key: cord-0790360-8y92vh5y
authors: Mishra, Bimal Kumar; Keshri, Ajit Kumar; Rao, Yerra Shankar; Mishra, Binay Kumar; Mahato, Buddhadeo; Ayesha, Syeda; Rukhaiyyar, Bansidhar Prasad; Saini, Dinesh Kumar; Singh, Aditya Kumar
title: COVID-19 created chaos across the globe: Three novel quarantine epidemic models
date: 2020-05-25
journal: Chaos Solitons Fractals
DOI: 10.1016/j.chaos.2020.109928
sha: 8619613050da13f44758552a0e44fb85e201baeb
doc_id: 790360
cord_uid: 8y92vh5y

The latest version of human coronavirus said to be COVID-19 came out as a sudden pandemic disease within human population and in the absence of vaccination and proper treatment till date, it daunting threats heavily to human lives, infecting more than 12, 11, 214 people and death more than 67, 666 people in 208 countries across the globe as on April 06, 2020, which is highly alarming. When no treatment or vaccine is available till date and to avoid COVID-19 to be transmitted in the community, social distancing is the only way to prevent the disease, which is well taken into account in our novel epidemic models as a special compartment, that is, home isolation. Based on the transmitting behaviour of COVID-19 in the human population, we develop three quarantine models of this pandemic taking into account the compartments: susceptible population, immigrant population, home isolation population, infectious population, hospital quarantine population, and recovered population. Local and global asymptotic stability is proved for all the three models. Extensive numerical simulations are performed to establish the analytical results with suitable examples. Our research reveals that home isolation and quarantine to hospitals are the two pivot force-control policies under the present situation when no treatment is available for this pandemic.

The history of human coronaviruses began in 1965 when Tyrrell and Bynoe 1 found that they could passage a virus named B814. It was found in human embryonic tracheal organ cultures obtained from the respiratory tract of an adult with a common cold [15] . Human Arabia and has approximately 2,500 cases and 800 deaths and still causes as sporadic cases [17] [18] [19] . On February 11, 2020, the WHO Director-General announced that the disease caused by this new CoV was a "COVID-19," which is the acronym of "coronavirus disease 2019" [20] . Coronavirus is the name given to group of viruses that infect mammals and birds.

The name Corona is derived from latin word meaning Crown that refers to its characteristic appearance where its surface is covered in the form of club shaped protein spikes [21] . The host and reservoir of coronavirus is depicted in Figure 1 .

5 Figure 1 : History of Coronavirus -Host & Intermediate Reservoirs [22] By 2020, coronavirus had gained popularity globally with respect to its nomenclature. This family of viruses seems to expand continuously according to Scientists and SARS-CoV-2 alternatively known as COVID-19 also belongs to this family of viruses. There is a probability that some of these viruses might have been missed by the scientists and hence there is still a lot to explore about COVID-19 and the response mechanism of the human immune system. By now 40 coronaviruses have got their names approved as per the International Committee for the Taxonomy of Viruses. The numbers of identified coronaviruses have reached to seven including the current COVID 19. Most of these viruses seem to affect the animals. Four among these seven viruses are acquired by community and transmit through population continually for long period of time. On the other hand, SARS-CoV, MERS-CoV and SARS-CoV-2 are recent outbreaks and are associated with very high mortality rates. Shereen et al. [22] states in addition to the above three, H5N1 influenza A and H1N1 2009 cause acute lung injury (ALI) and acute respiratory distress syndrome (ARDS) all leading to pulmonary failure and ultimate fatality. All corona viruses start in animals and pass on to humans through mutation, recombination and adaptation. Shereen et al. [22] supports that the originating source of COVD-19 is bats but the intermediate source before transmission to humans is not yet identified. 6 Many animal coronaviruses infect animals during a particular season. The reason why these animal coronaviruses do not cause symptoms in their host reservoir is because they continuously co-evolve and adapt over a long period of time with their reservoir host. The symptoms are mild even if they are shown. Symptoms in species vary like in birds it causes upper respiratory tract infections while in cows or pigs it results in Diarrhea. In humans it causes respiratory tract infections that could lead to common cold. The high infection rate, recombination rate and mutation rate of these animal coronaviruses increase the probability of mutant's ability to transmit into another host. It has been speculated that the severity of disease is significantly very high between the new host and the coronavirus at the new round of adaption. It is not yet proved but speculated that the new host will be able to fight the virus effectively only after a long period of adaption and co-evolution hence leading to milder symptoms. The seven human coronaviruses are transitioned into humans through other animals after originating in animals like bats and rodents. The four human coronaviruses including hCoV-OC43 and hCoV-229E cause mild common cold like symptoms. Although they cause infections across all age groups throughout the year but the rates are higher in winters and early spring. They cause multiple re-infections throughout human lifespan. These viruses are transmitted to human beings when they come in close in contact with the host.

These viruses during their incubation period in human beings develop or even mutate for further transmission to other human beings. Earlier form of Coronaviruses includes SARS and MERS that were lethal but were not able to Transmit in Exponential manner like COVID 19. The structure of 2019-nCoV S in the prefusion conformation is depicted in Figure 2 . COVID 19 is associated with the fever, severe respiratory illness, and pneumonia. It is found to be related with SARS-CoV and several bat coronaviruses and is characterized as new member of the betacoronavirus genus. Shereen et al. [22] states that the subgroups of coronaviruses family include alpha (α), beta (β), gamma (γ) and delta (δ) coronavirus and COVID 19 belongs to the β group of coronaviruses.Its infection rate is significantly higher than the severe acute respiratory syndrome coronavirus (SARS-CoV) hence raising the concern of international Public Health Emergency. It enters into the host cells through a densely glycosylated spike (S) protein. This protein undergoes substantial structural change in order 8 to form fusion between the membranes of the virus and the host cell. The protein subunit S1 binds with the host cell receptor. Prefusion trimer is destabilized by the receptor binder resulting in transition to subunit S2. The down (receptor inaccessible) conformation state is more stable than the up (receptor accessible) conformation state. S protein pays an indispensable function and the characterization the Prefusion structure provides vital information which is used in the design and development of the vaccine. COVID 19 overall structure S resembles that of SARS-CoV S resulting in a root mean square deviation (RMSD) of 3.8 Å over 959 Cα atoms. The position of the Receptor Binding Domains (RBD) in their respective down conformations is one of the larger differences between these two structures. The SARS-CoV RBD packs tightly against the N-terminal domain of the neighboring protomer, while COVID 19 RBD is angled closer to the central cavity of the trimer. COVID 19 S and SARS-CoV S, reflect the high structural homology between the two proteins despite the difference in the alignment. Angiotensin-converting enzyme 2 (ACE2) is the same functional host cell receptor shared between COVID 19 S and SARS-CoV S. The apparent ease with which COVID 19 spreads from human to human is mainly because of its very high affinity for human ACE2. Shereen et al. [22] supports that the transmission rate of SARS-CoV 2 is higher than the SRAS CoV owing to the events of genetic recombination in the RBD domain. Shereen et al. [22] states that a clinically approved antivirus is not yet available to fight against the COVID 19.The rapid spreading of COVID 19 indicates the urgent need for coronavirus vaccines. The information available on COVID 19 atomic-level structure will facilitate the additional protein-engineering efforts which could result in the improvement of antigenicity and protein expression for the development of vaccine. The structural data will also provide information on occurrence of the mutations when virus undergoes genetic drift; define whether residues map to known antibody epitopes sites for other coronavirus spike proteins. The structure also assures on whether the protein produced is homogeneous and in the Prefusion conformation. [22] The Structure of Respiratory Syndrome which causes coronavirus in Humans is depicted in Figure 3 . The designing and screening of small molecules with fusion-inhibiting potential are also facilitated by the atomic-level detail. The information is indispensable in supporting the precision design of the vaccine and discovering the antiviral measures [23] . Further, Shereen et al. [22] also describes the compartive analysis of the critical nature of the three outbreaks including COVID 19, SARS CoV and MERS CoV while emulating the approaches that are recommended in developing effective vaccines.

The COVID-19 virus affects different people in different ways. COVID-19 is a respiratory disease and most infected people develops mild to moderate symptoms and recover without requiring special treatment. People who have underlying medical conditions and those over 60 years old have a higher risk of developing severe disease and death. Common symptoms include fever, tiredness, and dry cough. Other symptoms include: shortness of breath, aches and pains, sore throat, and very few people reports diarrhoea, nausea or a runny nose [24].

Pneumonia of unknown cause detected in Wuhan Republic of China, the animal-to-human transmission was presumed as the main mechanism.

Nevertheless, subsequent cases were not associated with this exposure mechanism. Therefore, it was concluded that the virus could also be transmitted from human-to-human, and symptomatic people are the most frequent source of COVID-19 spread. The possibility of transmission before symptoms develop seems to be infrequent, although it cannot be excluded. Moreover, there are suggestions that individuals who remain asymptomatic could transmit the virus. This data suggests that the use of isolation is the best way to contain this epidemic [25] . Based on data from the first cases in Wuhan and investigations conducted by the Chinese Centre for Disease Control and Prevention (CDC) and local CDCs, the incubation time could be generally within 3 to 7 days and up to 2 weeks as the longest time from infection to symptoms was 12.5 days (95% confidence interval, 9.2 to 18) [26] . This data also showed that this novel epidemic doubled about every seven days, whereas the basic reproduction number (R 0 ) is 2.2. In other words, on average, each patient transmits the infection to an additional 2.2 individuals. Of note, estimations of the R 0 of the SARS-CoV epidemic in 2002-2003 were approximately 3 [27] . Study suggests that people may acquire the coronavirus through the air and after touching contaminated objects. Scientists discovered the virus is detectable for up to three hours in aerosols, up to four hours on copper, up to 24 hours on cardboard and up to two to three days on plastic and stainless steel [28] . An analysis of publicly available data on infections from the new coronavirus, SARS-CoV-2, that causes the respiratory illness, COVID-19 yielded an estimate of 5.1 days for the median disease incubation period, according to a new study led by researchers at Johns Hopkins Bloomberg School of Public Health [29] . This median time from exposure to onset of symptoms suggests that the 14-day quarantine period used by the U.S. Centers for Disease Control and Prevention for individuals with likely exposure to the coronavirus is reasonable. The analysis suggests that about 97.5 percent of people who develop symptoms of SARS-CoV-2 infection will do so within 11.5 days of exposure. Lauer et al. estimated that for every 10,000 individuals quarantined for 14 days, only about 101 would develop symptoms after being released from quarantine [29] .

Stage-1: Imported cases where those who travelled to other countries have contracted the infection.

Stage-2: Local transmission in which people coming in immediate contact of an infected person report infection.

Stage-3: 'Community transmission' when a person who has no travel history has contracted through domestic sources.

Stage-4: When a disease is declared an epidemic.

Based on the report of WHO, the transmission of COVID-19 in the top 10 countries across the globe is mentioned in Table The subsequent materials of this paper are structured as follows: Section 2 deals with basic terminologies and basic reproduction number of severe pandemic diseases. Three epidemic models of three different stages of COVID-19 and its stability are discussed in section 3.

Section 4 discusses on simulations and its analysis with examples. Finally the paper is concluded in Section 5 followed by references.

Nomenclature used in three epidemic models of the pandemic disease COVID-19 is given in 2. Some basic terminologies:

In epidemiology a susceptible individual (sometimes known simply as a susceptible) is a member of a population who is at risk of becoming infected by a disease.

It is the international movement of people to a destination country of which they are not natives or where they do not possess citizenship in order to settle or reside there, especially as permanent residents or naturalized citizens, or to take up employment as a migrant worker or temporarily as a foreign worker. Home Isolation (Q 1 (t)): It refers to the restriction of movement or separation of well persons who are susceptible or possibly exposed to a contagious disease (COVID-19 in our case), before it is known whether they will become ill. Isolation usually takes place in the home and 13 may be applied at the individual level or to a group or community of susceptible or exposed population.

have a contagious disease (COVID-19 in our case) in order to prevent its transmission to others. It typically occurs in a hospital setting, but under very special cases may be done at home under a special facility. Usually individuals are quarantined, but the practice may be applied in larger groups. Basic Reproduction number of some highly infectious disease is given in Table 3 . 

In this section we develop three epidemic models on three different stages of infection of COVID-19.

In this paper, our mathematical models for stage 1, 2 and 3 are based on the following hypothesis: 

Most of these hypotheses are common for all the three mathematical models.

Here, a mathematical model is developed to express the first stage of COVID-19. This model has six compartments: susceptible population -immigrant population -home isolation population -infectious population -hospital quarantine population -recovered population.

Any country or state is in stage 1, if person(s) are found COVID-19 positive that has recent international travel history. This model helps us to understand how this foreign return population spreads COVID-19 to other citizens of that country, if not handled properly. In the absence of vaccination and proper treatment, two force-control policies namely home isolation and hospital quarantine, are adopted by many countries to control this COVID-19

epidemic. The same is framed and analysed here with the help of our model as depicted in (1 )

where, the total population N=S + E+ I+ Q 1 + Q 2 + R Then,

In the absence of the disease,

This shows that population size N tends to carrying capacity

It shows that the solution of (1) exists in the region defined by   1  2  6  1  2  1 , , , , , :

Since all the solution remains bounded in the positively invariant region Ω in the maximal interval [0,∞). Thus the initial value problem is well posed.

For the sake of simplicity, and also satisfying the necessary constraints, we take the five classes for calculation of basic reproduction number.

On linearizing the model, we have,

where F is the rate of infection and V is the rate of infection by compartment to compartment. 

On calculation, the spectral radius of 1 2

Thus the basic reproduction number is obtained by

For the equilibrium points in the steady state of the system (1)

In this section we discuss the local stability of endemic equilibrium and disease free equilibrium of the system (1) (after dropping the last equation of (1)) & by analyzing the corresponding characteristic equations respectively.

If R 0 < 1 , the disease free equilibrium of the system (1) is locally asymptotically stable in the region Ω. If R 0 > 1 , it is unstable in the given region Ω.

The characteristic matrix at the diseases free equilibrium is

Clearly, the system (2) has four negative real roots and the root

Clearly all roots are negative real roots. So system (1) is disease free equilibrium and is locally asymptotically stable too.

The endemic equilibrium * * * * * * * 12 ( , , , , , )

, which is interior of Ω, can be obtained by taking all the equations of the system (2) equal to zero. Thus, 

Here the four eigen values are 

, so by Routh-Hurwitz criteria system (1) is stable.

Since all the eigen values have negative real parts so the system (1) is locally asymptotically stable at the endemic equilibrium *  , if R 0 > 1.

In this section, we prove the global stability for endemic equilibrium. We adopt the geometrical approach for the mapping : 

for some C > 0. Based on this procedure used by system (1) and then by (2), is used to prove for Bendixson condition ̅̅̅ . 21 Theorem-3:

If R 0 > 1, then Ω is globally asymptotically stable for the system (1)

The Jacobian matrix of the reduced system (1), leaving E and R compartments we have

The second compound additive Jacobian matrix is given by (2 ) 0 0 0 

The Lozinskii measure of the matrix M can be estimated as

,where g 1 and g 2 is defined as: 

Since all the populations are positive so adding all the total population N=S+E+I+Q 1 +Q 2 +R , , , , , :

Since all the solution remains bounded in the positively invariant region Ω, and in the maximal interval [0,∞), the initial value problem is well posed.

Here we consider only five classes for calculation of basic reproduction number for our sake of simplicity and also satisfying the constraints. where F is the rate of infection and V is the rate of infection from compartment to compartment. 

By calculating the spectral radius of

Thus the basic reproduction number is given by

For equilibrium points in the steady state of the system (3), we have, (1 ) 0

In this section, we discuss the local stability of endemic equilibrium and disease free equilibrium of the system (3) (after dropping the last equation of (3)), and by analyzing the corresponding characteristic equations.

If R 0 <1 , the disease free equilibrium of the system (3) is locally asymptotically stable in the region Ω. If R 0 >1 , it is unstable in the given region Ω.

The characteristic matrix at the disease free equilibrium is ( 

Routh-Hurwitz criteria, system (3) is in disease free equilibrium state and is locally asymptotically stable. 27 The endemic equilibrium * * * * * 

The endemic equilibrium * * * * * * * 12 ( , , , , , )

is locally asymptotically stable when R 0 > B  I  I  I  I  S  S I  S   B  I  I  I  I  I  I 

Hence by Routh-Hurwitz criteria system is stable.

Thus all the eigen values have negative real parts and hence the system (3) is locally asymptotically stable at the endemic equilibrium *  if R 0 > 1.

In order to prove the global stability of the region Ω we take S and I compartment, leave Q 1 population and make it free from the R population. Therefore, in the positive quadrant of the S-I plane we apply Dulac's criteria with multiplier D=1/I Consider, It shows that the solution of (5) exists in the region defined by   1  2  5  1  2  1 , , , , :

Since all the solution remains bounded in the positively invariant region Ω , and in the maximal interval [0,∞), the initial value problem is well posed.

For the basic reproduction number, we take the following system of equations, 1 1 2 where F is the rate of infection and V is the rate of infection by compartment to compartment.

We have, 

Calculating the spectral radius, we have, basic reproduction number as

For the equilibrium points in the steady state of the system (5), we have, 

In this section, we discuss the local stability of endemic equilibrium and disease free equilibrium of the system (5) (after dropping the last equation of (5)) and then by analyzing the corresponding characteristic equations.

If R 0 <1 , the disease free equilibrium of the system (5) is locally asymptotically stable in the region Ω. If R 0 >1 , it is unstable in the given region Ω.

Proof: The characteristic matrix at the disease free equilibrium is

One of the characteristic root or eigen vale is .

So by Routh-Hurwitz criteria the system (5) in disease free equilibrium is locally asymptotically stable.

The endemic equilibrium * * * * * * 12 ( , , , , )

, which is interior of Ω and can be obtained by taking all the equations of the system (5) equal to zero and solving it simultaneously. We have,

The endemic equilibrium * * * * * * 12 ( , , , , ) S I Q Q R  is locally asymptotically stable when R 0 >1

Proof:

On linearizing the system (6), we have, ** 1 ** 12

The two characteristic roots or Eigen values are And other three roots are by solving the cubic equation 3  2  2  2  2  3  1  1  1 2  2  1  1  1  1  1  1 2  1  1 2   1   3   2  3  2  2  2   D  I  S   D  I  I  I  I  S  S  S   D  S  I  I  I  I  I 

So by Routh-Hurwitz criteria the system is locally asymptotically stable. Thus all the Eigen values have negative real parts and hence the system (5) is locally asymptotically stable at the endemic equilibrium *  if R 0 >1.

In order to prove the global stability of the region Ω we take S and I compartment, leave Q 1 population and make it free from the R population. Therefore, in the positive quadrant of S-I plane, we apply Dulac's criteria with multiplier D=1/I is globally stable in the given region Ω for the system (5)

In figure 7(c) , it is evident that the more we have social distancing of the susceptible population, that is, home isolation, the larger we have the chances to escape from the infectious COVID-19. Figures 9 a, b, c, d  Figures 10 a, b(i), b(ii) , c, d, e The more we quarantine the infectious COVID-19 population in the hospital, the more is the recovery, which is very clearly reflected from figure 10(b)(ii). After 11.85 days, the recovery starts, but after 21 days, there is a sharp increase in the recovery population. Table 11 : Parameter values for the Figure 11 a, b, c, d and 12 a, b, c, d Figure 12 (b)(i) clearly reflects that, the more we quarantine the positive cases of COVID-19, the recovery population is more. figure 12(c) , it is evident that, the more we home isolate the susceptible population, the more is the chances of getting them infected from COVID-19. 

Three quarantine models are developed for three different stages of COVID-19. Reproduction number for all the three stages is obtained and the condition for local and global asymptotic stability is well established. When the reproduction number is greater than one, the systems developed in all the three stages of infection-endemic equilibrium in S-I phase plane is 55 globally asymptotic stable. The two vital pivot parameters of all the three models of COVID-19, home isolation and hospital quarantine is well analysed. From the numerical simulations it is very clear that the more we have home isolation (or complete lockdown or social distancing), the less we have the chances to be infected and the disease to be transmitted in the population, which is very evident from Figures 7(c), 8(c) , 9(c),10(c),11(c), 12(c). The general home isolation for our results is approximately 11.85 days. From the simulation results it is also evident from Figures 8(b) (i), 9(b), 10(b)(i)&(ii), 11(b), 12(b)(i) &(ii) that COVID-19 positive population when are subjected to hospital quarantine, the transmission of the disease is reduced and the rate of recovery also increases. Till date when no vaccine or treatment is available for COVID-19, then from our developed models, it is well established that social distancing and hospital quarantine (for positive cases) is the only best treatment.

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