Dynamical analysis of a discrete‐time COVID‐19 epidemic model

Abstract

In this paper, we explore local dynamics with topological classifications, bifurcation analysis, and chaos control in a discrete‐time COVID‐19 epidemic model in the interior of ${ℝ}_{+}^4$. It is explored that for all involved parametric values, discrete‐time COVID‐19 epidemic model has boundary equilibrium solution and also it has an interior equilibrium solution under definite parametric condition. We have explored the local dynamics with topological classifications about boundary and interior equilibrium solutions of the discrete‐time COVID‐19 epidemic model by linear stability theory. Further, for the discrete‐time COVID‐19 epidemic model, existence of periodic points and convergence rate are also investigated. It is also studied the existence of possible bifurcations about boundary and interior equilibrium solutions and proved that there exists no flip bifurcation about boundary equilibrium solution. Moreover, it is proved that about interior equilibrium solution, there exist Hopf and flip bifurcations, and we have studied these bifurcations by utilizing explicit criterion. Moreover, by feedback control strategy, chaos in the discrete COVID‐19 epidemic model is also explored. Finally, theoretical results are verified numerically.

1. INTRODUCTION

1.1. Motivation and literature survey

People and pandemics share an unpleasant history since the start of civilization. In recent times, however, no death‐delivering disease has caused such huge an alarm on a global scale as the COVID‐19. Since its first appearance in China's Wuhan City in December 2019, it has now gone on to spread across all the continents. In the face of millions of deaths in one‐and‐a‐half years, almost all branches of modern research have conducted studies with a mere aim of helping humans in pain. Mathematics being no exception, the present paper would also make a small contribution in the ongoing effort of uncovering its several dynamics.

That the risk is not yet over has been proven by the subsequent second and third waves of the global epidemic. In terms of lives lost, China, the United States, Italy, and India are among the top countries. Several scientific and medicinal studies show that the COVID‐19 initially spread in animal species (like bats and camels) which later infected homo sapiens as well. In Asia and Middle East, relatively less harmful kinds of the severe acute respiratory syndrome were reported. They are called MERS‐CoV.

At the time of writing these lines, different types of vaccines are already out. The process of vaccination, nevertheless, seems to be slow–understandable given the worldwide outbreak. They have been varying opinions as to the effectiveness itself. One thing is clear: It doesn't cure the disease; it only creates immunity against the viral attack hence enabling one to fight it better in future. Therefore, the preventive and precautionary measures (social distancing, self‐isolation, washing hands, wearing face masks, etc.) are still the only sure solutions.

In the meanwhile, many chemists, biologists, doctors, and mathematicians are trying to study the behavior of coronavirus. On the mathematical side, there are few models already discovered, to study it's behavior. Some of them are SIR (susceptible, infected, and recovered) model, SIQR (susceptible, infectious, quarantined, and recovered) model, SEIR (susceptible, exposed, infected, and recovered) model, and SIRS (susceptible, infected and recovered, susceptible) model. For instance, Çakan ¹ studied the dynamics of following continuous‐time COVID‐19 epidemic model:

$$\begin{array}{cc}\hfill \frac{dS}{dt}& =b-\mu S-\beta SI,\hfill \\ \hfill \frac{dE}{dt}& =\beta SI-\gamma S(t-\tau )I(t-\tau )e^{-\mu \tau }-(\delta +\mu )E,\hfill \\ \hfill \frac{dI}{dt}& =\gamma S(t-\tau )I(t-\tau )e^{-\mu \tau }-\left[{\mu }_1+{\mu }_2\left(1-c(t)\right)\right]I-\left[{\alpha }_1+{\alpha }_2\left(1-c(t)\right)\right]I-dI,\hfill \\ \hfill \frac{dR}{dt}& =\left[{\alpha }_1+{\alpha }_2\left(1-c(t)\right)\right]I+\delta E-\mu R,\hfill \end{array}$$ (1)

where $S$, $E$, $I$, and $R$, respectively, denote susceptible, exposed, infected, and recovered populations; $b$ and $d$ are, respectively, birth and natural death rates; $\beta$ is the effective contact rate between $S$ and $I$; $\gamma$ is the progression rate of $E$ into $I$; $\delta$ is the transition rate of $E$ to $R$; $c(t)$ has the values in the interval $[\mathrm{0,1}]$; ${\alpha }_1$ is the natural recovery rate of $I$; ${\alpha }_{2}c(t)$ is the contribution rate to recovery of health care system; and ${\mu }_1$ is the minimum disease‐induced death rate. Noor et al ² studied the computational dynamical analysis of following continuous‐time COVID‐19 epidemic model:

$$\begin{array}{cc}\hfill \frac{dS}{dt}& =a-cSI(1+\gamma I)-\mu S+\alpha R,\hfill \\ \hfill \frac{dI}{dt}& =cIS(1+\gamma I)-(\beta +\mu +\delta -b)I,\hfill \\ \hfill \frac{dR}{dt}& =\beta I-(\alpha +\mu )R,\hfill \end{array}$$ (2)

where $S$, $I$, and $R$ are, respectively, susceptible, infected, and recovered populations; $c$ is the convex incidence rate; $\delta$ is the death rate due to COVID‐19; $\alpha$ is the rate of $R$ to $S$ population; $\mu$ is the natural death rate; recovery rate is $\beta$; and $b$ is the rate of infected human's immigrant to one to another location. Adeniyi et al ³ proposed and studied the dynamics of following continuous‐time COVID‐19 epidemic model:

$$\begin{array}{cc}\hfill \frac{dS}{dt}& =\mathrm{\Lambda }-\lambda S-\mu S+{\alpha }_{2}Q+\delta R,\frac{dQ}{dt}=\lambda S-({\alpha }_1+{\alpha }_2+\mu )Q,\hfill \\ \hfill \frac{dI}{dt}& ={\alpha }_{1}Q-({\alpha }_3+\sigma +\mu )I,\frac{dR}{dt}={\alpha }_{3}I-(\delta +\mu )R,\hfill \\ \hfill \frac{dE}{dt}& =\frac{{\omega }_{0}I}{{\omega }_1+{\omega }_{2}I}-dE,\hfill \end{array}$$ (3)

where $S$, $Q$, $I$, $R$, and $E$, respectively, denote susceptible, quarantined, infected, recovered, and exposed populations; ${\alpha }_1$ is the rate at which symptoms show to infected population; ${\alpha }_2$ is the rate of $I$ to $S$ class; recovery rate is ${\alpha }_3$; death rate due to COVID‐19 is $\sigma$; $\mu$ is the natural death rate (not due to COVID‐19); and ${\omega }_1$, ${\omega }_2$, and ${\omega }_3$ are, respectively, information growth rate, half saturation point, and saturation constant of information. Sulaiman ⁴ used the following data‐driven SIR model to examine the dynamical analysis of the nature of disease:

$$\frac{dS}{dt}=-\frac{\beta }{N}SI,\frac{dI}{dt}=\frac{\beta }{N}SI-\gamma I,\frac{dR}{dt}=\gamma I,$$ (4)

where $S$, $I$, and $R$ are, respectively, susceptible, infected, and recovered populations; $\beta$ is the average number of contact person per day; $\gamma$ is the individual infected rate; and $N=S+I+R$. Rao et al ⁵ studied the dynamics of following $SEIVR$ epidemic model:

$$\begin{array}{cc}\hfill \frac{dS}{dt}& =\mathrm{\Lambda }-{\beta }_{1}SE-{\beta }_{2}SI-{\beta }_{3}SV-d_{1}S,\hfill \\ \hfill \frac{dE}{dt}& =W_E+{\beta }_{1}SE+{\beta }_{2}SI+{\beta }_{3}SV-d_{2}E-pE,\hfill \\ \hfill \frac{dI}{dt}& =W_I+pE-d_{3}I-\alpha I,\hfill \\ \hfill \frac{dV}{dt}& =W_V+{\gamma }_{1}E+{\gamma }_{2}I-d_{4}V,\hfill \\ \hfill \frac{dR}{dt}& =W_R+\alpha I-d_{5}R,\hfill \end{array}$$ (5)

where ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$ are, respectively, rates between $S$, $E$, and $I$; $p^{-1}$ is the incubation period between infection and it's symptoms; $\alpha$ is the recovery rate; $d_i(i=1,\text{⋯},5)$ are, respectively, death rate of $S$, $E$, $I$, $R$ and the amount of COVID‐19 in environment; $\mathrm{\Lambda }$ is the population influx; $W_E$ and $W_I$ are, respectively, the number of exposed and infected individual mobility; and $W_V$ represents the number of COVID‐19 from anywhere else. In 2021, Lu et al ⁶ also studied dynamic analysis of SIQR model with delay for pandemic COVID‐19.

1.2. Mathematical formulation of discrete COVID‐19 epidemic model

Here, first, we present continuous‐time COVID‐19 mathematical model in which total population $N$ is divided into four compartments with $N=S+I+Q+R$ where $S$, $I$, $Q$, and $R$, respectively, denote susceptible, infected, quarantined, and recovered people, and further, it is assumed that COVID‐19 infection attain permanent immunity. So based on Figure 1, we have following model equations in which individuals moves from one class to another class represented as class $S$ to $I$, $I$ to $Q$, $Q$ to $R$, and it may also flow from $I$ to $R$ directly:

$$\begin{array}{cc}\hfill \frac{dS}{dt}& =\mathrm{\Lambda }-\mu S-\beta SI,\frac{dI}{dt}=\beta SI-(\delta +\mu +{\alpha }_1+\gamma )I,\hfill \\ \hfill \frac{dQ}{dt}& =\delta I-ϵQ-{\alpha }_{2}Q-\mu Q,\frac{dR}{dt}=\gamma I+ϵQ-\mu R.\hfill \end{array}$$ (6)

FIGURE 1.

Flow diagram for continuous‐time COVID‐19 epidemic model

In the model (6), natural increase of population is $\mathrm{\Lambda }$; $\beta$ is the rate at which susceptible class tends to infected one; $\delta$ is the transition rate from $I$ to $Q$; cure rate of infected and quarantined persons are $\gamma$ and $ϵ$, respectively; ${\alpha }_1$ and ${\alpha }_2$ are COVID‐19 mortality rate of infected and quarantined persons, respectively; and $\mu$ is the natural death rate. On the other hand, by applying Euler forward formula, the continuous‐time COVID‐19 model (6) takes the following form:

$$\begin{array}{cc}\hfill \frac{S_{n+1}-S_n}{h}& =\mathrm{\Lambda }-\mu S_n-\beta S_n{I}_n,\frac{I_{n+1}-I_n}{h}=\beta S_n{I}_n-(\delta +\mu +{\alpha }_1+\gamma )I_n,\hfill \\ \hfill \frac{Q_{n+1}-Q_n}{h}& =\delta I_n-ϵQ_n-{\alpha }_2{Q}_n-\mu Q_n,\frac{R_{n+1}-R_n}{h}=\gamma I_n+ϵQ_n-\mu R_n.\hfill \end{array}$$ (7)

From (7), the simplification yields

$$\begin{array}{cc}\hfill S_{n+1}& =h\mathrm{\Lambda }+(1-h\mu )S_n-h\beta S_n{I}_n,I_{n+1}=(1-h\delta -h\mu -h{\alpha }_1-h\gamma )I_n+h\beta S_n{I}_n,\hfill \\ \hfill Q_{n+1}& =h\delta I_n+(1-hϵ-h{\alpha }_2-h\mu )Q_n,R_{n+1}=h\gamma I_n+hϵQ_n+(1-h\mu )R_n,\hfill \end{array}$$ (8)

with step size is $h$ and $t$ is represented by $n$.

1.3. Main contribution

The purpose of present study is to investigate dynamics characteristics of discrete COVID‐19 model (8) instead of continuous‐time COVID‐19 model (6) because discrete models are more appropriate and also gives more efficient computational results as compared to continuous once. So our main contribution in this paper includes:

• Study of feasible equilibrium solution with linearized form of discrete model (8).

• Study of local behavior at equilibrium solutions and periodic points of discrete COVID‐19 epidemic model (8).

• Study of convergence rate for COVID‐19 epidemic model (8).

• Study of detailed bifurcation analysis about equilibrium solutions of discrete COVID‐19 epidemic model (8).

• Study of chaos by state feedback method.

• Verification of theoretical results numerically.

1.4. Paper layout

The layout of the paper is as follows: The existence of equilibrium solutions of discrete COVID‐19 epidemic model (8) is explored in Section 2. In Section 3, we have constructed linearized form of discrete COVID‐19 epidemic model (8) whereas Section 4 is about the exploration of local behavior of COVID‐19 epidemic model (8) at equilibrium solutions. In Section 5, periodic points with period‐ $n$ of COVID‐19 epidemic model (8) is investigated. The convergence rate for COVID‐19 epidemic model (8) is studied in Section 6. In detail, bifurcation analysis about equilibrium solutions of discrete COVID‐19 epidemic model (8) is given in Section 7. Section 8 is about the study of chaos control of COVID‐19 epidemic model (8). Theoretical results are numerically verified in Section 9. The concluding remarks are given in Section 10.

2. EXISTENCE OF EQUILIBRIUM SOLUTIONS OF DISCRETE COVID‐19 EPIDEMIC MODEL (8)

In ${ℝ}_{+}^4$, existence of equilibrium solutions of COVID‐19 epidemic model (8) is studied in this section.

Lemma 2.1

For existence results regarding equilibrium solutions of discrete COVID‐19 epidemic model (8), following statements hold:

• (i) $\forall h,\beta ,\mathrm{\Lambda },\mu ,\delta ,{\alpha }_1,{\alpha }_2,\gamma ,ϵ>0$, discrete COVID‐19 epidemic model (8) has boundary equilibrium solution $E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)$.
• (ii)
  If $\beta >\frac{\nu \left(\delta +\mu +{\alpha }_1+\gamma \right)}{\mathrm{\Lambda }}$, then discrete COVID‐19 epidemic model (8) has an interior equilibrium solution $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right),$ where

  $$\begin{array}{cc}\hfill \mathcal{S}& =\frac{\delta +\mu +{\alpha }_1+\gamma }{\beta },\mathcal{I}=\frac{\beta \mathrm{\Lambda }-\mu \left(\delta +\mu +{\alpha }_1+\gamma \right)}{\beta \left(\delta +\mu +{\alpha }_1+\gamma \right)},\hfill \\ \hfill \mathcal{Q}& =\frac{\delta \left(\beta \mathrm{\Lambda }-\mu \left(\delta +\mu +{\alpha }_1+\gamma \right)\right)}{\beta \left(ϵ+{\alpha }_2+\mu \right)\left(\delta +\mu +{\alpha }_1+\gamma \right)},\hfill \\ \hfill \mathcal{R}& =\frac{\left(ϵ\delta +\gamma \left(ϵ+{\alpha }_2+\mu \right)\right)\left(\beta \mathrm{\Lambda }-\mu \left(\delta +\mu +{\alpha }_1+\gamma \right)\right)}{\mu \beta \left(ϵ+{\alpha }_2+\mu \right)\left(\delta +\mu +{\alpha }_1+\gamma \right)}.\hfill \end{array}$$ (9)

If equilibrium solution of COVID‐19 epidemic model (8) is $E_{SIQR}(S,I,Q,R)$, then

$$\begin{array}{cc}\hfill S& =h\mathrm{\Lambda }+(1-h\mu )S-h\beta SI,I=(1-h\delta -h\mu -h{\alpha }_1-h\gamma )I+h\beta SI,\hfill \\ \hfill Q& =h\delta I+(1-hϵ-h{\alpha }_2-h\mu )Q,R=h\gamma I+hϵQ+(1-h\mu )R.\hfill \end{array}$$ (10)

It is noted that for $E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)$, algebraic system (10) satisfied identically. Thus, one can obtain that the boundary solution of discrete COVID‐19 epidemic model (8) is $E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)$. For the interior equilibrium solution of discrete COVID‐19 epidemic model (8), one need to solve following algebraic system simultaneously:

$$S(\mu +\beta I)=\mathrm{\Lambda },\delta +\mu +{\alpha }_1+\gamma =\beta S,(ϵ+{\alpha }_2+\mu )Q=\delta I,\mu R=\gamma I+ϵQ.$$ (11)

The second equation of (11) yields

$$S=\frac{\delta +\mu +{\alpha }_1+\gamma }{\beta }.$$ (12)

Using (12) in the first equation of (11), one has

$$I=\frac{\beta \mathrm{\Lambda }-\mu \left(\delta +\mu +{\alpha }_1+\gamma \right)}{\beta \left(\delta +\mu +{\alpha }_1+\gamma \right)}.$$ (13)

Using (13) in the third equation of (11), one has

$$Q=\frac{\delta \left(\beta \mathrm{\Lambda }-\mu (\delta +\mu +{\alpha }_1+\gamma )\right)}{\beta \left(ϵ+{\alpha }_2+\mu \right)\left(\delta +\mu +{\alpha }_1+\gamma \right)}.$$ (14)

Using (13) and (14) in the fourth equation of (11), one obtains

$$R=\frac{\left(\beta \mathrm{\Lambda }-\mu (\delta +\mu +{\alpha }_1+\gamma )\right)\left(\gamma (ϵ+{\alpha }_2+\mu )+ϵ\delta \right)}{\mu \beta (ϵ+{\alpha }_2+\mu )(\delta +\mu +{\alpha }_1+\gamma )}.$$ (15)

Finally, from (12)–(15), it can be concluded that if $\beta >\frac{\mu (\delta +\mu +{\alpha }_1+\gamma )}{\mathrm{\Lambda }}$, then interior equilibrium solution of discrete COVID‐19 epidemic model (8) is $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$, where $\mathcal{S}:S=\frac{\delta +\mu +{\alpha }_1+\gamma }{\beta }$,    $\mathcal{I}:I=\frac{\beta \mathrm{\Lambda }-\mu \left(\delta +\mu +{\alpha }_1+\gamma \right)}{\beta \left(\delta +\mu +{\alpha }_1+\gamma \right)}$, $\mathcal{Q}:Q=\frac{\delta \left(\beta \mathrm{\Lambda }-\mu \left(\delta +\mu +{\alpha }_1+\gamma \right)\right)}{\beta \left(ϵ+{\alpha }_2+\mu \right)\left(\delta +\mu +{\alpha }_1+\gamma \right)}$, and $\mathcal{R}:R=\frac{\left(ϵ\delta +\gamma \left(ϵ+{\alpha }_2+\mu \right)\right)\left(\beta \mathrm{\Lambda }-\mu \left(\delta +\mu +{\alpha }_1+\gamma \right)\right)}{\mu \beta \left(ϵ+{\alpha }_2+\mu \right)\left(\delta +\mu +{\alpha }_1+\gamma \right)}$. Moreover, it is important here to mention that if $\beta >\frac{\mu (\delta +\mu +{\alpha }_1+\gamma )}{\mathrm{\Lambda }}$ , that is, $\frac{\beta \mathrm{\Lambda }}{\mu (\delta +\mu +{\alpha }_1+\gamma )}>1$, then discrete COVID‐19 epidemic model (8) has interior equilibrium solution, and hence, basic reproductive number is ${\mathcal{R}}_0=\frac{\beta \mathrm{\Lambda }}{\mu (\delta +\mu +{\alpha }_1+\gamma )}$. □

3. LINEARIZED FORM OF MODEL (8)

In the present section, linearized form of discrete COVID‐19 epidemic model (8) about equilibrium solution $E_{SIQR}(S,I,Q,R)$ is explored. The linearized form of discrete COVID‐19 epidemic model (8) about equilibrium solution $E_{SIQR}(S,I,Q,R)$ under the map

$$\left(F,G,H,K\right)\mapsto (S_{n+1},I_{n+1},Q_{n+1},R_{n+1}),$$ (16)

is

$${\mathrm{\Psi }}_{n+1}=J{|}_{E_{SIQR}(S,I,Q,R)}{\mathrm{\Psi }}_n,$$ (17)

where

$$J{|}_{E_{SIQR}(S,I,Q,R)}=\left(\begin{array}{cccc}1-h\mu -h\beta I& -h\beta S& 0& 0\\ h\beta I& 1-h(\delta +\mu +{\alpha }_1+\gamma )+h\beta S& 0& 0\\ 0& h\delta & 1-h(ϵ+{\alpha }_2+\mu )& 0\\ 0& h\gamma & hϵ& 1-h\mu \end{array}\right),$$ (18)

and

$$\begin{array}{cc}\hfill F& =h\mathrm{\Lambda }+(1-h\mu )S-h\beta SI,G=(1-h\delta -h\mu -h{\alpha }_1-h\gamma )I+h\beta SI,\hfill \\ \hfill H& =h\delta I+(1-hϵ-h{\alpha }_2-h\mu )Q,K=h\gamma I+hϵQ+(1-h\mu )R.\hfill \end{array}$$ (19)

4. LOCAL BEHAVIOR OF DISCRETE COVID‐19 EPIDEMIC MODEL (8) ABOUT EQUILIBRIUM SOLUTIONS

Local behavior of COVID‐19 epidemic model (8) about equilibrium solutions $E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)$ and $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ is explored in this section.

4.1. Local dynamic behavior of COVID‐19 epidemic model (8) about $E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)$

About $E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)$, (18) becomes

$$J{|}_{E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)}=\left(\begin{array}{cccc}1-h\mu & -\frac{h\beta \mathrm{\Lambda }}{\mu }& 0& 0\\ 0& 1-h(\delta +\mu +{\alpha }_1+\gamma )+\frac{h\beta \mathrm{\Lambda }}{\mu }& 0& 0\\ 0& h\delta & 1-h(ϵ+{\alpha }_2+\mu )& 0\\ 0& h\gamma & hϵ& 1-h\mu \end{array}\right),$$ (20)

with characteristic roots are

$${\lambda }_{\mathrm{1,2}}=1-h\mu ,{\lambda }_3=1-h(\delta +\mu +{\alpha }_1+\gamma )+\frac{h\beta \mathrm{\Lambda }}{\mu },{\lambda }_4=1-h(ϵ+{\alpha }_2+\mu ).$$ (21)

From (21) and by stability theory, ⁷ , ⁸ , ⁹ , ¹⁰ , ¹¹ one can conclude local dynamic behavior of discrete COVID‐19 epidemic model (8) about $E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)$ as follows.

Lemma 4.1

For local dynamic behavior of COVID‐19 epidemic model (8) about $E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)$, the following statements hold:

• (i)
  $E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)$ of discrete COVID‐19 model (8) is a sink if

  $$0<\mu <\min \left\{\frac{2}{h},\frac{2}{h}-ϵ-{\alpha }_2\right\}\text{and}\frac{h\beta \mathrm{\Lambda }}{h\beta \mathrm{\Lambda }+2\mu }<{\mathcal{R}}_0<1,$$ (22)

  with

  $$\frac{2}{\delta +\mu +{\alpha }_1+\gamma }<h<\frac{2}{ϵ+{\alpha }_2}.$$ (23)
• (ii)
  $E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)$ of discrete COVID‐19 model (8) is a source if (23) holds and additionally

  $$\mu >\max \left\{\frac{2}{h},\frac{2}{h}-ϵ-{\alpha }_2\right\}\text{and}{\mathcal{R}}_0<\frac{h\beta \mathrm{\Lambda }}{h\beta \mathrm{\Lambda }+2\mu }.$$ (24)
• (iii)
  $E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)$ of discrete COVID‐19 model (8) is a saddle if (23) holds and additionally

  $$\mu >\max \left\{\frac{2}{h},\frac{2}{h}-ϵ-{\alpha }_2\right\}\text{and}\frac{h\beta \mathrm{\Lambda }}{h\beta \mathrm{\Lambda }+2\mu }<{\mathcal{R}}_0<1,$$ (25)

  or

  $$\max \left\{0,\frac{2}{h}-ϵ-{\alpha }_2\right\}<\mu <\frac{2}{h}\text{and}\frac{h\beta \mathrm{\Lambda }}{h\beta \mathrm{\Lambda }+2\mu }<{\mathcal{R}}_0<1,$$ (26)

  or

  $$0<\mu <\min \left\{\frac{2}{h},\frac{2}{h}-ϵ-{\alpha }_2\right\}\text{and}{\mathcal{R}}_0<\frac{h\beta \mathrm{\Lambda }}{h\beta \mathrm{\Lambda }+2\mu },$$ (27)

  or

  $$\max \left\{0,\frac{2}{h}-ϵ-{\alpha }_2\right\}<\mu <\frac{2}{h}\text{and}{\mathcal{R}}_0<\frac{h\beta \mathrm{\Lambda }}{h\beta \mathrm{\Lambda }+2\mu }.$$ (28)
• (iv)
  $E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)$ of discrete COVID‐19 model (8) is non‐hyperbolic if

  $$\mu =\frac{2}{h},$$ (29)

  or

  $${\mathcal{R}}_0=\frac{h\beta \mathrm{\Lambda }}{h\beta \mathrm{\Lambda }+2\mu },$$ (30)

  or

  $$\mu =\frac{2}{h}-ϵ-{\alpha }_2.$$ (31)

4.2. Local dynamic behavior of discrete COVID‐19 epidemic model (8) about interior equilibrium solution $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$

In order to find local dynamic behavior of discrete COVID‐19 epidemic model (8) about interior equilibrium solution $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$, the following theorem is utilized which shows the fact that all roots of the characteristic equation of $J{|}_{E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)}$ whose absolute value less than one (see Theorem 1.2.4 of Camouzis and Ladas ¹¹ ).

Theorem 4.2

The necessary and sufficient conditions for roots of the following fourth‐degree polynomial

$$P(\lambda )={\lambda }^4+{\mathcal{H}}_1{\lambda }^3+{\mathcal{H}}_2{\lambda }^2+{\mathcal{H}}_3\lambda +{\mathcal{H}}_4,$$ (32)

satisfying $\left|{\lambda }_{\mathrm{1,2},\mathrm{3,4}}\right|<1$ are

$$\begin{array}{c}\hfill \left|{\mathcal{H}}_1+{\mathcal{H}}_3\right|<1+{\mathcal{H}}_2+{\mathcal{H}}_4,\left|{\mathcal{H}}_3-{\mathcal{H}}_1\right|<2(1-{\mathcal{H}}_4),{\mathcal{H}}_2-3{\mathcal{H}}_4<3,\\ \hfill {\mathcal{H}}_2+{{\mathcal{H}}_4}^2(1+{\mathcal{H}}_2)+{{\mathcal{H}}_3}^2+{\mathcal{H}}_4(1+{{\mathcal{H}}_1}^2)<1+2{\mathcal{H}}_2{\mathcal{H}}_4+{\mathcal{H}}_1{\mathcal{H}}_3(1+{\mathcal{H}}_4)+{{\mathcal{H}}_4}^3.\end{array}$$ (33)

Lemma 4.3

$E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ of discrete COVID‐19 epidemic model (8) is stable if

$$\begin{array}{c}\hfill \left|{\mathcal{H}}_1+{\mathcal{H}}_3\right|<1+{\mathcal{H}}_2+{\mathcal{H}}_4,\left|{\mathcal{H}}_3-{\mathcal{H}}_1\right|<2(1-{\mathcal{H}}_4),{\mathcal{H}}_2-3{\mathcal{H}}_4<3,\\ \hfill {\mathcal{H}}_2+{{\mathcal{H}}_4}^2(1+{\mathcal{H}}_2)+{{\mathcal{H}}_3}^2+{\mathcal{H}}_4(1+{{\mathcal{H}}_1}^2)<1+2{\mathcal{H}}_2{\mathcal{H}}_4+{\mathcal{H}}_1{\mathcal{H}}_3(1+{\mathcal{H}}_4)+{{\mathcal{H}}_4}^3,\end{array}$$ (34)

where

$$\begin{array}{cc}\hfill {\mathcal{H}}_1=& -4+h\left({\alpha }_2+ϵ+2\mu +\mu {\mathcal{R}}_0\right),\hfill \\ \hfill {\mathcal{H}}_2=& 6+h\left[-3\left({\alpha }_2-h\beta \mathrm{\Lambda }+ϵ\right)-\mu \left\{6+h({\alpha }_1-{\alpha }_2+\gamma +\delta -ϵ)\right\}\right.\hfill \\ \hfill & \left.+\mu {\mathcal{R}}_0\left\{-3+h\left(-2{\alpha }_1+{\alpha }_2-2\left(\gamma +\delta \right)+ϵ\right)\right\}\right],\hfill \\ \hfill {\mathcal{H}}_3=& -4+3h\left({\alpha }_2+ϵ+2\mu +\mu {\mathcal{R}}_0\right)\hfill \\ \hfill & +\frac{2h^2}{\delta +\mu +{\alpha }_1+\gamma }\left[\mu \left(\delta +\mu +{\alpha }_1+\gamma \right)\left({\alpha }_1-{\alpha }_2+\gamma +\delta -ϵ\right)\right.\hfill \\ \hfill & \left.-\beta \mathrm{\Lambda }\left\{{\alpha }_1+{\alpha }_2+\gamma +\delta +ϵ+3\mu \right\}\right]\hfill \\ \hfill & +\frac{h^3}{\delta +\mu +{\alpha }_1+\gamma }\left[-\mu \left({\alpha }_2+ϵ+2\mu \right){\left(\delta +\mu +{\alpha }_1+\gamma \right)}^2\right.\hfill \\ \hfill & \left.+\beta \mathrm{\Lambda }\left\{\left({\alpha }_1+\gamma +\delta \right)\left({\alpha }_2+ϵ\right)+2\mu \left({\alpha }_1+{\alpha }_2+\gamma +\delta +ϵ\right)+3{\mu }^2\right\}\right],\hfill \\ \hfill {\mathcal{H}}_4=& \frac{(1-h\mu )\left\{-1+h({\alpha }_2+ϵ+\mu )\right\}}{\delta +\mu +{\alpha }_1+\gamma }\left[h\beta \mathrm{\Lambda }-\gamma -\delta -\mu +h^2{\alpha }_1^2\mu \right.\hfill \\ \hfill & +h^2\left(\gamma +\delta +\mu \right)\left\{-\beta \mathrm{\Lambda }+\mu \left(\gamma +\delta +\mu \right)\right\}\hfill \\ \hfill & \left.+{\alpha }_1\left\{-1+h^2\left(-\beta \mathrm{\Lambda }+2\mu \left(\gamma +\delta +\mu \right)\right)\right\}\right].\hfill \end{array}$$ (35)

About $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$, (18) becomes

$$\begin{array}{c}\hfill J{|}_{E_{SIQR}^{+}}=\left(\begin{array}{cccc}1-h\mu {\mathcal{R}}_0& -h(\delta +\mu +{\alpha }_1+\gamma )& 0& 0\\ h\mu {\mathcal{R}}_0-h\mu & 1& 0& 0\\ 0& h\delta & 1-hϵ-h{\alpha }_2-h\mu & 0\\ 0& h\gamma & hϵ& 1-h\mu \end{array}\right).\end{array}$$ (36)

The characteristic polynomial of $J{|}_{E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)}$ about interior equilibrium solution $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ of discrete COVID‐19 epidemic model (8) is

$$P(\lambda )={\lambda }^4+{\mathcal{H}}_1{\lambda }^3+{\mathcal{H}}_2{\lambda }^2+{\mathcal{H}}_3\lambda +{\mathcal{H}}_4,$$ (37)

where ${\mathcal{H}}_1,{\mathcal{H}}_2$, ${\mathcal{H}}_3$, and ${\mathcal{H}}_4$ are depicted in (35). Now, Theorem 4.2 implies that interior equilibrium solution $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ of discrete COVID‐19 epidemic model (8) is a sink if $\left|{\mathcal{H}}_1+{\mathcal{H}}_3\right|<1+{\mathcal{H}}_2+{\mathcal{H}}_4,\left|{\mathcal{H}}_3-{\mathcal{H}}_1\right|<2(1-{\mathcal{H}}_4),{\mathcal{H}}_2-3{\mathcal{H}}_4<3,{\mathcal{H}}_2+{{\mathcal{H}}_4}^2(1+{\mathcal{H}}_2)+{{\mathcal{H}}_3}^2+{\mathcal{H}}_4(1+{{\mathcal{H}}_1}^2)<1+2{\mathcal{H}}_2{\mathcal{H}}_4+{\mathcal{H}}_1{\mathcal{H}}_3(1+{\mathcal{H}}_4)+{{\mathcal{H}}_4}^3$. □

5. PERIODIC POINTS WITH PERIOD‐ $n$ OF DISCRETE COVID‐19 EPIDEMIC MODEL (8)

Motivated from the work of Zhang, ¹² it is explored that equilibrium solutions $E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)$ and $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ of discrete COVID‐19 epidemic model (8) are periodic points with period‐ $n$ in this section.

Theorem 5.1

Equilibrium solutions $E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)$ and $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ of discrete COVID‐19 epidemic model (8) are periodic points of prime period‐1.

From (8), one denotes

$$\mathcal{P}\left(S,I,Q,R\right):=\left(F(S,I,Q,R),G(S,I,Q,R),H(S,I,Q,R),K(S,I,Q,R)\right),$$ (38)

where $F(S,I,Q,R),G(S,I,Q,R),H(S,I,Q,R)$, and $K(S,I,Q,R)$ are depicted in (19). From (38), the computation yields

$$\begin{array}{cc}\hfill \mathcal{P}{|}_{E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)}& =E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right),\hfill \\ \hfill \mathcal{P}{|}_{E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)}& =E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right).\hfill \end{array}$$ (39)

Therefore, Equation (39) implies that $E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)$ and $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ of discrete COVID‐19 epidemic model (8) are periodic points of prime period‐1. □

Theorem 5.2

Equilibrium solution $E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)$ of discrete COVID‐19 epidemic model (8) is a periodic point of period‐ $n$.

From (38), we have

$$\begin{array}{cc}\hfill {\mathcal{P}}^2=& \left(h\mathrm{\Lambda }+\left(1-h\mu \right)F(S,I,Q,R)\right.\hfill \\ \hfill & -h\beta F(S,I,Q,R)G(S,I,Q,R),\hfill \\ \hfill & (1-h\delta -h\mu -h{\alpha }_1-h\gamma )G(S,I,Q,R)\hfill \\ \hfill & +h\beta F(S,I,Q,R)G(S,I,Q,R),\hfill \\ \hfill & h\delta G(S,I,Q,R)\hfill \\ \hfill & +(1-hϵ-h{\alpha }_2-h\mu )H(S,I,Q,R),\hfill \\ \hfill & h\gamma G(S,I,Q,R)+hϵH(S,I,Q,R)\hfill \\ \hfill & \left.+(1-h\mu )K(S,I,Q,R)\right)\Rightarrow {\mathcal{P}}^2{|}_{E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)}=E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right),\hfill \end{array}$$ (40)

$$\begin{array}{cc}\hfill {\mathcal{P}}^3& =\left(h\mathrm{\Lambda }+\left(1-h\mu \right)F^2(S,I,Q,R)\right.\hfill \\ \hfill & -h\beta F^2(S,I,Q,R)G^2(S,I,Q,R),\hfill \\ \hfill & (1-h\delta -h\mu -h{\alpha }_1-h\gamma )G^2(S,I,Q,R)\hfill \\ \hfill & +h\beta F^2(S,I,Q,R)G^2(S,I,Q,R),\hfill \\ \hfill & h\delta G^2(S,I,Q,R)\hfill \\ \hfill & +(1-hϵ-h{\alpha }_2-h\mu )H^2(S,I,Q,R),\hfill \\ \hfill & h\gamma G^2(S,I,Q,R)+hϵH^2(S,I,Q,R)\hfill \\ \hfill & \left.+(1-h\mu )K^2(S,I,Q,R)\right)\Rightarrow {\mathcal{P}}^3{|}_{E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)}=E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right),\hfill \\ \hfill ⋮& \hfill \\ \hfill {\mathcal{P}}^n& =\left(h\mathrm{\Lambda }+\left(1-h\mu \right)F^{n-1}(S,I,Q,R)\right.\hfill \\ \hfill & -h\beta F^{n-1}(S,I,Q,R)G^{n-1}(S,I,Q,R),\hfill \\ \hfill & (1-h\delta -h\mu -h{\alpha }_1-h\gamma )G^{n-1}(S,I,Q,R)\hfill \\ \hfill & +h\beta F^{n-1}(S,I,Q,R)G^{n-1}(S,I,Q,R),\hfill \\ \hfill & h\delta G^{n-1}(S,I,Q,R)\hfill \\ \hfill & +(1-hϵ-h{\alpha }_2-h\mu )H^{n-1}(S,I,Q,R),\hfill \\ \hfill & h\gamma G^{n-1}(S,I,Q,R)+hϵH^{n-1}(S,I,Q,R)\hfill \\ \hfill & \left.+(1-h\mu )K^{n-1}(S,I,Q,R)\right)\Rightarrow {\mathcal{P}}^n{|}_{E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)}=E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right).\hfill \end{array}$$ (41)

Equations (40) and (41) imply that equilibrium solution $E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)$ of discrete COVID‐19 epidemic model (8) is a periodic point of period‐ $n$. □

Theorem 5.3

Equilibrium solution $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ of discrete COVID‐19 epidemic model (8) is a periodic point of period‐ $n$.

From (40) and (41), the following computation yields the required statement:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{P}}^2{|}_{E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)}& =E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right),\hfill \\ \hfill {\mathcal{P}}^3{|}_{E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)}& =E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right),\hfill \\ \hfill ⋮& \hfill \\ \hfill {\mathcal{P}}^n{|}_{E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)}& =E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right).\hfill \end{array}\end{array}$$

 □

6. CONVERGENCE RATE OF DISCRETE COVID‐19 EPIDEMIC MODEL (8)

Convergence rate of discrete COVID‐19 epidemic model (8) is studied for the completeness of this section.

Theorem 6.1

If $\{(S_n,I_n,Q_n,R_n)\}$ is a positive solution of discrete COVID‐19 epidemic model (8) such that $\underset{n\to \infty }{\lim }\{(S_n,I_n,Q_n,R_n)\}=E_{SIQR}(S,I,Q,R)$, then

$${\phi }_n=\left(\begin{array}{c}{\phi }_n^1\\ {\phi }_n^2\\ {\phi }_n^3\\ {\phi }_n^4\end{array}\right),$$ (42)

satisfying the following mathematical relation:

$$\begin{array}{ccc}\hfill \underset{n\to \infty }{\lim }\sqrt[n]{||{\phi }_n||}& =\hfill & \hfill \left|{\lambda }_{\mathrm{1,2},\mathrm{3,4}}J{|}_{E_{SIQR}(S,I,Q,R)}\right|,\\ \hfill \underset{n\to \infty }{\lim }\frac{||{\phi }_{n+1}||}{||{\phi }_n||}& =\hfill & \hfill \left|{\lambda }_{\mathrm{1,2},\mathrm{3,4}}J{|}_{E_{SIQR}(S,I,Q,R)}\right|.\end{array}$$ (43)

It is recalled that if $\{(S_n,I_n,Q_n,R_n)\}$ is a positive solution of COVID‐19 epidemic model (8) such that $\underset{n\to \infty }{\lim }\{(S_n,I_n,Q_n,R_n)\}=E_{SIQR}(S,I,Q,R)$, then for error terms one has

$$\begin{array}{cc}\hfill S_{n+1}-S& =\left(1-h\mu -h\beta I\right)\left(S_n-S\right)-h\beta S_n\left(I_n-I\right),\hfill \\ \hfill I_{n+1}-I& =h\beta I\left(S_n-S\right)+\left(1-h\left(\delta +\mu +{\alpha }_1+\gamma \right)+h\beta S_n\right)\left(I_n-I\right),\hfill \\ \hfill Q_{n+1}-Q& =h\delta \left(I_n-I\right)+\left(1-h\left(ϵ+{\alpha }_2+\mu \right)\right)\left(Q_n-Q\right),\hfill \\ \hfill R_{n+1}-R& =h\gamma \left(I_n-I\right)+hϵ\left(Q_n-Q\right)+\left(1-h\mu \right)\left(R_n-R\right).\hfill \end{array}$$ (44)

Set

$${\phi }_n^1=S_n-S,{\phi }_n^2=I_n-I,{\phi }_n^3=Q_n-Q,{\phi }_n^4=R_n-R.$$ (45)

In view of (45), (44) becomes

$$\begin{array}{cc}\hfill {\phi }_{n+1}^1& ={\alpha }_{11}{\phi }_n^1+{\alpha }_{12}{\phi }_n^2,\hfill \\ \hfill {\phi }_{n+1}^2& ={\alpha }_{21}{\phi }_n^1+{\alpha }_{22}{\phi }_n^2,\hfill \\ \hfill {\phi }_{n+1}^3& ={\alpha }_{32}{\phi }_n^2+{\alpha }_{33}{\phi }_n^3,\hfill \\ \hfill {\phi }_{n+1}^4& ={\alpha }_{42}{\phi }_n^2+{\alpha }_{43}{\phi }_n^3+{\alpha }_{44}{\phi }_n^4,\hfill \end{array}$$ (46)

where

$$\begin{array}{cc}\hfill {\alpha }_{11}& =1-h\mu -h\beta I,\hfill \\ \hfill {\alpha }_{12}& =-h\beta S_n,\hfill \\ \hfill {\alpha }_{21}& =h\beta I,\hfill \\ \hfill {\alpha }_{22}& =1-h\left(\delta +\mu +{\alpha }_1+\gamma \right)+h\beta S_n,\hfill \\ \hfill {\alpha }_{32}& =h\delta ,\hfill \\ \hfill {\alpha }_{33}& =1-h\left(ϵ+{\alpha }_2+\mu \right),\hfill \\ \hfill {\alpha }_{42}& =h\gamma ,\hfill \\ \hfill {\alpha }_{43}& =hϵ,\hfill \\ \hfill {\alpha }_{44}& =1-h\mu .\hfill \end{array}$$ (47)

From (47), one has

$$\begin{array}{cc}\hfill \underset{n\to \infty }{\lim }{\alpha }_{11}& =1-h\mu -h\beta I,\hfill \\ \hfill \underset{n\to \infty }{\lim }{\alpha }_{12}& =-h\beta S,\hfill \\ \hfill \underset{n\to \infty }{\lim }{\alpha }_{21}& =h\beta I,\hfill \\ \hfill \underset{n\to \infty }{\lim }{\alpha }_{22}& =1-h\left(\delta +\mu +{\alpha }_1+\gamma \right)+h\beta S,\hfill \\ \hfill \underset{n\to \infty }{\lim }{\alpha }_{32}& =h\delta ,\hfill \\ \hfill \underset{n\to \infty }{\lim }{\alpha }_{33}& =1-h\left(ϵ+{\alpha }_2+\mu \right),\hfill \\ \hfill \underset{n\to \infty }{\lim }{\alpha }_{42}& =h\gamma ,\hfill \\ \hfill \underset{n\to \infty }{\lim }{\alpha }_{43}& =hϵ,\hfill \\ \hfill \underset{n\to \infty }{\lim }{\alpha }_{44}& =1-h\mu ,\hfill \end{array}$$ (48)

that is,

$$\begin{array}{cc}\hfill {\alpha }_{11}& =1-h\mu -h\beta I+{\sigma }_{11},\hfill \\ \hfill {\alpha }_{12}& =-h\beta S+{\sigma }_{12},\hfill \\ \hfill {\alpha }_{21}& =h\beta I+{\sigma }_{21},\hfill \\ \hfill {\alpha }_{22}& =1-h\left(\delta +\mu +{\alpha }_1+\gamma \right)+h\beta S+{\sigma }_{22},\hfill \\ \hfill {\alpha }_{32}& =h\delta +{\sigma }_{32},\hfill \\ \hfill {\alpha }_{33}& =1-h\left(ϵ+{\alpha }_2+\mu \right)+{\sigma }_{33},\hfill \\ \hfill {\alpha }_{42}& =h\gamma +{\sigma }_{42},\hfill \\ \hfill {\alpha }_{43}& =hϵ+{\sigma }_{43},\hfill \\ \hfill {\alpha }_{44}& =1-h\mu +{\sigma }_{44},\hfill \end{array}$$ (49)

where ${\sigma }_{11},{\sigma }_{12},{\sigma }_{21},{\sigma }_{22},{\sigma }_{32},{\sigma }_{33},{\sigma }_{42},{\sigma }_{43},{\sigma }_{44}\to 0$ as $n\to \infty$. In view of existing literature (see Pituk ¹³ ), one has the following error system:

$${\phi }_{n+1}=(A+B_n){\phi }_n,$$ (50)

where $A=J{|}_{E_{SIQR}(S,I,Q,R)}$ and $B_n=\left(\begin{array}{cccc}{\sigma }_{11}& {\sigma }_{12}& 0& 0\\ {\sigma }_{21}& {\sigma }_{22}& 0& 0\\ 0& {\sigma }_{32}& {\sigma }_{33}& 0\\ 0& {\sigma }_{42}& {\sigma }_{43}& {\sigma }_{44}\end{array}\right)$. Therefore, one has the following limiting system of error terms:

$$\left(\begin{array}{c}{\phi }_{n+1}^1\\ {\phi }_{n+1}^2\\ {\phi }_{n+1}^3\\ {\phi }_{n+1}^4\end{array}\right)=\left(\begin{array}{cccc}1-h\mu -h\beta I& -h\beta S& 0& 0\\ h\beta I& 1-h(\delta +\mu +{\alpha }_1+\gamma )+h\beta S& 0& 0\\ 0& h\delta & 1-h(ϵ+{\alpha }_2+\mu )& 0\\ 0& h\gamma & hϵ& 1-h\mu \end{array}\right)\left(\begin{array}{c}{\phi }_n^1\\ {\phi }_n^2\\ {\phi }_n^3\\ {\phi }_n^4\end{array}\right),$$ (51)

which is the same as linearized system of discrete COVID‐19 epidemic model (8) about $E_{SIQR}(S,I,Q,R)$. Particularly, (51) implies that

$$\left(\begin{array}{c}{\phi }_{n+1}^1\\ {\phi }_{n+1}^2\\ {\phi }_{n+1}^3\\ {\phi }_{n+1}^4\end{array}\right)=\left(\begin{array}{cccc}1-h\mu & -\frac{h\beta \mathrm{\Lambda }}{\mu }& 0& 0\\ 0& 1-h(\delta +\mu +{\alpha }_1+\gamma )+\frac{h\beta \mathrm{\Lambda }}{\mu }& 0& 0\\ 0& h\delta & 1-h(ϵ+{\alpha }_2+\mu )& 0\\ 0& h\gamma & hϵ& 1-h\mu \end{array}\right)\left(\begin{array}{c}{\phi }_n^1\\ {\phi }_n^2\\ {\phi }_n^3\\ {\phi }_n^4\end{array}\right),$$ (52)

and

$$\left(\begin{array}{c}{\phi }_{n+1}^1\\ {\phi }_{n+1}^2\\ {\phi }_{n+1}^3\\ {\phi }_{n+1}^4\end{array}\right)=\left(\begin{array}{cccc}1-h\mu {\mathcal{R}}_0& -h(\delta +\mu +{\alpha }_1+\gamma )& 0& 0\\ h\mu ({\mathcal{R}}_0-1)& 1& 0& 0\\ 0& h\delta & 1-h\left(ϵ+{\alpha }_2+\mu \right)& 0\\ 0& h\gamma & hϵ& 1-h\mu \end{array}\right)\left(\begin{array}{c}{\phi }_n^1\\ {\phi }_n^2\\ {\phi }_n^3\\ {\phi }_n^4\end{array}\right),$$ (53)

which are the same as respective linearized system obtained at equilibrium solution $E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)$ and $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ of the discrete COVID‐19 epidemic model (8). □

7. BIFURCATIONS OF DISCRETE COVID‐19 EPIDEMIC MODEL (8)

The bifurcation analysis about $E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)$ and $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ of the discrete COVID‐19 epidemic model (8) are explored deeply in the section by bifurcation theory. ¹⁴ , ¹⁵ Actually, the bifurcation means small change in the variation of single parameter, namely, bifurcation parameter, drastically change in the behavior of solution of the system. In general, if one considers the following smooth parameter‐dependence map:

$$S\mapsto f(S,h),$$ (54)

where $S\in {ℝ}^n$. If $S=S_0$ is hyperbolic fixed point for $h=h_0$, then generally, there are the following three ways of violation for the hyperbolicity condition: $(i)+ve$ eigenvalues through the unit circle, $(ii)-ve$ eigenvalues through the unit circle, and $(iii)$. Pair of complex eigenvalues approach the unit circle. But our theoretical investigation for under‐consideration model reveals that there exist two types of bifurcations, namely, flip and Hopf bifurcations, whose basic definitions are stated below:

Definition 7.1

Bifurcation related to the existence of ${\lambda }_1=-1$ is known as flip bifurcation.

Definition 7.2

Bifurcation related to the existence of ${\lambda }_{\mathrm{1,2}}=e^{\pm \iota \theta }$ is known as Hopf bifurcation.

7.1. Bifurcation analysis about $E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)$

From (21), the computation yields ${\lambda }_{\mathrm{1,2}}{|}_{(29)}=-1$ but ${\lambda }_{\mathrm{3,4}}{|}_{(29)}=-1-h\left(\delta +{\alpha }_2+\gamma \right)+\frac{h^2\beta \mathrm{\Lambda }}{2},-1-hϵ-h{\alpha }_2\ne 1$ or −1, which implies that discrete COVID‐19 epidemic model (8) may undergo flip bifurcation if $(h,\beta ,\mathrm{\Lambda },\mu ,\delta ,{\alpha }_1,{\alpha }_2,\gamma ,ϵ)$ located in the set:

$$\mathcal{F}{|}_{E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)}=\left\{(h,\beta ,\mathrm{\Lambda },\mu ,\delta ,{\alpha }_1,{\alpha }_2,\gamma ,ϵ):\mu =\frac{2}{h}\right\}.$$ (55)

The following theorem guarantees the fact that if $(h,\beta ,\mathrm{\Lambda },\mu ,\delta ,{\alpha }_1,{\alpha }_2,\gamma ,ϵ)\in \mathcal{F}{|}_{E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)}$, then discrete COVID‐19 epidemic model (8) does not undergo flip bifurcation.

Theorem 7.3

If $(h,\beta ,\mathrm{\Lambda },\mu ,\delta ,{\alpha }_1,{\alpha }_2,\gamma ,ϵ)\in \mathcal{F}{|}_{E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)}$, then discrete COVID‐19 epidemic model (8) does not undergo flip bifurcation.

Since discrete COVID‐19 epidemic model (8) is invariant with respect to $I=Q=R=0$. Therefore, in order to determine bifurcation, discrete COVID‐19 epidemic model (8) is restricted on $I=Q=R=0$, where it becomes

$$S_{n+1}=h\mathrm{\Lambda }+(1-h\mu )S_n.$$ (56)

From (56), one denotes the map

$$f(S):=h\mathrm{\Lambda }+(1-h\mu )S.$$ (57)

Now, if $\mu ={\mu }^{\ast }=\frac{2}{h}$ and $S=S^{\ast }=\frac{\mathrm{\Lambda }}{\mu }$, then from (57), one gets

$${\left.\frac{\partial f}{\partial S}\right|}_{\mu ={\mu }^{\ast }=\frac{2}{h},S=S^{\ast }=\frac{\mathrm{\Lambda }}{\mu }}=-1,$$ (58)

$${\left.\frac{\partial f}{\partial \mu }\right|}_{\mu ={\mu }^{\ast }=\frac{2}{h},S=S^{\ast }=\frac{\mathrm{\Lambda }}{\mu }}=-\frac{h\mathrm{\Lambda }}{\mu }\ne 0,$$ (59)

and

$${\left.\frac{{\partial }^{2}f}{\partial S^2}\right|}_{\mu ={\mu }^{\ast }=\frac{2}{h},S=S^{\ast }=\frac{\mathrm{\Lambda }}{\mu }}=0.$$ (60)

It is noted that the condition obtained in (60) violates the non‐degenerate condition for the existence of flip bifurcation and hence one can say that discrete COVID‐19 epidemic model (8) does not undergo flip bifurcation if $(h,\beta ,\mathrm{\Lambda },\mu ,\delta ,{\alpha }_1,{\alpha }_2,\gamma ,ϵ)\in \mathcal{F}{|}_{E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)}$.

7.2. Bifurcation analysis about $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$

We will explore Hopf and flip bifurcations by choosing $h$ as a bifurcation parameter about interior equilibrium solution $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ of discrete COVID‐19 epidemic model (8) by utilizing explicit criterion in this section.

7.2.1. Hopf Bifurcation about $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$

By using following explicit criterion, ¹⁶ Hopf bifurcation for the discrete COVID‐19 epidemic model (8) about $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ is explored.

Lemma 7.4

Consider the following $n$‐dimensional discrete dynamical system:

$$X_{n+1}=f_h(X_n),$$ (61)

where $h\in ℝ$ is considered as a bifurcation parameter. Moreover, characteristic polynomial of $J{|}_X$ about $X$ of $n$‐dimensional discrete dynamical system, which is depicted in (61), is

$$P(\lambda )={\lambda }^n+{\mathcal{H}}_1{\lambda }^{n-1}+{\mathcal{H}}_2{\lambda }^{n-2}+\text{⋯}+{\mathcal{H}}_n.$$ (62)

Now, considering the determinants: ${\mathrm{\Delta }}_0^{\pm }(h)=1$, ${\mathrm{\Delta }}_1^{\pm }(h),\text{⋯},{\mathrm{\Delta }}_n^{\pm }(h)$, which can be expressed as

$$\begin{array}{c}\hfill {\mathrm{\Delta }}_j^{\pm }(h)=\left|\left(\begin{array}{ccccc}1& {\mathcal{H}}_1& {\mathcal{H}}_2& \text{⋯}& {\mathcal{H}}_{j-1}\\ 0& 1& {\mathcal{H}}_1& \text{⋯}& {\mathcal{H}}_{j-2}\\ 0& 0& 1& \text{⋯}& {\mathcal{H}}_{j-3}\\ \text{⋯}& \text{⋯}& \text{⋯}& \text{⋯}& \text{⋯}\\ 0& 0& 0& \text{⋯}& 1\end{array}\right)\pm \left(\begin{array}{ccccc}{\mathcal{H}}_{n-j+1}& {\mathcal{H}}_{n-j+2}& \text{⋯}& {\mathcal{H}}_{n-1}& {\mathcal{H}}_n\\ {\mathcal{H}}_{n-j+2}& {\mathcal{H}}_{n-j+3}& \text{⋯}& {\mathcal{H}}_n& 0\\ \text{⋯}& \text{⋯}& \text{⋯}& \text{⋯}& \text{⋯}\\ {\mathcal{H}}_{n-1}& {\mathcal{H}}_n& \text{⋯}& 0& 0\\ {\mathcal{H}}_n& 0& \text{⋯}& 0& 0\end{array}\right)\right|\end{array},$$ (63)

where $j=1,\text{⋯},n$. Furthermore, Hopf bifurcation occurs at critical value $h=h_0$ if the following parametric conditions hold:

• ${\mathrm{\Gamma }}_1:$ Eigenvalue assignment: $P_{h_0}(1)>0,{(-1)}^n{P}_{h_0}(-1)>0,{\mathrm{\Delta }}_{n-1}^{-}(h_0)=0,{\mathrm{\Delta }}_{n-1}^{+}(h_0)>0,{\mathrm{\Delta }}_j^{\pm }(h_0)>0$ where $j=n-3,n-5,\text{⋯},1$ (or 2), when $n$ is even (or odd, respectively).
• ${\mathrm{\Gamma }}_2:$ Transversality condition: $\frac{d}{dh}{\mathrm{\Delta }}_{n-1}^{-}(h_0)\ne 0$.
• ${\mathrm{\Gamma }}_3:$ Nonresonance condition: $\cos \left(\frac{2\pi }{l}\right)\ne 1-0.5P_h(1)\frac{{\mathrm{\Delta }}_{n-3}^{-}(h_0)}{{\mathrm{\Delta }}_{n-2}^{+}(h_0)}$ or resonance condition $\cos \left(\frac{2\pi }{l}\right)=1-0.5P_h(1)\frac{{\mathrm{\Delta }}_{n-3}^{-}(h_0)}{{\mathrm{\Delta }}_{n-2}^{+}(h_0)}$, where $l=\mathrm{3,4},\text{⋯}$.

Theorem 7.5

If

$$\begin{array}{cc}\hfill 1-{\mathcal{H}}_4-{\mathcal{H}}_2-{{\mathcal{H}}_3}^2+{{\mathcal{H}}_4}^3-{{\mathcal{H}}_4}^2\left(1+{\mathcal{H}}_2\right)& \hfill \\ \hfill +2{\mathcal{H}}_2{\mathcal{H}}_4-{{\mathcal{H}}_1}^2{\mathcal{H}}_4+{\mathcal{H}}_1{\mathcal{H}}_3\left(1+{\mathcal{H}}_4\right)=& 0,\hfill \\ \hfill 1+{\mathcal{H}}_2+{\mathcal{H}}_4-{{\mathcal{H}}_3}^2-{{\mathcal{H}}_4}^3-{{\mathcal{H}}_4}^2\left(1+{\mathcal{H}}_2\right)& \hfill \\ \hfill +{{\mathcal{H}}_1}^2{\mathcal{H}}_4-{\mathcal{H}}_1{\mathcal{H}}_3\left(1-{\mathcal{H}}_4\right)>& 0,\hfill \\ \hfill 1+{\mathcal{H}}_1+{\mathcal{H}}_2+{\mathcal{H}}_3+{\mathcal{H}}_4>& 0,\hfill \\ \hfill 1-{\mathcal{H}}_1+{\mathcal{H}}_2-{\mathcal{H}}_3+{\mathcal{H}}_4>& 0,\hfill \\ \hfill 1\pm {\mathcal{H}}_4>& 0,\hfill \\ \hfill \frac{d}{dh}\left(1-{\mathcal{H}}_4-{\mathcal{H}}_2-{{\mathcal{H}}_3}^2+{{\mathcal{H}}_4}^3-{{\mathcal{H}}_4}^2\left(1+{\mathcal{H}}_2\right)\right.& \hfill \\ \hfill \left.+2{\mathcal{H}}_2{\mathcal{H}}_4-{{\mathcal{H}}_1}^2{\mathcal{H}}_4+{\mathcal{H}}_1{\mathcal{H}}_3\left(1+{\mathcal{H}}_4\right)\right){|}_{h=h_0}\ne & 0,\hfill \\ \hfill \cos \frac{2\pi }{l}\ne 1-\frac{\left(1-{\mathcal{H}}_4\right)\left(1+{\mathcal{H}}_1+{\mathcal{H}}_2+{\mathcal{H}}_3+{\mathcal{H}}_4\right)}{2\left(1+{\mathcal{H}}_3-{\mathcal{H}}_4\left({\mathcal{H}}_1+{\mathcal{H}}_4\right)\right)},l=\mathrm{3,4},\text{⋯},& \hfill \end{array}$$ (64)

then about $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ discrete COVID‐19 epidemic model (8) undergoes a Hopf bifurcation at a critical value $h_0$ where ${\mathcal{H}}_1,{\mathcal{H}}_2,{\mathcal{H}}_3,{\mathcal{H}}_4$ are depicted in (35) and $h_0$ is the real root of $1-{\mathcal{H}}_4(h)-{\mathcal{H}}_2(h)-{{\mathcal{H}}_3}^2(h)+{{\mathcal{H}}_4}^3(h)-{{\mathcal{H}}_4}^2(h)\left(1+{\mathcal{H}}_2(h)\right)+2{\mathcal{H}}_2(h){\mathcal{H}}_4(h)-{{\mathcal{H}}_1}^2(h){\mathcal{H}}_4(h)+{\mathcal{H}}_1(h){\mathcal{H}}_3(h)\left(1+{\mathcal{H}}_4(h)\right)=0$.

By utilizing Lemma 7.4 for $n=4$, one gets

$$\begin{array}{cc}\hfill {\mathrm{\Delta }}_3^{-}(h)=1-{\mathcal{H}}_4-{\mathcal{H}}_2-{{\mathcal{H}}_3}^2+{{\mathcal{H}}_4}^3-{{\mathcal{H}}_4}^2\left(1+{\mathcal{H}}_2\right)& \hfill \\ \hfill +2{\mathcal{H}}_2{\mathcal{H}}_4-{{\mathcal{H}}_1}^2{\mathcal{H}}_4+{\mathcal{H}}_1{\mathcal{H}}_3\left(1+{\mathcal{H}}_4\right)=& 0,\hfill \\ \hfill {\mathrm{\Delta }}_3^{+}(h)=1+{\mathcal{H}}_2+{\mathcal{H}}_4-{{\mathcal{H}}_3}^2-{{\mathcal{H}}_4}^3-{{\mathcal{H}}_4}^2\left(1+{\mathcal{H}}_2\right)& \hfill \\ \hfill +{{\mathcal{H}}_1}^2{\mathcal{H}}_4-{\mathcal{H}}_1{\mathcal{H}}_3\left(1-{\mathcal{H}}_4\right)>& 0,\hfill \\ \hfill P_h(1)=1+{\mathcal{H}}_1+{\mathcal{H}}_2+{\mathcal{H}}_3+{\mathcal{H}}_4>& 0,\hfill \\ \hfill {(-1)}^4{P}_h(-1)=1-{\mathcal{H}}_1+{\mathcal{H}}_2-{\mathcal{H}}_3+{\mathcal{H}}_4>& 0,\hfill \\ \hfill {\mathrm{\Delta }}_j^{\pm }(h_0)=1\pm {\mathcal{H}}_4>& 0,\hfill \\ \hfill \frac{d}{dh}\left({\mathrm{\Delta }}_3^{-}(h)\right){|}_{h=h_0}=\frac{d}{dh}\left(1-{\mathcal{H}}_4-{\mathcal{H}}_2-{{\mathcal{H}}_3}^2+{{\mathcal{H}}_4}^3-{{\mathcal{H}}_4}^2\left(1+{\mathcal{H}}_2\right)\right.& \hfill \\ \hfill \left.+2{\mathcal{H}}_2{\mathcal{H}}_4-{{\mathcal{H}}_1}^2{\mathcal{H}}_4+{\mathcal{H}}_1{\mathcal{H}}_3\left(1+{\mathcal{H}}_4\right)\right){|}_{h=h_0}\ne & 0.\hfill \end{array}$$ (65)

Finally,

$$1-0.5P_h(1)\frac{{\mathrm{\Delta }}_1^{-}(h)}{{\mathrm{\Delta }}_2^{+}(h)}=1-\frac{\left(1-{\mathcal{H}}_4\right)\left(1+{\mathcal{H}}_1+{\mathcal{H}}_2+{\mathcal{H}}_3+{\mathcal{H}}_4\right)}{2\left(1+{\mathcal{H}}_3-{\mathcal{H}}_4\left({\mathcal{H}}_1+{\mathcal{H}}_4\right)\right)}.$$

7.2.2. Flip bifurcation about $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$

By using following explicit criterion, ¹⁶ , ¹⁷ flip bifurcation for the discrete COVID‐19 epidemic model (8) about $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ by choosing $h$ as a bifurcation parameter is explored.

Lemma 7.6

Consider the system (61) with $h\in ℝ$ is a bifurcation parameter. Moreover, characteristic polynomial of $J{|}_X$ about $X$ of (61) is of the form, which is depicted in (62). Now, considering the determinants: ${\mathrm{\Delta }}_0^{\pm }(h)=1$, ${\mathrm{\Delta }}_1^{\pm }(h),\text{⋯},{\mathrm{\Delta }}_n^{\pm }(h)$, which are depicted in (63) and $j=1,\text{⋯},n$. Furthermore, flip bifurcation occurs at critical value $h=h_0$ if the following parametric conditions hold:

• ${\mathrm{\Gamma }}_1:$ Eigenvalue assignment: $P_{h_0}(-1)=0,P_{h_0}(1)>0,{\mathrm{\Delta }}_{n-1}^{\pm }(h_0)>0,{\mathrm{\Delta }}_j^{\pm }(h_0)>0$ where $j=n-3,n-5,\text{⋯},1$ (or 2), when $n$ is even (or odd, respectively).
• ${\mathrm{\Gamma }}_2:$ Transversality condition: $\frac{\sum _{i=1}^n{(-1)}^{n-i}{\mathcal{H}}_i^{{}^{\prime }}}{\sum _{i=1}^n{(-1)}^{n-i}(n-i+1){\mathcal{H}}_{i-1}}\ne 0$ where ${\mathcal{H}}_i^{\prime }$ represent the derivative w.r.t. $h$ at $h=h_0$.

Theorem 7.6

If

$$\begin{array}{cc}\hfill 1-{\mathcal{H}}_4-{\mathcal{H}}_2-{{\mathcal{H}}_3}^2+{{\mathcal{H}}_4}^3-{{\mathcal{H}}_4}^2\left(1+{\mathcal{H}}_2\right)& \hfill \\ \hfill +2{\mathcal{H}}_2{\mathcal{H}}_4-{{\mathcal{H}}_1}^2{\mathcal{H}}_4+{\mathcal{H}}_1{\mathcal{H}}_3\left(1+{\mathcal{H}}_4\right)>& 0,\hfill \\ \hfill 1+{\mathcal{H}}_2+{\mathcal{H}}_4-{{\mathcal{H}}_3}^2-{{\mathcal{H}}_4}^3-{{\mathcal{H}}_4}^2\left(1+{\mathcal{H}}_2\right)& \hfill \\ \hfill +{{\mathcal{H}}_1}^2{\mathcal{H}}_4-{\mathcal{H}}_1{\mathcal{H}}_3\left(1-{\mathcal{H}}_4\right)>& 0,\hfill \\ \hfill 1+{\mathcal{H}}_1+{\mathcal{H}}_2+{\mathcal{H}}_3+{\mathcal{H}}_4>& 0,\hfill \\ \hfill 1-{\mathcal{H}}_1+{\mathcal{H}}_2-{\mathcal{H}}_3+{\mathcal{H}}_4=& 0,\hfill \\ \hfill 1\pm {\mathcal{H}}_4>& 0,\hfill \\ \hfill \frac{{\mathcal{H}}_1^{{}^{\prime }}-{\mathcal{H}}_2^{{}^{\prime }}+{\mathcal{H}}_3^{{}^{\prime }}-{{\mathcal{H}}_4}^{{}^{\prime }}}{4-3{\mathcal{H}}_1+2{\mathcal{H}}_2-{\mathcal{H}}_3}\ne 0,& \hfill \end{array}$$ (66)

then about $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ discrete COVID‐19 epidemic model (8) undergoes a flip bifurcation at a critical value $h_0$, where $h_0$ is the real root of $1-{\mathcal{H}}_1(h)+{\mathcal{H}}_2(h)-{\mathcal{H}}_3(h)+{\mathcal{H}}_4(h)=0$.

By utilizing Lemma 7.6 for $n=4$, one gets

$$\begin{array}{cc}\hfill {\mathrm{\Delta }}_3^{-}(h)=1-{\mathcal{H}}_4-{\mathcal{H}}_2-{{\mathcal{H}}_3}^2+{{\mathcal{H}}_4}^3-{{\mathcal{H}}_4}^2\left(1+{\mathcal{H}}_2\right)& \hfill \\ \hfill +2{\mathcal{H}}_2{\mathcal{H}}_4-{{\mathcal{H}}_1}^2{\mathcal{H}}_4+{\mathcal{H}}_1{\mathcal{H}}_3\left(1+{\mathcal{H}}_4\right)>& 0,\hfill \\ \hfill {\mathrm{\Delta }}_3^{+}(h)=1+{\mathcal{H}}_2+{\mathcal{H}}_4-{{\mathcal{H}}_3}^2-{{\mathcal{H}}_4}^3-{{\mathcal{H}}_4}^2\left(1+{\mathcal{H}}_2\right)& \hfill \\ \hfill +{{\mathcal{H}}_1}^2{\mathcal{H}}_4-{\mathcal{H}}_1{\mathcal{H}}_3\left(1-{\mathcal{H}}_4\right)>& 0,\hfill \\ \hfill P_{h_0}(1)=1+{\mathcal{H}}_1+{\mathcal{H}}_2+{\mathcal{H}}_3+{\mathcal{H}}_4>& 0,\hfill \\ \hfill P_{h_0}(-1)=1-{\mathcal{H}}_1+{\mathcal{H}}_2-{\mathcal{H}}_3+{\mathcal{H}}_4=& 0,\hfill \\ \hfill {\mathrm{\Delta }}_j^{\pm }=1\pm {\mathcal{H}}_4>& 0,\hfill \\ \hfill \frac{\sum _{i=1}^4{(-1)}^{4-i}{\mathcal{H}}_i^{{}^{\prime }}}{\sum _{i=1}^4{(-1)}^{4-i}(4-i+1){\mathcal{H}}_{i-1}}=\frac{{\mathcal{H}}_1^{{}^{\prime }}-{\mathcal{H}}_2^{{}^{\prime }}+{\mathcal{H}}_3^{{}^{\prime }}-{{\mathcal{H}}_4}^{{}^{\prime }}}{4-3{\mathcal{H}}_1+2{\mathcal{H}}_2-{\mathcal{H}}_3}\ne 0.& \hfill \end{array}$$ (67)

8. CHAOS CONTROL

Chaos control is the stabilization of one of these unstable periodic orbits by modest system perturbations. As a result, an otherwise chaotic motion becomes more steady and predictable, which is typically advantageous. To avoid major changes in the system's natural dynamics, the perturbation must be small in comparison to the overall size of the attractor. So, in this section, feedback control strategy is utilized in order to study chaos in discrete COVID‐19 epidemic model (8) about interior equilibrium solution $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$. First, we have given the basic concept of chaos as follows:

Definition 8.1

The orbit of the map $S\mapsto f(S)$ is called chaotic if

• (i)It possesses a positive Lyapunov exponent.
• (ii)It does not converge to a periodic orbit, that is, there does not exist a periodic orbit $y_n=y_n+T$ such that $\underset{n\to \infty }{\lim }\left|S_n-y_n\right|=0$.

By utilizing feedback control strategy, discrete COVID‐19 epidemic model (8) takes the form

$$\begin{array}{cc}\hfill S_{n+1}& =h\mathrm{\Lambda }+(1-h\mu )S_n-h\beta S_n{I}_n+\varrho (S_n-S),\hfill \\ \hfill I_{n+1}& =(1-h\delta -h\mu -h{\alpha }_1-h\gamma )I_n+h\beta S_n{I}_n+\varrho (I_n-I),\hfill \\ \hfill Q_{n+1}& =h\delta I_n+(1-hϵ-h{\alpha }_2-h\mu )Q_n+\varrho (Q_n-Q),\hfill \\ \hfill R_{n+1}& =h\gamma I_n+hϵQ_n+(1-h\mu )R_n+\varrho (R_n-R),\hfill \end{array}$$ (68)

with $\varrho$ is chosen as the control parameter. The $J^C{|}_{E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)}$ about interior equilibrium solution $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ of controlled discrete COVID‐19 epidemic model (68) is

$$\begin{array}{cc}\hfill & J{|}_{E_{SIQR}}^C=\left(\begin{array}{cccc}1-h\nu -h\beta I+\varrho & -h\beta S& 0& 0\\ h\beta I& 1-h(\delta +\mu +{\alpha }_1+\gamma )+h\beta S+\varrho & 0& 0\\ 0& h\delta & 1-h(ϵ+{\alpha }_2+\mu )+\varrho & 0\\ 0& h\gamma & hϵ& 1-h\mu +\varrho \end{array}\right).\hfill \end{array}$$ (69)

The characteristic polynomial of $J^C{|}_{E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)}$ about $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ is

$$P(\lambda )={\lambda }^4+{\mathcal{H}}_1^{\ast }{\lambda }^3+{\mathcal{H}}_2^{\ast }{\lambda }^2+{\mathcal{H}}_3^{\ast }\lambda +{\mathcal{H}}_4^{\ast },$$ (70)

where

$$\begin{array}{cc}\hfill {\mathcal{H}}_1^{\ast }=& h\left({\alpha }_2+ϵ+2\mu +\mu {\mathcal{R}}_0\right)-4\left(1+\varrho \right),\hfill \\ \hfill {\mathcal{H}}_2^{\ast }=& \frac{1}{\delta +\mu +{\alpha }_1+\gamma }\left[-h^2{\alpha }_1^2\mu +6{\left(1+\varrho \right)}^2\left(\gamma +\delta +\mu \right)\right.\hfill \\ \hfill & +h^2\left\{\mu \left(\gamma +\delta +\mu \right)\left({\alpha }_2-\gamma -\delta +ϵ\right)+\beta \mathrm{\Lambda }\left({\alpha }_2+\gamma +\delta +ϵ+3\mu \right)\right\}\hfill \\ \hfill & -3h\left(1+\varrho \right)\left\{\beta \mathrm{\Lambda }+\left(\gamma +\delta +\mu \right)\left({\alpha }_2+ϵ+2\mu \right)\right\}\hfill \\ \hfill & +{\alpha }_1{h}^2\left\{\beta \mathrm{\Lambda }-{\mu }^2+\mu \left({\alpha }_2-2\left(\gamma +\delta \right)+ϵ\right)\right\}\hfill \\ \hfill & \left.-3h{\alpha }_1\left(1+\varrho \right)\left({\alpha }_2+ϵ+2\mu \right)+6{\alpha }_1{\left(1+\varrho \right)}^2\right],\hfill \\ \hfill {\mathcal{H}}_3^{\ast }=& \frac{h^3}{\delta +\mu +{\alpha }_1+\gamma }\left[-\mu {\left(\delta +\mu +{\alpha }_1+\gamma \right)}^2\left({\alpha }_2+ϵ+2\mu \right)\right.\hfill \\ \hfill & \left.+\beta \mathrm{\Lambda }\left\{\left({\alpha }_2+ϵ\right)\left({\alpha }_1+\gamma +\delta \right)+2\mu \left({\alpha }_1+{\alpha }_2+\gamma +\delta +ϵ\right)+3{\mu }^2\right\}\right]\hfill \\ \hfill & -\frac{2h^2\left(1+\varrho \right)}{\delta +\mu +{\alpha }_1+\gamma }\left[-\mu \left({\alpha }_1-{\alpha }_2+\gamma +\delta -ϵ\right)\left(\delta +\mu +{\alpha }_1+\gamma \right)\right.\hfill \\ \hfill & \left.+\beta \mathrm{\Lambda }\left({\alpha }_1+{\alpha }_2+\gamma +\delta +ϵ+3\mu \right)\right]-4{\left(1+\varrho \right)}^3\hfill \\ \hfill & +\frac{3h{\left(1+\varrho \right)}^2\left\{\beta \mathrm{\Lambda }+\left(\delta +\mu +{\alpha }_1+\gamma \right)\left({\alpha }_2+ϵ+2\mu \right)\right\}}{\delta +\mu +{\alpha }_1+\gamma },\hfill \end{array}$$ (71)

$$\begin{array}{cc}\hfill {\mathcal{H}}_4^{\ast }=& \frac{\left(1-h\mu +\varrho \right)\left\{-1+h\left({\alpha }_2+ϵ+\mu \right)-\varrho \right\}}{\delta +\mu +{\alpha }_1+\gamma }\hfill \\ \hfill & \times \left[h^2{\alpha }_1^2\mu +h^2\left(\gamma +\delta +\mu \right)\left\{-\beta \mathrm{\Lambda }+\mu \left(\gamma +\delta +\mu \right)\right\}\right.\hfill \\ \hfill & +h\beta \mathrm{\Lambda }\left(1+\varrho \right)-\left(\gamma +\delta +\mu \right){\left(1+\varrho \right)}^2\hfill \\ \hfill & \left.-{\alpha }_1\left\{h^2\left(\beta \mathrm{\Lambda }-2\mu \left(\gamma +\delta +\mu \right)\right)+{\left(1+\varrho \right)}^2\right\}\right].\hfill \end{array}$$ (72)

Based on linear stability theory, the local dynamics of controlled discrete COVID‐19 epidemic model (68) about $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ can be stated as following Lemma:

Lemma 8.2

$E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ of controlled discrete COVID‐19 epidemic model (68) is a sink if

$$\begin{array}{c}\hfill \left|{\mathcal{H}}_1^{\ast }+{\mathcal{H}}_3^{\ast }\right|<1+{\mathcal{H}}_2^{\ast }+{\mathcal{H}}_4^{\ast },\left|{\mathcal{H}}_3^{\ast }-{\mathcal{H}}_1^{\ast }\right|<2(1-{\mathcal{H}}_4^{\ast }),{\mathcal{H}}_2^{\ast }-3{\mathcal{H}}_4^{\ast }<3,\\ \hfill {\mathcal{H}}_2^{\ast }+{\mathcal{H}}_4^{\ast 2}(1+{\mathcal{H}}_2^{\ast })+{\mathcal{H}}_3^{\ast 2}+{\mathcal{H}}_4^{\ast }(1+{\mathcal{H}}_1^{\ast 2})<1+2{\mathcal{H}}_2^{\ast }{\mathcal{H}}_4^{\ast }+{\mathcal{H}}_1^{\ast }{\mathcal{H}}_3^{\ast }(1+{\mathcal{H}}_4^{\ast })+{\mathcal{H}}_4^{\ast 3},\end{array}$$ (73)

where ${\mathcal{H}}_1^{\ast },{\mathcal{H}}_2^{\ast },{\mathcal{H}}_3^{\ast }$, and ${\mathcal{H}}_4^{\ast }$ are depicted in (71) and (72).

9. NUMERICAL SIMULATIONS

Theoretical results are illustrated numerically in this section. In this regard, following cases are presented to discuss the correctness of obtained theoretical results about equilibrium solution for discrete COVID‐19 epidemic model (8):

Case I:

If $h=0.447,\mathrm{\Lambda }=1.43,\mu =1.03,\beta =5,\delta =0.4,{\alpha }_1=0.42,\gamma =1.05,ϵ=0.6,{\alpha }_2=0.4$, then from (34), computation yields $\left|{\mathcal{H}}_1+{\mathcal{H}}_3\right|=2.0362676817098855<1+{\mathcal{H}}_2+{\mathcal{H}}_4=2.3837796498198314,\left|{\mathcal{H}}_3-{\mathcal{H}}_1\right|=1.0239199044970109<2(1-{\mathcal{H}}_4)=1.9270855802155096,{\mathcal{H}}_2-3{\mathcal{H}}_4=1.2379508102508503<3,{\mathcal{H}}_2+{{\mathcal{H}}_4}^2(1+{\mathcal{H}}_2)+{{\mathcal{H}}_3}^2+{\mathcal{H}}_4(1+{{\mathcal{H}}_1}^2)=1.7284646940943127<1+2{\mathcal{H}}_2{\mathcal{H}}_4+{\mathcal{H}}_1{\mathcal{H}}_3(1+{\mathcal{H}}_4)+{{\mathcal{H}}_4}^3=1.9010170884696964$, which implies that equilibrium solution $E_{SIQR}^{+}(0.\mathrm{58,0}.\mathrm{2871034482758619,0}.\mathrm{056572108034652596,0}.32563289855383165)$ of discrete COVID‐19 epidemic model (8) is a sink. In this case, plots for discrete COVID‐19 epidemic model (8) with initial values $(S_0,I_0,Q_0,R_0)=(1.\mathrm{44,0}.\mathrm{0056,0}.\mathrm{046,0}.05)$ are drawn in Figure 2 which shows that $E_{SIQR}^{+}\left(0.\mathrm{58,0}.\mathrm{2871034482758619,0}.\mathrm{056572108034652596,0}.32563289855383165\right)$ is a sink.

Case II:

Now, in this case, it is proved that at $h=1.0426472848436514$, discrete COVID‐19 epidemic model (8) undergoes a Hopf bifurcation if $\mathrm{\Lambda }=1.33,\mu =1.003,\beta =3,\delta =0.0014,{\alpha }_1=1,\gamma =0.36,ϵ=0.15,{\alpha }_2=0.4$ and $h\in [0.\mathrm{1,1}.11]$ with initial values $(S_0,I_0,Q_0,R_0)=\left(0.\mathrm{48,0}.\mathrm{00084,0}.\mathrm{0052,0}.0053\right)$. If $\mathrm{\Lambda }=1.33,\mu =1.003,\beta =3,\delta =0.0014,{\alpha }_1=1,\gamma =0.36,ϵ=0.15,{\alpha }_2=0.4$ and $h=1.0426472848436514$, then from (37), one gets

$${\lambda }^4+0.42450682654919447{\lambda }^3+0.8684116401585866{\lambda }^2+0.65818938970299\lambda +0.0283454500857471=0,$$ (74)

whose roots are ${\lambda }_{\mathrm{1,2}}=0.12024981675558936\pm 0.9927436635759357\iota ,{\lambda }_3=-0.6192312333621909,{\lambda }_4=-0.04577522669818224$ where $|{\lambda }_{\mathrm{1,2}}|=|0.12024981675558936\pm 0.9927436635759357\iota |=1$. This implies that for said parametric values, the eigenvalues criterion for the existence of Hopf bifurcation holds, and hence, discrete COVID‐19 epidemic model (8) may undergo Hopf bifurcation. In the rest of simulation, it is proved that discrete COVID‐19 epidemic model (8) must undergo Hopf bifurcation. For instance, if $\mathrm{\Lambda }=1.33,\mu =1.003,\beta =3,\delta =0.0014,{\alpha }_1=1,\gamma =0.36,ϵ=0.15,{\alpha }_2=0.4$, and $h=1.0426472848436514$, then from (64), the computation yields

$$\begin{array}{cc}\hfill 1-{\mathcal{H}}_4-{\mathcal{H}}_2-{{\mathcal{H}}_3}^2+{{\mathcal{H}}_4}^3-{{\mathcal{H}}_4}^2\left(1+{\mathcal{H}}_2\right)& \hfill \\ \hfill +2{\mathcal{H}}_2{\mathcal{H}}_4-{{\mathcal{H}}_1}^2{\mathcal{H}}_4+{\mathcal{H}}_1{\mathcal{H}}_3\left(1+{\mathcal{H}}_4\right)=& 0,\hfill \\ \hfill 1+{\mathcal{H}}_2+{\mathcal{H}}_4-{{\mathcal{H}}_3}^2-{{\mathcal{H}}_4}^3-{{\mathcal{H}}_4}^2\left(1+{\mathcal{H}}_2\right)& \hfill \\ \hfill +{{\mathcal{H}}_1}^2{\mathcal{H}}_4-{\mathcal{H}}_1{\mathcal{H}}_3\left(1-{\mathcal{H}}_4\right)=1.195641858530397>& 0,\hfill \\ \hfill 1+{\mathcal{H}}_1+{\mathcal{H}}_2+{\mathcal{H}}_3+{\mathcal{H}}_4=2.9794533064965183>& 0,\hfill \\ \hfill 1-{\mathcal{H}}_1+{\mathcal{H}}_2-{\mathcal{H}}_3+{\mathcal{H}}_4=0.8140608739921492>& 0,\hfill \\ \hfill 1+{\mathcal{H}}_4=1.0283454500857472>& 0,\hfill \\ \hfill 1-{\mathcal{H}}_4=0.9716545499142529>& 0,\hfill \\ \hfill \frac{d}{dh}\left(1-{\mathcal{H}}_4-{\mathcal{H}}_2-{{\mathcal{H}}_3}^2+{{\mathcal{H}}_4}^3-{{\mathcal{H}}_4}^2\left(1+{\mathcal{H}}_2\right)\right.& \hfill \\ \hfill \left.+2{\mathcal{H}}_2{\mathcal{H}}_4-{{\mathcal{H}}_1}^2{\mathcal{H}}_4+{\mathcal{H}}_1{\mathcal{H}}_3\left(1+{\mathcal{H}}_4\right)\right){|}_{h=1.0426472848436514}=-2.5455536627459776\ne & 0,\hfill \\ \hfill 1-\frac{\left(1-{\mathcal{H}}_4\right)\left(1+{\mathcal{H}}_1+{\mathcal{H}}_2+{\mathcal{H}}_3+{\mathcal{H}}_4\right)}{2\left(1+{\mathcal{H}}_3-{\mathcal{H}}_4\left({\mathcal{H}}_1+{\mathcal{H}}_4\right)\right)}=0.12024981675559898.& \hfill \end{array}$$ (75)

Moreover, $\cos \frac{2\pi }{l}=0.12024981675559898$ implies $l=\pm 4.33246986002$. Thus, from (75), all conditions of Theorem 7.5 hold, and hence, it can be concluded that discrete COVID‐19 epidemic model (8) undergoes Hopf bifurcation. So Hopf bifurcation diagrams and maximum Lyapunov exponent are drawn in Figure 3. Moreover, it is cleared from Figure 4 that the interior equilibrium solution $E_{SIQR}^{+}\left(0.\mathrm{7881333333333332,0}.\mathrm{228177240173687,0}.\mathrm{000205697447677503,0}.081928874456310\right)$ of model (8) with initial values $(S_0,I_0,Q_0,R_0)=\left(0.\mathrm{48,0}.\mathrm{0084,0}.\mathrm{0052,0}.053\right)$ is stable focus. Finally, Figure 5 implies that $E_{SIQR}^{+}\left(0.\mathrm{7881333333333332,0}.\mathrm{228177240173687,0}.000205697447677503,\right.\left.0.081928874456310\right)$ of model (8) with initial values $(S_0,I_0,Q_0,R_0)=\left(0.\mathrm{48,0}.\mathrm{0084,0}.\mathrm{0052,0}.053\right)$ is unstable focus.

Case III:

Now, it is proved that at $h=0.5676520238972457$, discrete COVID‐19 epidemic model (8) undergoes a flip bifurcation if $\mathrm{\Lambda }=1.04,\mu =1,\beta =5,\delta =0.00014,{\alpha }_1=0.06,\gamma =0.05,ϵ=0.015,{\alpha }_2=0.4$, and $h\in [0.\mathrm{05,1}]$ with $(S_0,I_0,Q_0,R_0)=(0.\mathrm{00017,0}.\mathrm{000384,0}.\mathrm{0182,0}.00012)$. If $\mathrm{\Lambda }=1.04,\mu =1,\beta =5,\delta =0.00014,{\alpha }_1=0.06,\gamma =0.05,ϵ=0.015,{\alpha }_2=0.4$, and $h=0.5676520238972457$, then from (37), one gets

$$\begin{array}{c}\hfill {\lambda }^4+0.029815019074268534{\lambda }^3-0.6705401415646195{\lambda }^2+0.2706290592501235\lambda -\\ \hfill 0.029015780110985375=0,\end{array}$$ (76)

whose one root is ${\lambda }_1=-1$ but rest of roots are ${\lambda }_{\mathrm{2,3},4}=0.19677238618540321$, 0.3410646186375577, $0.4323479761027687\ne 1\text{or}-1$. This implies that for said parametric values the eigenvalues criterion for the existence of flip bifurcation holds, and hence, discrete COVID‐19 epidemic model (8) may undergo flip bifurcation. In the rest of simulation, it is proved that discrete COVID‐19 epidemic model (8) must undergo flip bifurcation. For instance, if $\mathrm{\Lambda }=1.04,\mu =1,\beta =5,\delta =0.00014,{\alpha }_1=0.06,\gamma =0.05,ϵ=0.015,{\alpha }_2=0.4$, and $h=0.5676520238972457$, then from (66), the computation yields

$$\begin{array}{cc}\hfill 1-{\mathcal{H}}_4-{\mathcal{H}}_2-{{\mathcal{H}}_3}^2+{{\mathcal{H}}_4}^3-{{\mathcal{H}}_4}^2\left(1+{\mathcal{H}}_2\right)& \hfill \\ \hfill +2{\mathcal{H}}_2{\mathcal{H}}_4-{{\mathcal{H}}_1}^2{\mathcal{H}}_4+{\mathcal{H}}_1{\mathcal{H}}_3\left(1+{\mathcal{H}}_4\right)=1.6727869992601545>& 0,\hfill \\ \hfill 1+{\mathcal{H}}_2+{\mathcal{H}}_4-{{\mathcal{H}}_3}^2-{{\mathcal{H}}_4}^3-{{\mathcal{H}}_4}^2\left(1+{\mathcal{H}}_2\right)& \hfill \\ \hfill +{{\mathcal{H}}_1}^2{\mathcal{H}}_4-{\mathcal{H}}_1{\mathcal{H}}_3\left(1-{\mathcal{H}}_4\right)=0.21862231553905934>& 0,\hfill \\ \hfill 1+{\mathcal{H}}_1+{\mathcal{H}}_2+{\mathcal{H}}_3+{\mathcal{H}}_4=0.6008881566487836>& 0,\hfill \\ \hfill 1-{\mathcal{H}}_1+{\mathcal{H}}_2-{\mathcal{H}}_3+{\mathcal{H}}_4=& 0,\hfill \\ \hfill 1+{\mathcal{H}}_4=0.9709842198890147>& 0,\hfill \\ \hfill 1-{\mathcal{H}}_4=1.0290157801109854>& 0,\hfill \\ \hfill \frac{{\mathcal{H}}_1^{{}^{\prime }}-{\mathcal{H}}_2^{{}^{\prime }}+{\mathcal{H}}_3^{{}^{\prime }}-{{\mathcal{H}}_4}^{{}^{\prime }}}{4-3{\mathcal{H}}_1+2{\mathcal{H}}_2-{\mathcal{H}}_3}=3.523285244838702\ne 0,& \hfill \end{array}$$ (77)

Thus, from (77), all conditions of Theorem 7.6 hold, and hence, it can be concluded that discrete COVID‐19 epidemic model (8) undergoes flip bifurcation. So flip bifurcation diagrams and maximum Lyapunov exponent are drawn in Figure 6.

Case IV:

Finally, the numerical simulation will be provided in order to verify result of Lemma 8.2 for controlled discrete COVID‐19 epidemic model (68). If $h=0.567,\mathrm{\Lambda }=1.33,\mu =1.03,\beta =2.8,\delta =0.14,{\alpha }_1=1,\gamma =0.36,ϵ=0.15,{\alpha }_2=0.4,\varrho =0.2$, then from (73), computation yields $\left|{\mathcal{H}}_1^{\ast }+{\mathcal{H}}_3^{\ast }\right|=3.513036111281745<1+{\mathcal{H}}_2^{\ast }+{\mathcal{H}}_4^{\ast }=3.5751745908671846$, $\left|{\mathcal{H}}_3^{\ast }-{\mathcal{H}}_1^{\ast }\right|=1.4580476041332746<2(1-{\mathcal{H}}_4^{\ast })=1.7010122010284772$, ${\mathcal{H}}_2^{\ast }-3{\mathcal{H}}_4^{\ast }=1.9771989929241387<3,{\mathcal{H}}_2^{\ast }+{\mathcal{H}}_4^{\ast 2}(1+{\mathcal{H}}_2^{\ast })+{\mathcal{H}}_3^{\ast 2}+{\mathcal{H}}_4^{\ast }(1+{\mathcal{H}}_1^{\ast 2})=4.631038704690054<1+2{\mathcal{H}}_2^{\ast }{\mathcal{H}}_4^{\ast }+{\mathcal{H}}_1^{\ast }{\mathcal{H}}_3^{\ast }(1+{\mathcal{H}}_4^{\ast })+{\mathcal{H}}_4^{\ast 3}=4.664259336487028$, which implies that interior equilibrium solution of controlled discrete COVID‐19 epidemic model (68) is a sink. In this case, plots for controlled discrete COVID‐19 epidemic model (68) with initial values $(S_0,I_0,Q_0,R_0)=(1.\mathrm{5,0}.\mathrm{12,1}.\mathrm{3,2}.2)$ are drawn in Figure 7 which shows that interior equilibrium solution is a sink.

Case V:

Finally, we will fit real data obtained from published materials to our under‐consideration COVID‐19 epidemic model (8) in order to verify our mathematical analysis regarding stability and bifurcation. The collected real data for India are depicted in Table 1 whereas corresponding dynamical analysis of discrete COVID‐19 epidemic model (8) is given in Figure 8. In Figure 8A, the plot of susceptible population shows a curve that represents the number of susceptible individuals increase rapidly with the passage of time, whereas Figure 8B shows a curve that represents the infection rate is initially too much high, but with the passage of time it reduced to the minimum. Also, Figure 8C represents the quarantined population which is increasing initially up to a peak value but with the passage of time decreased due to vaccination. Finally, Figure 8D represents that curve goes upward slowly which means that the number of recovered individuals were small initially due to less medical care and facilities, but with the passage of time, the recovery rate increases due to proper medication and vaccination.

Case VI:

Now, in this case, it is proved that at $h=0.748849603298569$, discrete COVID‐19 epidemic model (8) undergoes a Hopf bifurcation with parametric values depicted in Table 1 and $h\in [0.\mathrm{4,1}.0]$ with initial values $(S_0,I_0,Q_0,R_0)=\left(1.\mathrm{49,0}.\mathrm{0084,0}.\mathrm{052,0}.53\right)$. If $h=0.748849603298569$, then from (37), one gets

$${\lambda }^4-0.7066781302903977{\lambda }^3+1.1002524701225678{\lambda }^2-0.21732602057080364\lambda -0.007011747173420613=0,$$ (78)

whose roots are ${\lambda }_{\mathrm{1,2}}=0.24297239386389796\pm 0.9700332034626471\iota ,{\lambda }_3=-0.028170505328934784,{\lambda }_4=0.24890384789153663$ where $|{\lambda }_{\mathrm{1,2}}|=|0.24297239386389796\pm 0.9700332034626471\iota |=1$. This implies that for said parametric values, the eigenvalues criterion for the existence of Hopf bifurcation holds, and hence, discrete COVID‐19 epidemic model (8) may undergo Hopf bifurcation. In the rest of simulation, it is proved that discrete COVID‐19 epidemic model (8) must undergo Hopf bifurcation. For instance, if $h=0.748849603298569$ with parametric values of Table 1, then from (64), the computation yields

$$\begin{array}{cc}\hfill 1-{\mathcal{H}}_4-{\mathcal{H}}_2-{{\mathcal{H}}_3}^2+{{\mathcal{H}}_4}^3-{{\mathcal{H}}_4}^2\left(1+{\mathcal{H}}_2\right)& \hfill \\ \hfill +2{\mathcal{H}}_2{\mathcal{H}}_4-{{\mathcal{H}}_1}^2{\mathcal{H}}_4+{\mathcal{H}}_1{\mathcal{H}}_3\left(1+{\mathcal{H}}_4\right)=& 0,\hfill \\ \hfill 1+{\mathcal{H}}_2+{\mathcal{H}}_4-{{\mathcal{H}}_3}^2-{{\mathcal{H}}_4}^3-{{\mathcal{H}}_4}^2\left(1+{\mathcal{H}}_2\right)& \hfill \\ \hfill +{{\mathcal{H}}_1}^2{\mathcal{H}}_4-{\mathcal{H}}_1{\mathcal{H}}_3\left(1-{\mathcal{H}}_4\right)=1.8877491792388077>& 0,\hfill \\ \hfill 1+{\mathcal{H}}_1+{\mathcal{H}}_2+{\mathcal{H}}_3+{\mathcal{H}}_4=1.1692365720879476>& 0,\hfill \\ \hfill 1-{\mathcal{H}}_1+{\mathcal{H}}_2-{\mathcal{H}}_3+{\mathcal{H}}_4=3.0172448738103466>& 0,\hfill \\ \hfill 1+{\mathcal{H}}_4=0.9929882528265794>& 0,\hfill \\ \hfill 1-{\mathcal{H}}_4=1.0070117471734206>& 0,\hfill \\ \hfill \frac{d}{dh}\left(1-{\mathcal{H}}_4-{\mathcal{H}}_2-{{\mathcal{H}}_3}^2+{{\mathcal{H}}_4}^3-{{\mathcal{H}}_4}^2\left(1+{\mathcal{H}}_2\right)\right.& \hfill \\ \hfill \left.+2{\mathcal{H}}_2{\mathcal{H}}_4-{{\mathcal{H}}_1}^2{\mathcal{H}}_4+{\mathcal{H}}_1{\mathcal{H}}_3\left(1+{\mathcal{H}}_4\right)\right){|}_{h=0.748849603298569}=-1.9436399137532254\ne & 0,\hfill \\ \hfill 1-\frac{\left(1-{\mathcal{H}}_4\right)\left(1+{\mathcal{H}}_1+{\mathcal{H}}_2+{\mathcal{H}}_3+{\mathcal{H}}_4\right)}{2\left(1+{\mathcal{H}}_3-{\mathcal{H}}_4\left({\mathcal{H}}_1+{\mathcal{H}}_4\right)\right)}=0.2429723938639281.& \hfill \end{array}$$ (79)

FIGURE 2.

Plots for discrete COVID‐19 epidemic model (8) [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 3.

Bifurcation diagrams and maximum Lyapunov exponents for discrete COVID‐19 epidemic model (8) if $\mathrm{\Lambda }=1.33,\mu =1.003,\beta =3,\delta =0.0014,{\alpha }_1=1,\gamma =0.36,ϵ=0.15,{\alpha }_2=0.4$, and $h\in [0.\mathrm{1,1}.11]$ with initial values $(S_0,I_0,Q_0,R_0)=\left(0.\mathrm{48,0}.\mathrm{00084,0}.\mathrm{0052,0}.0053\right)$ [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 4.

Stable focus for discrete COVID‐19 epidemic model (8) if $\mathrm{\Lambda }=1.33,\mu =1.003,\beta =3,\delta =0.0014,{\alpha }_1=1,\gamma =0.36,ϵ=0.15,{\alpha }_2=0.4$, and $h=0.75$ with initial values $(S_0,I_0,Q_0,R_0)=\left(0.\mathrm{48,0}.\mathrm{0084,0}.\mathrm{0052,0}.053\right)$ [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 5.

Unstable focus for discrete COVID‐19 epidemic model (8) if $\mathrm{\Lambda }=1.33,\mu =1.003,\beta =3,\delta =0.0014,{\alpha }_1=1,\gamma =0.36,ϵ=0.15,{\alpha }_2=0.4$, and $h=1.01$ with initial values $(S_0,I_0,Q_0,R_0)=\left(0.\mathrm{48,0}.\mathrm{0084,0}.\mathrm{0052,0}.053\right)$ [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 6.

Bifurcation diagrams and maximum Lyapunov exponents for discrete COVID‐19 epidemic model (8) if $\mathrm{\Lambda }=1.04,\mu =1,\beta =5,\delta =0.00014,{\alpha }_1=0.06,\gamma =0.05,ϵ=0.015,{\alpha }_2=0.4$, and $h\in [0.\mathrm{05,1}]$ with $(S_0,I_0,Q_0,R_0)=(0.\mathrm{00017,0}.\mathrm{000384,0}.\mathrm{0182,0}.00012)$ [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 7.

Plots for controlled discrete COVID‐19 epidemic model (68) [Colour figure can be viewed at wileyonlinelibrary.com]

TABLE 1.

Real data for India

Parameter	Interpretation	Value	Source
$\mathrm{\Lambda }$	Natural increase in population	0.98	18
$\mu$	Natural death rate	0.73	19
$\beta$	Rate to join $S$ to $I$	1.54	20
$\delta$	Transmission rate from $I$ to $Q$	0.85	Estimated
${\alpha }_1$	Mortality rate of infected persons	0.25	20
$\gamma$	Cure rate of infected persons	0.547	20
$ϵ$	Cure rate of quarantined persons	0.25	20
${\alpha }_2$	Mortality rate of quarantined persons	0.12	20
$h$	Step size	0.25	Estimated

FIGURE 8.

Plots for fitting results of discrete COVID‐19 epidemic model (8) [Colour figure can be viewed at wileyonlinelibrary.com]

Moreover, $\cos \frac{2\pi }{l}=0.2429723938639281$ implies $l=\pm 4.74071203027$. Thus, from (79), all conditions of Theorem 7.5 hold, and hence, it can be concluded that discrete COVID‐19 epidemic model (8) undergoes Hopf bifurcation. So Hopf bifurcation diagrams and maximum Lyapunov exponent are drawn in Figure 9.

FIGURE 9.

Bifurcation diagrams and maximum Lyapunov exponents for discrete COVID‐19 epidemic model (8) with parametric values of Table 1, $h\in [0.\mathrm{4,1}.0]$ and initial values $(S_0,I_0,Q_0,R_0)=\left(1.\mathrm{49,0}.\mathrm{0084,0}.\mathrm{052,0}.53\right)$ [Colour figure can be viewed at wileyonlinelibrary.com]

10. CONCLUSION

This works is about the local dynamical properties, bifurcation, and control in a discrete‐time COVID‐19 epidemic model in ${ℝ}_{+}^4$. Algebraically, it is proved that discrete COVID‐19 epidemic model (8) has boundary equilibrium solution $E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)\forall h,\beta ,\mathrm{\Lambda },\mu ,\delta ,{\alpha }_1,{\alpha }_2,\gamma ,ϵ>0$, but it has interior equilibrium solution $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ if ${\mathcal{R}}_0>1$, where $\mathcal{S},\mathcal{I},\mathcal{Q}$ and $\mathcal{R}$ are depicted in (9). Further local dynamical characteristics with topological classifications about equilibrium solutions $E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)$ and $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ of discrete COVID‐19 epidemic model (8) are explored. It is investigated that $E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)$ of discrete COVID‐19 epidemic model (8) is a sink if $0<\mu <\min \left\{\frac{2}{h},\frac{2}{h}-ϵ-{\alpha }_2\right\}\text{and}\frac{h\beta \mathrm{\Lambda }}{h\beta \mathrm{\Lambda }+2\mu }<{\mathcal{R}}_0<1$ with $\frac{2}{\delta +\mu +{\alpha }_1+\gamma }<h<\frac{2}{ϵ+{\alpha }_2}$; source if (23) holds and additionally $\mu >\max \left\{\frac{2}{h},\frac{2}{h}-ϵ-{\alpha }_2\right\}\text{and}{\mathcal{R}}_0<\frac{h\beta \mathrm{\Lambda }}{h\beta \mathrm{\Lambda }+2\mu }$; saddle if (23) holds and additionally $\mu >\max \left\{\frac{2}{h},\frac{2}{h}-ϵ-{\alpha }_2\right\}\text{and}\frac{h\beta \mathrm{\Lambda }}{h\beta \mathrm{\Lambda }+2\mu }<{\mathcal{R}}_0<1$, or $\max \left\{0,\frac{2}{h}-ϵ-{\alpha }_2\right\}<\mu <\frac{2}{h}\text{and}\frac{h\beta \mathrm{\Lambda }}{h\beta \mathrm{\Lambda }+2\mu }<{\mathcal{R}}_0<1$, or $0<\mu <\min \left\{\frac{2}{h},\frac{2}{h}-ϵ-{\alpha }_2\right\}\text{and}{\mathcal{R}}_0<\frac{h\beta \mathrm{\Lambda }}{h\beta \mathrm{\Lambda }+2\mu }$, or $\max \left\{0,\frac{2}{h}-ϵ-{\alpha }_2\right\}<\mu <\frac{2}{h}\text{and}{\mathcal{R}}_0<\frac{h\beta \mathrm{\Lambda }}{h\beta \mathrm{\Lambda }+2\mu }$; non‐hyperbolic if $\mu =\frac{2}{h},$ or ${\mathcal{R}}_0=\frac{h\beta \mathrm{\Lambda }}{h\beta \mathrm{\Lambda }+2\mu },$ or $\mu =\frac{2}{h}-ϵ-{\alpha }_2$; and moreover, interior equilibrium solution $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ of discrete COVID‐19 epidemic model (8) is a sink if $\left|{\mathcal{H}}_1+{\mathcal{H}}_3\right|<1+{\mathcal{H}}_2+{\mathcal{H}}_4,\left|{\mathcal{H}}_3-{\mathcal{H}}_1\right|<2(1-{\mathcal{H}}_4),{\mathcal{H}}_2-3{\mathcal{H}}_4<3,{\mathcal{H}}_2+{{\mathcal{H}}_4}^2(1+{\mathcal{H}}_2)+{{\mathcal{H}}_3}^2+{\mathcal{H}}_4(1+{{\mathcal{H}}_1}^2)<1+2{\mathcal{H}}_2{\mathcal{H}}_4+{\mathcal{H}}_1{\mathcal{H}}_3(1+{\mathcal{H}}_4)+{{\mathcal{H}}_4}^3$ where ${\mathcal{H}}_1,{\mathcal{H}}_2,{\mathcal{H}}_3$, and ${\mathcal{H}}_4$ are depicted in (35). It is shown that $E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)$ and $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ of discrete COVID‐19 epidemic model (8) are periodic points of period‐ $n$. We have also studied convergence rate for discrete COVID‐19 epidemic model (8). Further, in order to understand dynamics of discrete COVID‐19 epidemic model (8) deeply, we have studied the possible bifurcation scenarios. It is proved that about $E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)$, there exist no flip bifurcation if $(h,\beta ,\mathrm{\Lambda },\mu ,\delta ,{\alpha }_1,{\alpha }_2,\gamma ,ϵ)\in \mathcal{F}{|}_{E_{S000}\left(\frac{\mathrm{\Lambda }}{\mu },\mathrm{0,0},0\right)}=\left\{(h,\beta ,\mathrm{\Lambda },\mu ,\delta ,{\alpha }_1,{\alpha }_2,\gamma ,ϵ):\mu =\frac{2}{h}\right\}$, but discrete COVID‐19 epidemic model (8) undergoes both Hopf and flip bifurcations about $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ by choosing $h$ as bifurcation parameter. We have studied Hopf and flip bifurcations about $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ of discrete COVID‐19 epidemic model (8) by utilizing explicit criterion. By feedback control strategy, chaos in discrete COVID‐19 epidemic model (8) about $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ is also explored. For controlled system (68), it is proved that $E_{SIQR}^{+}\left(\mathcal{S},\mathcal{I},\mathcal{Q},\mathcal{R}\right)$ is a sink if $\left|{\mathcal{H}}_1^{\ast }+{\mathcal{H}}_3^{\ast }\right|<1+{\mathcal{H}}_2^{\ast }+{\mathcal{H}}_4^{\ast },\left|{\mathcal{H}}_3^{\ast }-{\mathcal{H}}_1^{\ast }\right|<2(1-{\mathcal{H}}_4^{\ast }),{\mathcal{H}}_2^{\ast }-3{\mathcal{H}}_4^{\ast }<3,{\mathcal{H}}_2^{\ast }+{\mathcal{H}}_4^{\ast 2}(1+{\mathcal{H}}_2^{\ast })+{\mathcal{H}}_3^{\ast 2}+{\mathcal{H}}_4^{\ast }(1+{\mathcal{H}}_1^{\ast 2})<1+2{\mathcal{H}}_2^{\ast }{\mathcal{H}}_4^{\ast }+{\mathcal{H}}_1^{\ast }{\mathcal{H}}_3^{\ast }(1+{\mathcal{H}}_4^{\ast })+{\mathcal{H}}_4^{\ast 3}$ where ${\mathcal{H}}_1^{\ast },{\mathcal{H}}_2^{\ast },{\mathcal{H}}_3^{\ast }$, and ${\mathcal{H}}_4^{\ast }$ are depicted in (71). Next, in order to verify theoretical results, some simulations are also presented, and so our main finding of numerical simulations are concluded in the following subsection:

10.1. Numerical findings for theoretical results of discrete COVID‐19 model (8)

Figure 2 showed that interior equilibrium solution $E_{SIQR}^{+}(0.\mathrm{58,0}.\mathrm{2871034482758619,0}.\mathrm{0565721034652596,0}.32563289855383165)$ of discrete COVID‐19 model (8) is a sink. More precisely, Figure 2A indicated that in the beginning of disease outbreak, the number of susceptible individuals was more, decreased with the passage of time and attained a steady state. Figure 2B showed a plot for infected population indicated that disease progress slowly and after some time attain a peak represented that disease spread its maximum level and after attained an equilibrium solution. Figure 2C showed that the number of quarantined individuals was more in the beginning and then decreased and further increased and decreased and stable at the end, whereas Figure 2D showed that in the beginning, recovered population was less than infected one, but with the passage of time, recovered population increased and reached at steady state. Further, Hopf bifurcation diagrams of susceptible, infected, quarantined, recovered, susceptible and infected, infected and quarantined, quarantined and recovered, and recovered and susceptible populations are, respectively, shown in Figure 3A–H whereas maximum Lyapunov exponents for the existence of Hopf bifurcation are given in Figure 3I. In Figure 4, few phase portraits are given, and moreover, Figure 5 clearly showed the closed invariant curves which implies that discrete COVID‐19 model (8) undergoes supercritical Hopf bifurcation. This phenomenon also indicates that there exist periodic or quasi‐periodic oscillations between susceptible, infected, quarantined, and recovered populations. Similar conclusion can also be obtained regarding existence of flip bifurcation diagrams from Figure 6. On the other hand, for controlled discrete COVID‐19 epidemic model (68), Figure 7A showed that the number of susceptible individuals was more at the beginning and with the passage of time decreased and got in steady state. Figure 7B indicated that infection spread with the passage of time and then attained a steady state, whereas Figure 7C indicated that quarantined individuals increases with the passage of time. Moreover, Figure 7D showed that recovered population increases with the passage of time due to proper medication. Finally, Figure 8 has been drawn for collected real data of India in which Figure 8A indicated that the number of susceptible individuals increases rapidly with the passage of time, whereas Figure 8B showed that infection rate is initially too much high, but with the passage of time, it reduced to the minimum. Also, Figure 8C represented that quarantined population increased initially up to a peak value but with the passage of time decreased due to vaccination and other cares. Figure 8D indicated that recovered individuals were small initially due to less medical care and facilities, but with the passage of time, the recovery rate increased due to proper medication and vaccination. Finally, Figure 9 has been drawn for collected real data of India which indicates that discrete COVID‐19 model (8) undergoes Hopf bifurcation.

CONFLICT OF INTEREST

This work does not have any conflicts of interest.

Qadeer Khan A, Tasneem M, Younis BAI, Ibrahim TF. Dynamical analysis of a discrete‐time COVID‐19 epidemic model. Math Meth Appl Sci. 2022;1‐26. doi: 10.1002/mma.8806