wave. This is a suitable approach to utilize the health care facilities for asymptomatic or symptomatic individuals
for the time period
.
5.4. Scenario IV: To Contain the Epidemic through Appropriate Combination of Lockdown, Quarantine and Other Preventative Measures
Another scenario for containing the outbreak is to use a mix of lockdown, quarantine and other preventive measures. This may be accomplished by reducing the value of
by adjusting the effective contact rate between susceptible and asymptomatic/infected
and the segregation rate of asymptomatic/infected individuals
in the right proportions. We started with bench mark values
,
and
, as indicated in the red solid line in
Figure 4. The EP is on day 47 with 417,530 infected individuals. The parameters were then adjusted by a fixed proportion
. The EP, FOI and ROI are then assessed for new parameter values and compared with those at benchmark parameter values.
We start by looking at consequences of entirely ignoring lockdown, quarantine and other preventive measures. This indicates there is a higher
in the effective contact rate between susceptible and asymptomatic/infected. Asymptomatic/infected people are isolated at a lower rate
. When we increase
to
and reduce
to
. This is depicted as a blue dashed line in
Figure 4a–c. The value of
is
, the EP is on day 27 with 635,663 number of asymtomatic/infected individuals
. The FOI and ROI rise at a faster pace, as seen in
Figure 4b,c in the blue dashed line. This implies that a lockdown, quarantine and other preventative measures are necessary to contain the outbreak.
Next, we examine the impact of lockdown, quarantine and other preventive measures. This indicates that the effective contact rate
between susceptible and asymptomatic/infected should be reduced. Asymptomatic/infected people should isolate at a higher rate
. As a result, we lower
to
and raise
to
. This is provided in
Figure 4a–c in the green dotted line. With a value of
, the EP occurs on day 128, with a total of 148,149 asymtomatic/infected individuals
. The FOI and ROI rise at a slower pace for a longer duration, as seen in
Figure 4b,c in the green dotted line. The pandemic peak shrinks and the time it takes to achieve there increases. The slow disease propagation is useful because the government will have ample time to strengthen the country’s healthcare infrastructure in order to meet the needs of a significant number of patients expected in the near future. This is the optimal scenario for reducing EP while extending the time it takes to reach epidemic peak.
Finally, we investigate the dynamics of epidemic transmission by increasing quarantine and other preventative measures without enforcing lockdown. This indicates that the asymptomatic/infected isolation rate
should be increased while keeping the effective contact rate
between susceptible and asymptomatic/infected at a fixed level. This is established by increasing
to
while keeping
constant. This is graphically represented in
Figure 4a–c in the black long-dash line. With a value of
, the EP appears on day 56, with a total of 285833 asymtomatic/infected individuals. The strength of the FOI remains the same, but the time span to get at a specific point increases, as seen in
Figure 4b in the black long-dash line. The transfer rate of susceptible individuals to asymptomatic/infected class is lowered, as seen in
Figure 4c in the black long-dash line. This is another optimal scenario for reducing EP while extending the time it takes to reach EP without enforcing lockdown.
6. Conclusions
We established a SAIQ model with a QA incidence to analyze the propagation and containment of COVID-19. The closed-form solutions of the systems of four nonlinear first-order ODEs were established after reducing the system to a single ordinary differential equation in terms of the variable S. We provided the expressions for the important indicators of disease in the closed form. The propagation pattern and containment strategies of COVID19 are discussed using the numerical values of parameters from the literature in closed-form expressions of total number of infections , FOI and ROI .
In the first scenario, a lockdown policy is implemented to prevent the spread of a virus. We concluded that lockdown is a good strategy for a short period in lowering FOI and ROI but at the expense of social and economic losses. In the long run, this is not a viable method for disease control. In the second scenario, the virus is contained through quarantine and other preventative measures. The EP gets smaller, and the time it takes to reach there gets longer. The strength of the FOI remains the same as the community adheres to the quarantine, social distancing and other preventive measures, but the time span to get at a specific point increases. The transfer rate of the susceptible to asymptomatic/infected class is lowered. This is an excellent technique for preventing disease transmission. The third scenario for controlling disease transmission is to fix the value of close to the value where disease propagation is moderate. The EP remains the same since the basic reproduction number is fixed, but the time it takes to reach it increases as and drop. The decreasing and slows down the FOI and ROI. Instead of enforcing a strict lockdown to immediately flatten the curve, it is vital to slow down disease propagation to prevent a new epidemic wave. This is a suitable approach to utilize the healthcare facilities for asymptomatic or symptomatic individuals for the time period . Another scenario for containing the outbreak is to use a mix of lockdown, quarantine and preventive measures. This may be accomplished by reducing the value of the by adjusting the effective contact rate between susceptible and asymptomatic/infected and the segregation rate of asymptomatic/infected individuals in the right proportions.
In the majority of countries, government-implemented lockdown measures have significant social and economic implications. We discovered that quarantine and preventative measures are more successful in slowing disease propagation than harsh lockdown. Slow disease propagation can aid a country’s healthcare infrastructure. Implementing a partial shutdown with sufficient quarantine and preventative measures is the best way to cope with the pandemic.
It is worth mentioning that by including a compartment of vaccinated individuals to the model, the model could be modified to take vaccine effectiveness into consideration and forecast the spread of infection to demonstrate how vaccination could control an epidemic. This is a significant topic of epidemic control research and will be explored in a future article.