A Study on the Transmission Dynamics of the Omicron Variant of COVID-19 Using Nonlinear Mathematical Models

1  Introduction

Since COVID-19 is a newly discovered virus, little is known about how it spreads. As a result, health authorities must thoroughly understand the incubation and recovery periods to implement more efficient quarantine procedures for those suspected of carrying the virus. As of November 24, 2021, Omicron has been found in countries, and it continues to be the most popular variant all over the world. The transmission dynamics and the potential roles of various intervention strategies have been better understood by recent COVID-19 studies [1–7]. These methodologies incorporate relief and concealment to dial back the spread of the pandemic, decreasing pinnacle medical care to safeguard the people who are most in danger from contaminations, lessening the number of infective cases to a low level, implementing lockdown to a district of exceptionally infective cases, confining suspect cases at home, isolating those residing in a similar family at home. Some authors developed an Omicron variant model with variable population size [8–14].

After becoming infected, a strengthening of the immune system may cause a delay in entering the infectious stage, and a significant amount of delay may even result in the disease being stopped at the exposure level. As a result, the effect of time delay on studying the dynamics of disease spread is significant. In addition, the effect of quarantine on preventing disease spread and the transmission of infection from both the exposed and infected groups are taken into consideration. On the one hand, people who were exposed have the virus, but unlike an asymptomatic patient, they do not show any symptoms right away. There is a latency period before an exposed person becomes infected, and it can take up to 14 days for some people to become infected. By developing the integer model, the current study aims to investigate the effects of the latency period. Using the delay differential equations model, newly infected individuals are given some time before contracting the disease.

To prevent COVID-19 infection in the host population, some authors developed delay-type models. Liu et al. proposed a time delay model and utilised the methodology to analyse the COVID-19 pandemic in China [15]. A new form of disease model based on a time delay dynamics was developed in [16]. They fitted model parameters based on the total number of reported cases in Beijing and Wuhan, China. Using mathematical and statistical modelling, Sedighe et al. developed a model to determine the epidemic trend and forecast the number of patients hospitalised due to COVID-19 in Iran [17]. The SEIQR COVID-19 propagation model with two delays was investigated by Fangfung et al. in [18]. Their model took supply chain transmission and hierarchical quarantine rate into account. A modified SIR model which combines suitable delay parameters and generates a more reliable forecasts of COVID-19 real-time data was proposed in [19]. Where the authors compared the predictions of the recently constructed SIR model to actual data collected from Germany, Italy, Kuwait, and Oman. Shidong et al. created a delay SEIR model based on the feedback linearization technique to manage the effects of COVID-19 [20]. The authors in [21] proposed a SIRDV model to investigate the impact of vaccination campaigns during the pandemic in Israel and Great Britain. In [22], the authors introduced a time delay model considering the migration of individuals from susceptible to infected class. The Omicron model can be mathematically modeled in a way that is reasonably accurate to the occurrences that have been observed when delay factors are included in the system of differential equations.

In this paper, two delay mathematical models are proposed. The work is significant, because it contains the mathematical modeling with a real-data of the Omicron variant from Chennai, Tamil Nadu, India. In the form of sections, the delayed SQIRV model is proposed and its stability is examined in Section 2. The delayed SEIQIc RVW model is proposed and steady-state solution existence is tested in Section 3. In order to confirm and strengthen our theoretical findings regarding Omicron B.1.1.529 SARS-Cov-2, computational simulations are carried out from the real data which is collected from Tamilnadu in Section 4. In Section 5, we summarise our findings.

2  Delayed SQIRV Mathematical Model

Here we define the delay-type version of the integer-order SQIRV model proposed in [23]. The disease is considered to have an incubation time of τ>0, because the virus moved from the susceptible phase to the incubation phase. The time between becoming susceptible to the virus and experiencing its symptoms is referred to as the incubation period. Based on the policy decisions made by the government, a set of parameters have been obtained to forecast the pandemic trend. Table 1 lists the non-negative parameters that are used in this model.

Considering the given aspects, the delay SQIRV mathematical model is derived as follows:

dSdt=Γ−η21S−η3S(t−τ)I(t−τ)+η4Q+η5R+η6V,dQdt=η7I−η22Q−η9QV,dIdt=η3S(t−τ)I(t−τ)−(η23)I(t),dRdt=η12I(t)+η8Q−η24R,dVdt=η11R+η2S−η25V+η9QV,(1)

where η21=η1+η2, η22=η1+η4+η8, η23=η1+η7+η10+η12, η24=η1+η5+η11, and η25=η1+η6

Subject to initial conditions S(0)=S0,Q(0)=Q0,I(0)=I0,R(0)=R00,V(0)=V0.

As in the case of Omicron, a susceptible individual is assumed to interact with an infected individual in the equation system but does not enter the infected compartment until after a predetermined incubation period. The incubation period τ is just while moving from the powerless compartment to the contaminated compartment.

There are two steady-state solutions to the model under consideration. Time-independent solutions are obtained when the model system (1) is made static. The steady-state solution, I=0, when there are no infections is given by

E0=(S,Q,I,R,V)=(Γ(η1+η6)η1(η1+η2+η6),0,0,0,Γη2η1(η1+η2+η6))(2)

Also, the steady-state solution when infection is persistent i.e., I≠0 is given by

E∗ =(S∗,Q∗,I∗,R∗,V∗) =(ν33η3,η7I∗η22+η9V∗,η3Γ−η21ν33+η6Cν33−η4A−η5B−η6C∗,η8Q∗+η12I∗η1+η5+η11,η2η11R∗+η2η23η2(η1+η6−η9Q∗))(3)

where A=η7η22+η9V∗, B=η12η1+η5+η11+η7η8(η22+η9V∗)(η1+η5+η11), C=η11Bη1+η6−η9Q∗.

The fundamental reproduction number R0 is calculated by using the next generation operator matrix as folows [24,25]:

R0 is the largest eigenvalue of the spectral radius given by

R0(FV−1)=η3(Γ(η1+η6)η1(η1+η2+η6))(1(η1+η10+η12+η7))(4)

2.1 Stability Analysis of the Delayed SQIRV Model

The following theorem applies Rouche’s theorem [26] to characterise the local stability of the SQIRV system (1) for the infection-free steady state solution (2). The output is determined by the reproduction number R0.

Theorem 2.1. The infection free steady state E0 is locally asymptotically stable if R0<1 and unstable if R0>1 for τ≥0.

Proof. The characteristic equation of system (1), for the equilibrium point E0, is given by

Δ(λ)=|λId5×5−J001−J011e−τλ|(5)

where J001

 =(−(η1+η2)η40η5η60−(η1+η4+η8+η9V)η70000−(η1+η7+η10+η12)000η8η12−(η1+η5+η11)0η2η9V0η11−(η1+η6))

and

J011=(00−η3Se−tτ000000000η3Se−tτ000000000000),

C(λ) =(λ+(η1+η4+η7+η8))(λ+(η1+η4+η11))(λ−η3Se−λτ+(η1+η7+η10+η12)) (λ+12((η1+η2)+(η1+η6)±(η1+η2)2+4η2η6−η1η2(η1+η6)+(η1+η6)2)).(6)

Then from the Jacobian matrix, the eigen values are −(η1+η4+η7+η8),−(η1+η4+η11),η3Se−λτ−(η1+η7+η10+η12), and 12(−(η1+η2)−(η1+η6)±(η1+η2)2+4η2η6−η1η2(η1+η6)+(η1+η6)2).

When τ=0, the System (1) is stable iff η3S−(η1+η7+η10+η12)<0, and η1(η1+η2+η6)>1.

Then clearly the infection free steady state E0 (2) is locally asymptotically stable if R0<1.

Let τ>0. In this case, we will use Rouches s theorem to prove that all roots of the characteristic Eq. (5) cannot intersect the imaginary axis, i.e., the characteristic equation cannot have pure imaginary roots.

Suppose for the opposite, that is, suppose there exists w∈R such that λ=wi is a solution of (19).

Consider the term η3Se−iwτ−(η1+η7+η10+η12)=0

 ⟹ wi+(η1+η7+η10+η12)=η3Se−iwτ

 ⟹ wi+(η1+η7+η10+η12)=η3S(cos⁡wτ−isin⁡wτ)

Equating the real and imaginary parts we get

w=−iη3Ssin⁡wτ,(η1+η7+η10+η12)=η3Scos⁡wτ

Squaring and adding we get  ⟹ (w)2+(η1+η7+η10+η12)2=μsS2

 ⟹ w2=μsS2−(η1+η7+η10+η12)2

If R0<1, then w2<0, which is a contradiction.

Thus the infection free consistent state E0 is locally asymptotically stable if R0<1 for τ≥0.

The Ruth-Hurwitz stability theory and Rouche’s theorem are used in the following theorem to characterize the local stability of the SQIRV system (1) for the infectious persistent steady state solution (3). The consequence is determined by the reproduction number R0.

Theorem 2.2. The infection persistent steady state solution E∗ of (1) is locally asymptotically stable if R0>1 for τ≥0.

Proof.

The characteristic equation of system (1), for the equilibrium point E∗, is given by

Δ(λ)=|λId5×5−J101−J111e−τλ|(7)

where J101=

(−(η1+η2)η40η5η60−(η1+η4+η8+η9V∗)η70−η9Q∗00−(η1+η7+η10+η12)000η8η12−(η1+η5+η11)0η2η9V∗0η11−(η1+η6)+η9Q∗)

and

J111=(−η3I∗e−tτ0−η3S∗e−tτ0000000η3∗e−tτ0η3S∗e−tτ000000000000)

The characteristic polynomial is

λ5 +(e−λτη3I∗+F0)λ4+(e−λτη3I∗F1+F2)λ3+(e−λτη3I∗F3+F4)λ2 +(e−λτη3I∗F5+F6)λ+(e−λτη3I∗F7+F8)=0(8)

where F0=η21−η9Q∗+η22+η9V∗+η24+η25, F1=η22+η9V∗+η24+η25−η9Q∗, F2=η21(η22+η24+η25+η9V∗−η9Q∗)−η2η6,

F3=(η24+η25+η3S∗−η9Q∗)(η22+η9V∗)−η4η7−η3S∗η9Q∗−η5η12−η9Q∗η24+η24η25+η9Q∗η9V∗+η3S∗η24+η3S∗η25,

F4=(η21−η9Q∗)(η22+η9V∗)η24−[η9Q∗((η21)(η22+η9V∗)+η21η24)+η2η6((η22+η9V∗)+η24)]+η21((η22+η9V∗)η25+η24η25+η9Q∗η9V∗)+η9Q∗(η2η4+η8η11)+η24((η22+η9V∗)η25+η9Q∗η9V∗),

F5=η5η9Q∗η12−η3S∗η9Q∗(η22+η9V∗+η24)+η4η7(η24+η25)+η5η12(η22+η9V∗+η25)+η5(η7η8+η6η9V∗)+η6η12η11,

F5=(η9Q∗(η8η11+η24η9V∗)+(η22+η9V∗)η24(η25−η9Q∗))−(η3S∗η9Q∗(η22+η9V∗+η24)+η4η7(η24+η25)+η5η12(η22+η9V∗+η25)+η5(η7η8+η6η9V∗)+η6η12η11),

F6=(η9Q∗(η8η11+η24η9V∗)+(η22+η9V∗)η24(η25−η9Q∗))+η4η9Q∗(η5η7+η2η24)+η5η9Q∗(η3I∗η12+η2η8),

F7=η5η9Q∗(η7η8+η12(η22+η9V∗))+η4η9Q∗η12η11,

F8=(η8η9V∗+η24η9V∗)(η3S∗η9Q∗−η6η7)+η24(η9Q∗+η25)(η4η7−η3S∗(η22+η9V∗))−[η12(η22+η9V∗)(η6η24+η5η25) + η5(η7η8η25+η9Q∗η12η11)]

If τ=0, then by using the rule of Descartes of sign, we can get there are no positive real roots.

Also by Routh-Hurwitz stability criterion, the real parts of the complex roots are also negative if η3I∗(Fi)+Fj>0 for i=1,3,5,7;j=0,2,4,6,8, (R0−1)>0,R0>1. Then the infection persistent steady state (S∗,Q∗,I∗,R∗,V∗) is locally stable when R0>1.

If τ>0, then by using Rouch’s theorem, we have to prove that all roots of the characteristic Eq. (6) cannot have pure imaginary roots.

Suppose that there exists w∈R such that λ=wi is a solution of (6).

Now Eq. (22) becomes

iw5 +(e−iwτη3I∗+F0)w4−i(e−iwτη3I∗F1+F2)w3−(e−iwτη3I∗F3+F4)w2+i(e−iwτη3I∗F5+F6)w −(e−iwτη3I∗F7+F8)=0(9)

Then

iw5 +F0w4−iF2w3−F4w2+iF6w+F8 =η3I∗(−w4+iF1w3+F3w2−iF5w−F7)(cosτw−isinτw)(10)

Equating the real and imaginary parts of (10) we get

F0w4−F4w2+F8=η3I∗(−w4+F3w2−F7)cosτw+(F1w3−F5w)sinτw(11)

w5−F2w3+F6w=(F1w3−F5w)cosτw−η3I∗(−w4+F3w2−F7)sinτw(12)

Squaring both Eqs. (12), (12) and adding we get

w10 +(F02−2F2−η3I∗2)w8+[F22+2F6−2F0F4−η3I∗2(F12−2F3)]w6+[F42+2F0F8−2F0F6 −η3I∗2(F32−2F1F5)]w4+[F62−2F7F8−η3I∗2(F52−2F3F7)]w2+(F82−η3I∗2F72)=0.(13)

Let z=w2 in (13)

F(z) =z5+(F02−2F2−η3I∗2)z4+[F22+2F6−2F0F4−η3I∗2(F12−2F3)]z3+[F42+2F0F8−2F0F6 −η3I∗2(F32−2F1F5)]z2+[F62−2F7F8−η3I∗2(F52−2F3F7)]z+(F82−η3I∗2F72)=0.(14)

If R0>1, then from Eq. (14) we can see that F82−η3I∗2F72 is strictly negtive which implies F(0)>0. Thus we can get atleast one positive real root. Hence, if R0>1 all the real parts of the roots of (8) are negative. Thus the equilibrium position E∗ is stable when R0>1 for τ≥0.

3  Delayed SEIQIcRVW Model Formulation

This section is focused on constructing delay SEIQIcRVW model for our problem formulation. The delayed SEIQIcRVW model can be formulated from the integer-order model form given in [27]. It is considered that the disease has an incubation time of the virus τ>0 transferred from susceptible period to an incubation period. The incubation period is the delay time that passes between being susceptible and showing symptoms of the virus. The suitable parameters are used to formulate the Omicron delayed SEIQIcRVW Model, which are described in Table 2.

Considering the given aspects, the SEIQIcRVW delay mathematical model is derived as follows:

dSdt=P−δnS−μsS(t−τ)I(t−τ)+ρrR+ρvV,dEdt=μsS(t−τ)I(t−τ)−(ξ1)E,dIdt=γiE−(ξ2)I,dQdt=kE+ζI+ζqIc−(ξ3)Q,dIcdt=δcE−(ξ4)Ic,dRdt=νrI+γrQ+γcIc−(ξ5)R,dVdt=ηvQ+αvR−(ξ6)VdWdt=ωcI+ζwIc−δnW,(15)

where ξ1=δn+k+γi+δc, ξ2=δn+δe+νr+ζ+ωc, ξ3=γr+ηv+δn, ξ4=ζq+γc+ζw+δn, ξ5=δn+ρr+αv and ξ6=δn+ρv.

Subject to initial conditions: S(0)=S0,E(0)=E0,I(0)=I0,Q(0)=Q0,Ic(0)=Ic0,R(0)=R00,V(0)=V0.

3.1 Steady State Solutions the Delayed SEIQIcRVW Model

The system (15) is found static, i.e., the solutions of time independent are obtained. The steady state solutions in the infection free state, when I=0 is given by

Eq0 =(S0,E0,I0,Q0,Ic0,R0,V0,W0)=(Pδn,0,0,0,0,0,0,0).(16)

Also, when infection is persistent the steady state solutions, i.e., I≠0 is given by

Eq∗=(S∗,E∗,I∗,Q∗,Ic∗,R∗,V∗,W∗)(17)

where

S∗ =ξ2γi(Pμsγi−δnξ1ξ2μs(ξ1ξ2−ρvγiJ−γiG)),E∗=ξ1ξ2μsγi(Pμsγi−δnξ1ξ2μs(ξ1ξ2−ρvγiJ−γiG))I∗=Pμsγi−δnξ1ξ2μs(ξ1ξ2−ρvγiJ−γiG)=δnξ1ξ2(R0−1)μs(ξ1ξ2−ρvγiJ−γiG)Q∗=ξ2ξ4+ζξ4γi+ζqδcξ2ξ3ξ4γi(Pμsγi−δnξ1ξ2μs(ξ1ξ2−ρvγiJ−γiG))Ic∗=δcξ2ξ4γi(Pμsγi−δnξ1ξ2μs(ξ1ξ2−ρvγiJ−γiG))R∗=νrγiξ3ξ4+γr(ξ2ξ4+ζξ4γi+ζqδcξ2)+γiδcξ2ξ3γiξ3ξ4ξ5(Pμsγi−δnξ1ξ2μs(ξ1ξ2−ρvγiJ−γiG))V∗=ηvγiξ5(ξ2ξ4+ζξ4γi+ζqδcξ2)ξ3ξ4γiξ5 +αv[νrγiξ3ξ4+γr(ξ2ξ4+ζξ4γi+ζqδcξ2)+γiδcξ2ξ3]ξ3ξ4γiξ5(Pμsγi−δnξ1ξ2μs(ξ1ξ2−ρvγiJ−γiG))W∗=γiωcξ4+ζwδcξ2δnγiξ4(Pμsγi−δnξ1ξ2μs(ξ1ξ2−ρvγiJ−γiG))

with

J =νrγiξ3ξ4+γr(ξ2ξ4+ζξ4γi+ζqδcξ2)+γiδcξ2ξ3γiξ3ξ4ξ5,G =ηvγiξ5(ξ2ξ4+ζξ4γi+ζqδcξ2)ξ3ξ4γiξ5 +αv[νrγiξ3ξ4+γr(ξ2ξ4+ζξ4γi+ζqδcξ2)+γiδcξ2ξ3]ξ3ξ4γiξ5.

The basic reproduction number R0 is

R0(GV−1)=Pμsγiδn(δn+k+γi+δc)(δn+δe+νr+ζ+ωc).(18)

3.2 Stability Analysis of the Delayed SEIQI cRVW Model

The local stability of the SEIQI cRVW system (15) for the infection-free steady state solution (16) is examined in the next theorem applying Rouche’s theorem. The reproduction number R0 determines the result.

Theorem 3.1. The infection free consistent state E0 (16) is locally asymptotically stable if R0<1 and unstable if R0>1 for the time delay τ≥0.

Proof. The characteristic equation of system 15, for the equilibrium point E0, is given by

Δ(λ)=|λId8×8−J00−J01e−τλ|(19)

where

J00=(−δn0000ρrρv00−ξ10000000γiξ2000000kζ−ξ3ζq0000δc00−ξ400000νrγrγc−ξ500000ηv0αv−ξ6000ωc0ζw00−δn),

and

J01=(00−μsS0000000μsS00000000000000000000000000000000000000000000000000000).

C1(λ) =(λ+δn)(λ+δn)(λ+ξ1+ξ2+ξ12+4μsSγi−2ξ1ξ2+ξ22)(λ+ξ1+ξ2−ξ12+4μsSγi−2ξ1ξ2+ξ22)(λ+ξ3)(λ+ξ4)(λ+ξ5)(λ+ξ6).(20)

When τ = 0, the eigenvalues are −δn, −δn, 12(−ξ1−ξ2−ξ12+4μsSγi−2ξ1ξ2+ξ22), 12(−ξ1−ξ2+ξ12+4μsSγi−2ξ1ξ2+ξ22), −ξ3, −ξ4, −ξ5, −ξ6.

The given system (15) is stable when −ξ1−ξ2+ξ12+4μsSγi−2ξ1ξ2+ξ22<0

or ξ12+4μsSγi−2ξ1ξ2+ξ22<(ξ1+ξ2) or ξ12+4μsSγi−2ξ1ξ2+ξ22<(ξ1+ξ2)2

or μsSγi<ξ1ξ2. i.e., μsSγiξ1ξ2<1.

That is R0<1. Clearly infection free steady state E0 is locally asymptotically stable if R0<1 when τ=0.

Let τ>0. In this case, we will use Rouche’s theorem to prove that all roots of the characteristic Eq. (19) cannot intersect the imaginary axis, i.e., the characteristic equation cannot have pure imaginary roots.

Suppose for the opposite, that is, suppose there exists w∈R such that λ=wi is a solution of (19).

Consider the term wi+ξ1+ξ2−ξ12+4μsSe−τwiγi−2ξ1ξ2+ξ22=0

 ⟹ wi+ξ1+ξ2=ξ12+4μsSe−τwiγi−2ξ1ξ2+ξ22=0

 ⟹ (wi+ξ1+ξ2)2=ξ12+4μsSe−τwiγi−2ξ1ξ2+ξ22=0

 ⟹ (wi)2+(ξ1+ξ2)2+wi(ξ1+ξ2)−ξ12+2ξ1ξ2−ξ22=4μsS(cos⁡τw−isin⁡τw)γi

 ⟹ −w2+wi(ξ1+ξ2)+4ξ1ξ2=4μsS(cos⁡τw−isin⁡τw)γi

By equating the real and imaginary part, we get

4ξ1ξ2−w2=4μsγiScos⁡τw, w(ξ1+ξ2)=−4μsγiSsin⁡τw

If R0<1, then μsSγi−ξ1ξ2>0. Hence w2<0, which is a contradiction.

Thus the infection free consistent state E0 is locally asymptotically stable if R0<1 for τ≥0.

Now suppose that R0>1 from the characteristic polynomial (20), it is enough to consider the term (λ+ξ1+ξ2−ξ12+4μsSγi−2ξ1ξ2+ξ22). It is easy to see that C1(0)<0. On the other hand, limλ→+∞C1(λ)=+∞. Therefore, by continuity of C1(λ), there is at least one positive root of the characteristic Eq. (20). Hence, we conclude that Σ1 is unstable, for any τ≥0.

The local stability of the SEIQI cRVW system (15) for the infection’s persistent steady state solution (17) is determined using Rouche’s theorem and the Routh-Hurwitz technique in the next theorem. The result is governed by the reproduction number R0.

Theorem 3.2. If R0>1, then the endemic equilibrium point E∗ is locally asymptotically stable for τ≥0.

Proof. The characteristic equation of system 15, for the equilibrium point E∗ 17 is given by

Δ(λ)=|λId8×8−J10−J11e−τλ|.(21)

where the Jacobian matrices of the model at infection persistent steady state solution are

J(10)=(−δn0000ρrρv00−ξ10000000γiξ2000000kζ−ξ3ζq0000δc00−ξ400000νrγrγc−ξ500000ηv0αv−ξ6000ωc0ζw00−δn).

and

J(11)=(−μsI∗0−μsS∗00000μsI∗0μsS∗00000000000000000000000000000000000000000000000000000).