key: cord-0798658-kgu72a7w
authors: Naz, Rehana; Al‐Raeei, Marwan
title: Analysis of transmission dynamics of COVID‐19 via closed‐form solutions of a susceptible‐infectious‐quarantined‐diseased model with a quarantine‐adjusted incidence function
date: 2021-05-10
journal: Math Methods Appl Sci
DOI: 10.1002/mma.7481
sha: 0ddcb567eb76ad33ab6015f4b90f8281a71d3647
doc_id: 798658
cord_uid: kgu72a7w

We analyze the disease control and prevention strategies in a susceptible‐infectious‐quarantined‐diseased (SIQD) model with a quarantine‐adjusted incidence function. We have established the closed‐form solutions for all the variables of SIQD model with a quarantine‐adjusted incidence function provided [Formula: see text] by utilizing the classical techniques of solving ordinary differential equations (ODEs). The epidemic peak and time required to attain this peak are provided in closed form. We have provided closed‐form expressions for force of infection and rate at which susceptible becomes infected. The management of epidemic perceptive using control and prevention strategies is explained as well. The epidemic starts when ρ(0) > 1, the peak of epidemic appears when number of infected attains peak value when [Formula: see text] , and the disease dies out ρ(0) < 1. We have provided the comparison of estimated and actual epidemic peak of COVID‐19 in Pakistan. The forecast of epidemic peak for the United states, Brazil, India, and the Syrian Arab Republic is given as well.

We consider a model of COVID-19 transmission in a constant population of size N which consists of four compartments susceptible S(t), infected I(t), quarantined Q(t), deceased Δ(t), and total population N = S(t) + Q(t) + I(t) + Δ(t). In literature, several types of incidence functions are considered to study the dynamics of infectious diseases; see, for example, Bailey, 16 Jacquez et al, 17 and Watson. 18 Hethcote et al 13 studied the SIQS and SIQR models by taking simple mass action incidence, the standard incidence, and the quarantine-adjusted incidence. We follow the idea of Hethcote et al 13 and consider a quarantine-adjusted incidence function. The actively mixing population which is the pool of people with whom a susceptible can meet for our SIQD model is N − Q − D = S + I. The expression for the quarantine-adjusted incidence function is ( SI∕S + I), and it is obtained by replacing the denominator N in the standard incidence ( SI∕N) by actively mixed population N − Q − D = S + I. The FOI is I/S + I. The model can be expressed as a system of following four ODEs:

.

where dot above a variable denotes time differentiation and initial conditions are S(0) = S 0 , I(0) = I 0 , Q(0) = Q 0 , and Δ(0) = N − S(0) − I(0) − Q(0). The parameters are defined as follows: > 0 is contact rate of susceptible individuals with infected individuals, > 0 is isolation rate of infected individuals, > 0 is the virus-induced mortality rate of infected class, and > 0 is the virus-induced mortality rate of quarantined class.

Next, we compute the basic reproduction number 0 for this model by utilizing the next generation operator approach by van den Driessche and Watmough. 15 It is an essential dimensionless quantity which provides the threshold in the analysis of a disease to predict outbreak and to evaluate its control strategies. Let 0 = (S * , 0, 0, 0) represent the disease free equilibrium (DFE). The associated F and V matrices of SIQD model with quarantine-adjusted incidence functions (1)-(4) for the next generation operator approach are given as

and

The the basic reproduction number 0 is the only eigenvalue of FV −1 at the DFE 0 = (S * , 0, 0, 0) and is given by

It is important to mention here that the epidemic starts iḟI| t=0 > 0. Setting t = t 0 = 0 in Equation (2), we havė

provided S 0 /S 0 + I 0 − − > 0. This condition in terms of the basic reproduction number is 0 > 1 + I 0 /S 0 . The epidemic starts when 0 > 1 at initial time t 0 , the peak of epidemic appears when I(t) attains peak value I p at time t p when 0 = 1, and the disease dies out 0 < 1 at time t d .

In this section, we establish the closed-form solutions of system of Equations (1)-(4). It is important to mention here that Equations (1) and (2) form a coupled system of nonlinear ODEs in terms of variables S and I. Equations (3) and (4) are linear ODEs for variables Q and Δ, respectively. We can solve Equations (3) and (4) if variable I is known. First, we solve Equations (1) and (2) for variables S and I. Then we use value of I in ODEs (3) and (4) to find S(t) and Δ(t).

With the aid of Equations (1) and (2), Equation (10) takes following form:

Equations (1) and (2) can be expressed as a system of linear ODEs in terms of variable Z(t)

where variable Z satisfies a linear ODE given by (11) . It is straightforward to derive Z(t) from the linear ODE (11) and is given by

where Z 0 = S(0)∕I(0) and ≠ + . We substitute the expression for the variable Z(t) from (14) in ODE (12) , and resulting the ODE can be expressed in variable separable as follows:

After some simplifications and using initial condition S(0) = S 0 , Equation (15) yields

Substituting expression for Z(t) from (14) and S(t) from (16) in Equation (9), the closed-form solution for I(t) is given by

It is worthy to mention here that, alternatively, we can establish closed-form solution for the variable I(t) given in (17) by solving ODE (13) subject to initial condition I(0) = I 0 . We substitute the expression for the variable I(t) from (17) in ODE (3), and we arrive at a linear ODE which finally yields

and finally, Δ is given by

Alternatively, same expression for Δ(t) can be derived by solving ODE (4). The closed-form solutions for all variables of model can be summarized as follows:

and ≠ + . *

The model attains its equilibrium point when 0 < 1, and thus, < + . We take limit t → ∞ for closed-form solutions of all variables in Equation (20) and arrive at the following equilibrium point for the model provided < + :

The endemic will end as I(t) → 0 with S(t) approaching some positive value if t → ∞.

* The closed-form solutions for all variables of model for the special case when = 0 are given in Appendix A1.

In this section, we have provided the closed-form expressions for the epidemic peak and maximum number of infected individuals. The closed-form expressions for the FOI and rate at which susceptibles become infected are also given in analytical form.

It is worthy to mention here that we can find analytical expression for time t where peak of infected curve occurs and it is termed as the epidemic peak. Differentiate I(t) given in (20) with respect to t; we get

Setting dI∕dt = 0, we get

and t p > 0 as / + > 1 and S 0 > I 0 . We check second-order derivative at this peak value of t, and it is given by

which is negative as / + > 1. The maximum number of infected cases I p is reported for this value of t and is given by

Equations (23) and (25) can expressed in terms of the basic reproduction number as follows:

and

The FOI Ω = I∕S + I is the rate at which the susceptibles are infected. We utilize the closed-form solutions for S(t) and I(t) from (20) to find following closed-form expressions for the FOI:

The expression for FOI given in (28) can also be expressed in terms of the basic reproduction number 0 and is given by The rate at which susceptibles become infected is SI/S + I, which is Ω × S(t), and is given as

This can be further simplified as follows:

The rate at which susceptibles † become infected given (31) can be expressed as

In this section, we use the specific values of parameters and closed-form solutions of all variables provided in (20) to explore the results derived in Sections 2 and 3. The important indicators of understanding the transmission dynamic of disease which are peak time of epidemic (23), epidemic peak (25), FOI (28) , and rate at which susceptibles become infected (31) are studied with the aid of simulations. We also provide an effective framework to manage the epidemic. It is important to provide the range of parameters of SIQD model with a quarantine-adjusted incidence function in the context of the existing literature. The literature on the estimates of the basic reproduction number for the COVID-19 suggests that it varies from 1.4 to 5.7. According to WHO, 19 it can take values between 1.4 and 2.5. Zhao et al 20 provided the preliminary estimates of the basic reproduction number of novel coronavirus between 3.6 and 4.0 and between 2.24 and 3.58. In China, 0 was 5.6015 before sealing Wuhan city, and it was reduced to 3.4094 as of February 25, 2020. 21 Musa et al 22 estimated 0 as 2.37 in Africa.

The estimated range of the parameters , , and for the SIQD model with a quarantine-adjusted incidence function is presented in Table 1 . The reader is refereed to see previous studies 10,23-28 and references therein for further details. ‡

The following comparative static analysis of the basic reproduction number,

reveals that it has a direct relationship with infection rate and inverse relationship with rate of isolation . The epidemic control strategies suggest to lower the value of the reproduction number and reduce it to less than one, that is, 0 < 1. This can be achieved in different ways. When government enforces strict control measures in the form of lockdown (social distancing), then infection rate declines. The height of the epidemic peak I p reduces, the time span t p to attain this height increases, and 0 decreases. This slows down the transmission dynamics of the disease. And when community † It is worthy to mention here that the closed-form solutions of all variables (20) , peak time of epidemic (23), epidemic peak (25), force of infection (28) , and rate at which susceptibles become infected (31) are all important indicators of understanding the transmission dynamic of disease. These are valid to apply for any country's real data and forecast epidemic peak. These will provide insight for flattening curve and appropriate strategies to get rid of this pandemic. ‡ Carcione et al 6 in a study of SEIRD model provided high values of = 10 3 as the uncertainties are related to parameter , and it varies with time.

The graphs of S(t), I(t), Q(t), and Δ(t) for SIQD model with a quarantine-adjusted incidence function

Effect of change of parameters on epidemic peak I p and time span t p follows proper prevention measures (quarantine, self isolation, mask, disinfection of surfaces, and hands hygiene) then increases, and thus, it lowers the basic reproduction number. We can reduce the basic reproduction number either by reducing or by increasing . One can also use appropriate combination of reducing and increasing to lower the value of 0 .

The comparison of control strategies by government (lockdown) and prevention strategies (quarantine) is presented using simulations.

We consider the following parameters as an example to explore different control strategies mentioned as above: N is 10 million, = 0.02, = 0.6 = 0.2, = 0.001, and initial conditions are S(0) = N − 5, I(0) = 5, Q(0) = 0, Δ(0) = 0. The graphical analysis of all variables S(t), I(t), Q(t), and Δ(t) using closed-from solutions (20) is given in Figure 1 . § Figure 2A represents the effect of change of while keeping other parameters as fixed. The effect of change of while keeping other parameters as fixed is given in Figure 2B . A closer look at Figure 2A ,B shows that when reduces (lockdown) or increases (quarantine and prevention), the epidemic peak reduces, and time span to attain this peak increases. This is an effective strategy to control spread of disease. One can also use appropriate combination of reducing and increasing to lower the value of 0 . We have set initially = 0.4 and = 0.1, and this is shown in Figure 3 in green solid line. The value of 0 = 3.33; the epidemic peak is on Day 55 with 4 178 373 number of infected individuals. Then we changed parameters by a fixed proportion = 1.5 according to following three scenarios:

(i) Increase to and reduce to / . This is shown in Figure 3 in blue dashed line. The value of 0 = 6.92; the epidemic peak is on Day 32 with 6 171 341 number of infected individuals. (ii) Reduce to / and increase to . This is shown in Figure 3 in red dotted line. The value of 0 = 1.6; the epidemic peak is on Day 144 with 1 642 341 number of infected individuals. Scenario 2 is the best scenario to reduce epidemic peak and increase time span to attain the epidemic peak. (iii) Increase to with no change in . This is shown in Figure 3 in cyan long dash line. The value of 0 = 2.35; the epidemic peak is on Day 65 with 3 054 914 number of infected individuals. Scenario 2 or 3 is the best scenario to reduce epidemic peak and increase time span to attain the epidemic peak. We can conclude that increasing (quarantine and prevention) strategy is an effective way to slow down transmission dynamics than the strategy of reducing (lockdown). The prolonged lockdown results in social and economic loss. One should use appropriate combination of reducing (lockdown) and increasing (quarantine and prevention) to lower the value of 0 .

Another important strategy is to set the value of basic reproduction number at some reasonable fixed level close to the threshold value where transmission dynamics is slow. And then change parameters and to reduce epidemic peak and increase the time span to attain this epidemic peak. This scenario is presented in Figure 4 . This is a good strategy if the health care facilities are adequate to treat only I * infected for time period t * and is an effective way to develop herd immunity. To prevent second epidemic wave, it is necessary to slow down transmission dynamics instead of trying to quickly flatten the curve by strict lockdown measures. One should use appropriate combination of reducing and increasing to maintain 0 which will eventually help to attain the DFE. 

Next, we analyze the effect of change of parameters on the FOI and rate of infection. Figure 5A ,B represents the effect of change of while keeping other parameters as fixed. The FOI and rate of infection slow down as reduces. This means that lockdown works efficiently for a short period in reducing force and rate of infection but at the cost of social and economics loss. But this is not an effective strategy in long run.

The effect of change of while keeping other parameters as fixed on the FOI and rate of infection is given in Figure 5C ,D. A closer look at Figure 5C shows that when increases (quarantine and prevention), the time span to arrive at a certain point increases, but intensity of force remains same. The rate at which susceptible moves to infected class is reduced with an increase in .

The control and prevention policies include lockdown by government, quarantine, self isolation, social distancing, mask, disinfection of surfaces, and hands hygiene and are helpful in containment of corona virus. The effective control and prevention policies help in slow down pace of epidemic. The slow transmission dynamics will be helpful for a country to improve health care facilities by increasing capacity of beds, intensive care units, ventilators, and hiring more health care staff to deal with large number of patients.

Lockdown policies implemented by government in most of countries have negative impacts in terms of social and economic activities. The saving lives strategy has resulted in social and economic losses: decline in earnings, depreciation in gross domestic product (GDP), high unemployment rates, stagnant revenue, increased costs, and loss of social capital. We conclude that quarantine and prevention strategy measures are more efficient to slow down transmission dynamics than strict lockdown. The prolonged lockdown results in social and economic loss. One should use appropriate combinations of lockdown and prevention measures to flatten the curve of epidemic. The partial lockdown with effective quarantine and prevention measures is best way to handle the epidemic.

In this section, we illustrate the simulation of the SIQD model with a quarantine-adjusted incidence function by utilizing the closed-form solutions of all variables provided in (20) for the new coronavirus disease. We had concluded in Section 4 that reducing infection rate (lockdown) and increasing isolation rate (quarantine and prevention) in a fixed proportion are best ways to handle the epidemic. This can be done by reducing to / and increasing to for some fixed proportion . First, we analyze the estimated and actual epidemic peak for Pakistan. Then we apply the model for the United States, Brazil, India, and the Syrian Arab Republic. First, we fit the collected data of the new coronavirus disease to find the infection rate , isolation rate , virus-induced mortality rate of infected individuals , and scaling factor for the model. We assume that the virus-induced mortality rate of quarantined class is 10 −7 days −1 for the pandemic. We apply the least square method for the fitting of the collected cases with the the aid of closed-form solutions of all variables provided in (20) . It is worthy to mention here that we have used the least square method with a general function form instead of a linear function. In Figure 6 , we illustrate the flowchart of the fitting.

We study the epidemic peak of Pakistan which has already arrived on July 1, 2020. We use the collected cases of the new coronavirus disease up to date August 2, 2020, to fit the parameters of the new coronavirus disease. 29,30 The start date of pandemic in Pakistan is February 26, 2020, and we take this as t 0 . The initial values of the susceptible, infected, quarantine, and death cases are

The values of parameters estimated by utilizing the closed-form solution and least square method on actual data of Pakistan are given in Table 2 .

The basic reproduction number for Pakistan for set of parameters given in Table 2 is 0 = 1.0013. A closer look at Figure 7 reveals that the infection cases increase up to the Day 126, and total number of infected cases are 108 193. Now, we compare these estimated values with actual data of coronavirus cases in Pakistan. The actual peak occurred on Day 127, and number of infected at this peak value are 108 642. We conclude that our model provided close estimates to the actual data.

First, we fit the collected cases of the pandemic for the United States where we use the collected cases of the new coronavirus disease up to date August 2, 2020, to fit the parameters of the new coronavirus disease. 31,32 The first case of the pandemic in the United States appeared on January 13, 2020, and from this day, we use the initial values where the ini- Table 3 . A similar procedure is carried out for the collected cases of the pandemic for Brazil, 33 India, 34 and the Syrian Arab republic, 35,36 and collected cases of the new coronavirus disease up to date August 2, 2020 are considered to fit the parameters of the new coronavirus disease. The initial conditions, parameter values, and scaling factor for the SIQD model with a quarantine-adjusted incidence function to forecast COVID-19 transmission dynamics for the United States, Brazil, India, and the Syrian Arab Republic are summarized in Table 3 . The graphs for the active cases of infected individuals for United States, Brazil, India, and the Syrian Arab republic are presented in Figure 8 . The peak time t p (days), number of infected at peak I p , and basic reproduction number are provided in Table 4 .

In this section, we discussed the forecasting of the new coronavirus disease based on the closed-form solutions (20) of the SIQD model with a quarantine-adjusted incidence function for five countries, namely, Pakistan, the United States, Brazil, India, and the Syrian Arab Republic. We used the collected data of the infectious cases, the total cases, and the mortality cases up to date August 2 in the five countries to calculate the infection rate , isolation rate , virus-induced mortality rate of infected individuals , and scaling factor for the model. We assume that the virus-induced mortality rate of quarantined class is 10 −7 days −1 for the pandemic. We used the least square method with the general function for this estimating. Also, we computed the peak of the infection function and the predict date of this peak for the five countries. We have compared the estimated and actual epidemic peak t p and the number of infected individuals I p at the peak for Pakistan. We conclude that our model provided close estimates to the actual data. Moreover, we have computed the basic reproduction number based on the fitting parameter for this model which lies in the range [1.0013-1.0299]. The smallest value of the basic reproduction number is for Pakistan, and the highest value is for the United States. It is important to mention here that highest number of the new coronavirus appeared in the United States up to date of writing this work. 

We have developed an SIQD model with a quarantine-adjusted incidence function. The closed-form solutions for all variables of model are established by utilizing the classical techniques of solving ODEs, and these hold provided ≠ + . We have provided the closed-form expressions for FOI, rate at which susceptible becomes infected, the epidemic peak and time required to attain this peak. The management of epidemic perceptive using control and prevention strategies is explained as well. The epidemic starts when 0 > 1, the peak of epidemic appears when number of infected attains peak value when 0 = 1, and the disease dies out 0 < 1. The epidemic control strategies suggest to lower the value of the reproduction number and reduce it to less than, that is, 0 < 1. This can be achieved in different ways. The epidemic peak reduces, and time span to attain this peak increases by reducing infection rate or increasing isolation rate of infected individuals . Another effective strategy to reduce epidemic peak and increase the time span to attain this epidemic peak is to set the value of basic reproduction number at some fixed level close to the threshold value and then change parameters and in a fixed proportion. The effect of change of parameter on the FOI and rate of infection is also presented in detail. We have provided the forecasting of the new coronavirus disease based on the closed-form solutions of SIQD model with a quarantine-adjusted incidence function and the scaling factor for five countries, namely, Pakistan, the United States, Brazil, India, and the Syrian Arab Republic.

Contributions to the mathematical theory of epidemics. II. The problem of endemicity

Predicting COVID-19 Spread in Pakistan using the SIR model

Studying the progress of COVID-19 outbreak in India using SIRD model

The forecasting of COVID-19 with mortality using SIRD epidemic model for the United States, Russia, China, and the Syrian Arab Republic

Estimation of COVID-19 dynamics on a back-of-envelope: does the simplest SIR model provide quantitative parameters and predictions?

A simulation of a COVID-19 epidemic based on a deterministic SEIR model. Front Public Health

Why is it difficult to accurately predict the COVID-19 epidemic?

A new SAIR model on complex networks for analysing the 2019 novel coronavirus (COVID-19)

Analysis on novel coronavirus (covid-19) using machine learning methods

The role of asymptomatic class, quarantine and isolation in the transmission of COVID-19

Analytic solution of the SEIR epidemic model via asymptotic approximant. Phys D: Nonlinear Phenom

Accurate closed-form solution of the SIR epidemic model. Phys D: Nonlinear Phenom

Effects of quarantine in six endemic models for infectious diseases

Global stability of an SIQ epidemic model

Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission

The mathematical theory of infectious diseases and its applications: Charles Griffin & Company Ltd, 5a Crendon Street

Modeling and analyzing HIV transmission: the effect of contact patterns

On the spread of a fatal disease

Statement on the meeting of the International Health Regulations. Emergency Committee regarding the outbreak of novel coronavirus (2019-nCoV) -WHO

Preliminary estimation of the basic reproduction number of novel coronavirus (2019-nCoV) in China, from 2019 to 2020: a data-driven analysis in the early phase of the outbreak

Basic reproduction number and predicted trends of coronavirus disease 2019 epidemic in the mainland of China

Estimation of exponential growth rate and basic reproduction number of the coronavirus disease 2019 (COVID-19) in Africa

Modelling the epidemic trend of the 2019 novel coronavirus outbreak in China

Novel coronavirus 2019-nCoV: early estimation of epidemiological parameters and epidemic predictions

To mask or not to mask: modeling the potential for face mask use by the general public to curtail the COVID-19 pandemic

A novel coronavirus outbreak of global health concern

Impact of non-pharmaceutical interventions (NPIs) to reduce COVID19 mortality and healthcare demand

Clinical course and risk factors for mortality of adult inpatients with COVID-19 in Wuhan, China: a retrospective cohort study

Analysis of transmission dynamics of COVID 19 via closed-form solutions of a susceptible-infectious-quarantined-diseased model with a quarantine-adjusted incidence function

The effective control and prevention policies help in slow down pace of epidemic. The slow transmission dynamics will be helpful for a country to improve health care facilities by increasing capacity of beds, intensive care units, ventilators, and hiring more health care staff to deal with large number of patients. One can also use appropriate combination of reducing infection rate (lockdown) and increasing quarantine rate to lower the value of the reproduction number to its threshold value. This will help in epidemic end. This model can be extended by adding exposed or asymptotic compartment and will be considered in a future work.

There are no funders to report for this submission.

https://orcid.org/0000-0001-6232-4957 Marwan Al-Raeei https://orcid.org/0000-0003-0984-2098

The closed-form solutions for = 0 case for all the variables of model can be summarized as follows:and ≠ + . The closed-form solution given in (A2) can be rewritten in terms of the basic reproduction number as follows:and ≠ + .For this case, we will arrive at the following equilibrium point for the model provided < + :The endemic will end as I → 0 with S approaching some positive value if t → ∞.