Assessing the role of quarantine and isolation as control strategies for COVID-19 outbreak: A case study.

Life style of people almost in every country has been changed with arrival of corona virus. Under the drastic influence of the virus, mathematicians, statisticians, epidemiologists, microbiologists, environmentalists, health providers, and government officials started searching for strategies including mathematical modeling, lock-down, face masks, isolation, quarantine, and social distancing. With quarantine and isolation being the most effective tools, we have formulated a new nonlinear deterministic model based upon ordinary differential equations containing six compartments (susceptible S ( t ) , exposed E ( t ) , quarantined Q ( t ) , infected I ( t ) , isolated J ( t ) and recovered R ( t ) ). The model is found to have positively invariant region whereas equilibrium points of the model are investigated for their local stability with respect to the basic reproductive number R 0 . The computed value of R 0 = 1.31 proves endemic level of the epidemic. Using nonlinear least-squares method and real prevalence of COVID-19 cases in Pakistan, best parameters are obtained and their sensitivity is analyzed. Various simulations are presented to appreciate quarantined and isolated strategies if applied sensibly.

Introduction

COVID-19 (Coronavirus disease 2019 or novel coronavirus) is a transmissible disease caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Common symptoms include fever, cough, fatigue, body ache, shortness of breath, and loss of appetite, taste and smell. Some people may experience complications such as pneumonia, multi-organ failure, acute respiratory distress syndrome, septic shock, arrhythmias, myocarditis, blood clots, heart failure, seizure, encephalitis, stroke and Guillain Barré syndrome. The incubation period ranges from two to fourteen days.

The novel coronavirus spreads between people via the droplets produced while sneezing, coughing or talking. The droplets exhaled by infected people are inhaled into other people’s lungs causing new infections. When the droplets fall on surfaces they can cause virus spread if people touch the contaminated surfaces and then their faces with unwashed hands. While saliva and sputum are major carriers of virus, recent research suggests the spread of the disease via faecal-oral routes. The aerosol-generating procedures (AGPs) also result in more easier than normal virus transmission.

SARS-CoV-2 has features closely related to those of severe acute respiratory syndrome coronavirus (SARS-CoV or SARS-CoV-1) which is an enveloped, single stranded RNA virus that most affects the lungs as the virus enters the host cell using peplomer to bind to angiotensin-converting-enzyme 2 (ACE2) which is most abundantly present in alveolar type II (AT2) cells of the lungs. It is regarded as a zoonotic disease which is transmitted from bats as biological carriers to humans. Genetic analysis has shown that the virus clusters with the Betacoronavirus genus together with two bat-derived strains.

First identified in Wuhan, the capital of China’s Hubei province, on 31st December 2019, COVID-19 has since spread worldwide. The World Health Organization declared the outbreak as a Public Health Emergency of International Concern (PHEIC) on 30 January 2020 and a pandemic on 11 March 2020. As of 20 September 2020, 30.6 million confirmed cases and 950,000 deaths have been reported across 217 countries and territories including Pakistan. In Pakistan, the first COVID-19 case was reported on 26 February 2020. As of 20 September 2020, there are a total of 306,304 confirmed positive cases, 6420 deaths and 292,869 recoveries in the country.

COVID-19 has adversely affected economy, food supplies, oil prices, arts and culture sectors, religious events, sports, cinema, travel and tourism, education, health systems and politics on a global level. Mathematical modellers and epidemiologists are investigating many of the details relating to transmission, prevention and management of this new disease. Several studies have been published to comprehend the disease dynamics and outbreak responses via mathematical models. Tang et al. [1] proposed a compartmental model by stratifying the population into susceptible (S), exposed (E), pre-symptomatic (A), symptomatic (I), quarantined susceptible (), isolated exposed (), isolated infected (), hospitalized (H) and recovered (R) compartments. They estimated the basic reproduction number of the disease using data of laboratory confirmed COVID-19 cases during the month of January in mainland China. It was found that the reproduction number can be effectively reduced by adopting public health interventions. In another study [2], Tang et al. analyzed a modified version of their previous model by assuming time-dependent contact and diagnosis rates. This led to a smaller reproduction number in comparison to one estimated in their previous study. An SIRD (D - dead persons) model is discussed in Anastassopoulou et al. [3] while in Yang and Wang [4], the authors investigate an SEIRV (V - virus concentration in the environment) model for coronavirus transmission. Li et al. [5] extended the traditional SEIR model to SEIQDR (Q - diagnosed and quarantined, D - suspected cases of infection) model to assess the spread of COVID-19 in China. In [6], the coronavirus transmission is discussed using an SEIR framework for the number of cases in Wuhan and also the number of exported cases. It is shown that introducing control measures such as travel restrictions can be crucial to comprehend the outbreak dynamics and the likelihood of containing its transmission. Another interesting work is [7] in which the transmission rate is considered to be a function of time. The model is formulated by appending three more compartments to the SEIR framework. Other important contributions towards modeling strategies for the novel coronavirus can be found in Zeb et al. [8], Anirudh [9], Mandal et al. [10], Yousefpour et al. [11], ud Din et al. [12], Lu et al. [13], Chen et al. [14], Prem et al. [15], Khan and Atangana [16], Ivorra et al. [17], Fanelli and Piazza [18], D’Arienzo and Coniglio [19], Ndairou et al. [20], Torrealba-Rodriguez et al. [21], Sweilam et al. [22].

Mathematical models are highly important tools to understand the spread of an infectious disease, make prediction and devise strategies for its control and elimination. Motivated by the aforementioned works, we intend to develop a deterministic model to analyze the effect of quarantine and isolation in controlling the spread of coronavirus disease in Pakistan. Pakistan is a developing country facing economical, environmental and energy crisis. The healthcare system in the country is still not efficient and facing numerous challenges. The limited testing capacity is leading to continuous surge in the COVID-19 cases in the country. In such situation, a strategic plan based on measures such as quarantine and isolation may help fighting spread of the disease. The findings of this case study establish the significance of these control strategies. To the best of our knowledge, such study has not been found in literature. The model proposed in this study is validated using real time data of total confirmed positive coronavirus cases for the month of May 2020.

The rest of the paper is organized as follows. Section 2 describes the model formulation based on a number of assumptions. Section 3 presents the equilibrium points, basic reproduction number and stability analysis of the model. Estimation of model parameters, sensitivity analysis and numerical simulations are performed in Section 4, while conclusion and future work are given in Section 5.

Model formulation

We formulate a mathematical model to examine the dynamics of six classes: susceptible exposed quarantined infected isolated and recovered individuals based on the following assumptions (Fig. 1 ):

Fig. 1

Flow diagram.

The susceptible class increases in size via recruitment (birth) of individuals at a constant rate and by inflow of individuals from the quarantined and recovered classes at rates with is fraction of individuals transferred to isolated class due to showing clinical symptoms of the disease, and respectively. The susceptible individuals acquire infection and become carriers of the virus at the rate resulting in a decrease in the susceptible population which is also decreased by natural death at the rate .

The population in the exposed class is increased because of the susceptible individuals who have not yet developed clinical symptoms but are carriers of the virus. The exposed population is reduced by development of clinical symptoms at the rate quarantine at the rate and natural death.

The quarantined individuals increase as a result of quarantine of individuals from the exposed class at the rate . The population is decreased following isolation of individuals showing symptoms at the rate transfer of individuals who do not show symptoms while in quarantine back to the susceptible class at the rate and natural death.

The infected population occurs due to development of COVID-19 symptoms by individuals in the exposed class. It decreases by isolation of infected individuals at the rate recovery at the rate and natural death.

The isolated class comprises of individuals who have developed COVID-19 symptoms and have therefore been isolated to seek medical attention. These individuals come from both the infected and quarantined classes at the rates and respectively. The isolated population diminishes as a result of recovery at the rate and natural death.

There is currently no evidence that individuals develop life-long immunity against COVID-19. Therefore, it is assumed that the recovered individuals tend to be susceptible at a rate . The population is increased with recovery of infected and isolated individuals at the rates and respectively and is diminished by natural death.

The resulting COVID-19 transmission model comprising of a nonlinear system of ordinary differential equations is given by:subject to the non-negative initial conditions . It can be shown that the solution of the model (2.1) subject to these initial conditions exists and is non-negative for all andis the positively invariant region for the model (2.1).

Stability analysis

The disease-free equilibrium point of the model (2.1) is given asUsing the next generation matrix technique [23], the basic reproductive number of the proposed model is obtained as follows:

The spectral radius of the matrix is the basic reproduction number :

The disease-free equilibrium is locally asymptotically stable if and unstable if .

The Jacobian matrix of the system (2.1) at is

The characteristic equation of is

Four of the eigenvalues of are and . Remaining two eigenvalues and are roots of the quadratic equationsatisfying the following criteria:

The two conditions stated in (3.5) imply that both and have negative real parts provided . This proves that is locally asymptotically stable. If one of and has positive real part. In this case, is unstable. □

For the proposed model also has an endemic equilibrium point in addition to the disease-free equilibrium:where and .

We state, without proof, the following result:

The endemic equilibrium is locally asymptotically stable if .

The proof to (3.2) is straightforward and can be established following the approach used in Memon et al. [24].

Estimation of model parameters and numerical results

Estimation of model parameters

When it comes to validation of a newly designed epidemic model then it is important to have real data for the epidemic being investigated. In addition, having real data greatly assists to obtain some biological parameters otherwise unknown for the model. Keeping in view importance of real data and model fitting, we have employed an approach called nonlinear least-squares curve fitting approach while using a routine called fminsearch from MATLAB R2020a Optimization Toolbox. This routine takes into account the proposed theoretical model that includes unknown parameters of the model and a sequence of real data values . Latter, the major objective of the routine is set to get unknown parameters for which the evaluated erroraccomplishes a minimum value.

There are 11 biological parameters in the proposed model. Available literature helps to get estimated values for some of the parameters whereas remaining have been obtained via nonlinear least-squares curve fitting approach. As can be seen in the Table 1 , the parameters and are best fitted using the approach discussed above while the parameters and have been estimated [24]. Moreover, the parameter (fraction of quarantined population made isolated) is assumed to be .

Table 1

Estimated and fitted parameters for the COVID-19 model (2.1).

Parameter	Description	Value	Source
Π	Recruitment rate	8972.105	Estimated
μ	Natural mortality rate	4.073e05	Estimated
α	Contact rate	8.52633418e09	Fitted
β_1	Exposure rate towards infection	1.41108474e01	Fitted
β_2	Quarantine rate	2.19739872e02	Fitted
γ_1	Isolation rate	1.07159236e01	Fitted
γ_2	Rate of developing clinical symptoms during quarantine	1.91550376e + 00	Fitted
δ_1	Recovery rate in symptomatic individuals	1.14130348e + 00	Fitted
δ_2	Recovery rate in isolated individuals	2.70939355e + 01	Fitted
ξ	Rate of being susceptible after recovery	1.20325287e + 02	Fitted
θ	Fraction of quarantined population made isolated	0.68	Assumed

Using the available information for total population of Pakistan during May 2020 the initial conditions for the state variables of the model (2.1) are taken to be . The initial conditions, estimated and fitted biological parameters yielded the best curve fitting. Moreover, real COVID-19 cases from Pakistan and the fitted curve have been shown in the Fig. 2 wherein it is observed that the model fits the real cases well enough. Finally, the basic reproductive number is computed to be about using the parameters’ values listed in the following Table 1.

Fig. 2

COVID-19 cases in Pakistan in May 2020 vs. model fitting.

Sensitivity analysis

In this section, it has been attempted to determine the biological parameters of the proposed model (2.1) which are highly sensitive for the spread of the COVID-19 epidemic. This should be carried out since the information about the estimated parameters has mostly some chances of inaccuracy putting system’s results in doubt.

Sensitivity index is the term commonly used to measure the sensitivity analysis of a varying quantity with respect to parameters of the model thereby causing one to quantify the relative change in the varying quantity whenever a model’s parameter varies.

In the present research work, it has been achieved via computation of the sensitivity indices of known as elasticity (normalized forward sensitivity) indices with the help of following formula:where stands for the model’s parameter. The normalized sensitivity index for with respect to the proposed COVID-19 epidemic model’s biological parameters is shown by the following equationsIt can be noted that the basic reproductive number is highly sensitive to the natural mortality rate that requires the most attention in order to stop the transmission of COVID-19 epidemic. On the other hand, the parameter (isolation rate) is found to be least effective parameter for the basic reproductive number while carrying out the forward sensitivity analysis. The normalized sensitivity/elasticity index of with respect to every parameter in the model (2.1) is computed in the Table 2 .

Table 2

Elasticity index values for .

Parameters	Elasticity indices
Π	+1
α	+1
β_1	+0.1349576172
β_2	0.1347079286
μ	1.000282312
γ_1	0.08583014836
δ_1	0.9141372286

Numerical results

In this section, various numerical simulations of the newly proposed corona virus model (2.1) have been carried out while using the initial conditions chosen to be . These conditions are also based upon real information collected at official level as stated in the Section 4.1. The unit of time is taken to be a day and ODE MATLAB solver called ode113 is used for simulation purposes. Finally, values of the working parameters involved in the proposed model have been used as listed in the Table 1. Profiles for the dynamics of the model’s state variables are shown in the Fig. 3 wherein it shows that the susceptible population decreases because such a population is transferring to other compartments within arbitrarily selected time interval of [0,500] whereas remaining state variables show increasing trend with infected population being on the surge. This was indeed the case during data collection for the present research study.

Fig. 3

Dynamical behavior for the state variables in the proposed COVID-19 model (2.1).

In order to further analyze the model (2.1), various biological parameters have been chosen to observe their impacts on the most indispensable compartments including quarantined infected and isolated . Dynamical behavior of these compartments dramatically declined when the contact rate of the disease is slightly decreased as shown by the Fig. 4 . Moving on, Fig. 5 shows that the decreasing values of the quarantine rate would bring an increase in the above discussed population, particularly in the symptomatic compartment On the other hand, when the isolation rate is slightly increased then there has been observed a substantial reduction in each population leading to prove that the isolation of susceptible population from the symptomatic ones should be considered an effective tool in order to control the epidemic as depicted in the Fig. 6 . Finally, as conceived, we have witnessed that the more the time taken by symptomatic population to get recovered the quicker will be the transmission of the virus as shown in the Fig. 7 .

Fig. 4

Dynamical behavior for the (a) quarantined, (b) symptomatic and (c) isolated individuals in the proposed COVID-19 model (2.1) for decreasing values of contact rate .

Fig. 5

Dynamical behavior for the (a) quarantined (b) symptomatic and (c) isolated individuals individuals in the proposed COVID-19 model (2.1) for decreasing values of quarantine rate .

Fig. 6

Dynamical behavior for the (a) quarantined (b) symptomatic and (c) isolated individuals individuals in the proposed COVID-19 model (2.1) for increasing values of the isolation rate .

Fig. 7

Dynamical behavior for the (a) quarantined (b) symptomatic and (c) isolated individuals individuals in the proposed COVID-19 model (2.1) for decreasing values of recovery rate .

Since the basic reproductive number is extremely important while analysing transmission dynamics of an epidemic, it is therefore mandatory to look into its profile under the influence of some biological parameters of the model. Shown by the Fig. 8 is the behavior of for the parameters (contact rate), (exposure rate towards infection), and (quarantine rate). It is observed that the increasing level of would accelerate while is said to be responsible to combat with the epidemic. Similar sort of analysis can be interpreted about from the Figs. 9 and 10 under the influence of remaining parameters.

Fig. 8

Dynamics for the basic reproductive number based upon (a) and and (b) and from the proposed COVID-19 model (2.1).

Fig. 9

Dynamics for the basic reproductive number based upon (a) and and (b) and from the proposed COVID-19 model (2.1).

Fig. 10

Dynamics for the basic reproductive number based upon (a) and and (b) and from the proposed COVID-19 model (2.1).

Conclusion & future directions

A new model related with ongoing outbreak of coronavirus is proposed in the present research study with major focus given on the dynamics of quarantined, infected and isolated classes. Deterministic approach was used while formulating the model whose solution is shown to have existence and boundedness within positively invariant region . Using next generation matrix technique, the basic reproductive number is computed to be 1.31 based on parameters best fitted with the nonlinear least-squares curve fitting method. Disease-free and endemic equilibria are shown to have local asymptotic stability when and respectively. Real COVID-19 cases in Pakistan for the month May, 2020 have been used to prove the validity of the proposed model whereas sensitivity of the parameters is also examined. Finally, various simulations (time series and contour plots) are carried out to assess the transmission dynamics of the virus based on different values of the parameters with conclusion that the contacts among individuals and elongation in the quarantine period are the most effective strategies to combat with the pandemic. This study will be revisited in future with introduction of tools concerning optimal control theory in order to identify best controls among the available management strategies.Other possible directions of interest are to construct stochastic [25] or fractional-order [26], [27], [28], [29] versions of the model proposed in the present study.

Research data for this article

The data used in this study is taken from Government of Pakistan website for COVID-19 (http://covid.gov.pk/stats/pakistan).

Role of the funding source

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

CRediT authorship contribution statement

Zaibunnisa Memon: Conceptualization, Methodology, Writing - original draft. Sania Qureshi: Software, Writing - original draft. Bisharat Rasool Memon: Software, Writing - review & editing.

Declaration of Competing Interest

Authors declare that they have no conflict of interest.