Multi-zonal transmission dynamics of pandemic COVID-19 and its stability

Abstract

At the end of 2019, the novel coronavirus (COVID-19) outbreak was identified in Wuhan city of China. Because of its profoundly infectious nature, it was converted into a pandemic in a very short period. Globally, 6,291,764 COVID-19 confirmed cases, and a total of 374,359 deaths are reported as of 1st June 2020; nevertheless, the circumstances become more and more critical over time. Controlling this global pandemic has necessitated extensive strategies putting into practice. Based on the intensity of the epidemic, the model discriminates area into three different types of zones, and this distinction is crucial to construct various effective strategies for all three types of zones separately. The threshold value of the COVID-19 based on the data from zone-wise distribution of Indian districts from 15th April to 3rd May 2020 is calculated. Furthermore, the model is modified in a multi-group model to analyse the global transmission of COVID-19. Optimal control theory is applied to the model. Five control strategies are included based on the level of intensity of COVID-19 .

1. Introduction

In early December 2019, the outbreak of a large family of viruses called coronavirus (COVID-19) which is suspected to be started from Wuhan city of China [1]. Rapidly the COVID-19 outbreak has been transformed into one of the most severe hazards to human health as a result of its rapid transmission rate and lack of an appropriate vaccine or specific treatment for the disease. As of this writing (2nd June 2020), almost all the countries and regions around the world have been affected by the virus and has taken lives beyond three lakhs people of different age group and 53,403 number of cases are still at a critical stage (worldometers.info). Beyond several natural death-traps and human-created disasters, it seems like the world is unable to handle this epidemic situation. To come out from this difficult situation created by the COVID-19 outbreak, almost all infected countries implementing harsh policies to maintain social distancing, and improving medicinal and hospitalization facilities.

The first case of COVID-19 in India originated from China, was reported on 30^th January 2020, in Kerala state situated on the south-western Malabar Coast [2]. Currently, in Asia, India has the largest number of COVID-19 confirmed cases, as on 3^rd June 2020, a total of 216,824 infected cases, 104,071 recoveries, and 6,088 deaths are reported in the country (worldometers.info). The Ministry of Health and Family Welfare (MOHFW) has declared the outbreak as an epidemic in many Indian states and districts and took harsh actions including complete lockdown, quarantine the suspected cases, and sympathetic awareness of the spatial footprint of COVID-19 to predict the possible spread and risk in a region.

A complete lockdown for long-time generates complexity and devastation in the Gross domestic product (GDP) of the country which is inescapable for any country. Centre for Monitoring Indian Economy (CMIE) mentioned that following nationwide lockdown results in unemployment of 27 million youth of 20–30 years age group, also the employment rate has fallen from 40% in February to 26% in June 2020 [3]. In this time, various efforts are being made by the government to design the possibilities of opening the lockdown in a strategic manner so that the COVID-19 transmission rate also not accelerate. On April 15, MOHFW had declared that depending on the number of existing and new cases and transmission rate of COVID-19, districts of India are classifying into three types of zones (red, orange, and green). Red zones have a pandemic situation with a large number of cases and a high transmission rate, orange zones have a comparatively controlled situation with few cases and green zones have not had any cases in the last 21 days. That means an infection-free zone is considered as a green zone, the zone in which infection is at the initial stage is considered as an orange zone and highly contaminated zones are considered as red zones. Moreover, the list of these zones was to be updated based on the incubation period of COVID-19 in the following manner, the red zone will be earmarked as orange zones if no COVID-19 case is reported for two weeks, and again, if no case is reported for next two weeks, the orange zone districts will become green zones [4]. On 15^th April 2020, districts are distributed as zone, 170 red zones, 230 orange zones, and 333 green zones; this distribution of districts is updated on 3^rd May 2020 as 130 red zones, 284 orange zones, and 319 green zones [5], [5]. The authorities have implemented various zone-wise lockdown strategies in which some relaxation is given in the green zone. The goal of this zone wise lockdown strategy is to reopen the economy in parts without raising COVID-19 infection cases.

Observing this strategy of the Government of India, we have made a mathematical model to analyse the transmission rate of COVID-19 at a geographic level. At this time, the collection of scientific literature on mathematical modelling to study the transmission dynamics of the ongoing pandemic is available most extensively [6], [7], [8], [9], [10], [11]. However, the present GOR-model suggest a strategic study of zone wise transmission, this idea contrasts with our previous works [12], [13], [14], [15] and other mathematical models available on the COVID-19 outbreak. In the present model the intensity of the infection in respective zones is considered to develop effective strategies associated with lockdown and medication. Stage wise lockdown strategies are used to fulfill medical emergency and stock of food. Lockdown and medical combined strategies are also used to regulate infected zones.

The manuscript is organized as follows. In Section 2, a compartmental model for the COVID-19 outbreak is proposed. In Section 3, the basic reproduction number for the GOR-model is calculated. In Section 4, local and global stability of disease-free and endemic equilibrium point is proved and examined. In Section 5, the multi-group GOR-model is proposed and the reproduction number for the model is framed. In Section 6, the optimal control theory is applied to the GOR-model which includes certain strong control strategies; and the effect of these control strategies is seen graphically in the next Section 7. The paper is ended with Section 8 of the conclusions.

2. Mathematical modeling

As we have discussed in previous Section 1, to construct the compartmental model, the transmission of COVID-19 infection is organized by subdividing countries in finitely many zones. According to the number of COVID-19 active cases, a transmission rate of the infection, and the extent of testing and surveillance, the zones are mainly classified into three categories as green, orange, and red zones. The following graphical scheme Fig. 1 representing the flow and fluctuations of the infection among three mutually exclusive compartments: the compartment of green zone$\left(G\right)$, the compartment of orange zone$\left(O\right)$ and the compartment of red zone$\left(R\right)$.

Fig. 1.

Graphical scheme representing the interactions among different zones.

The dynamical system is made up of three non-linear ordinary differential equations, describing the evolution of the zones through three different compartments over time:

$$\begin{array}{c}{G}^{\prime }=B-{\beta }_{1}GO-{\beta }_{2}GR+{\beta }_{5}O-\mu G\hfill \\ O^{\prime }={\beta }_{1}GO-{\beta }_{5}O+{\beta }_{3}R-{\beta }_{4}O-\mu O\hfill \\ R^{\prime }={\beta }_{2}GR-{\beta }_{3}R+{\beta }_{4}O-\mu R\hfill \end{array}$$ (1)

Here, $B$is the growth rate and $\mu$ is the rate at which zone become normal and release from this outbreak phase. ${\beta }_1$ and ${\beta }_2$ are the rates at which green zone is converting into orange and red zone respectively in 14 days of the COVID-19 outbreak. It is also observed that as a positive result of lockdown and medication facility, several orange (red) zone are converted into green (orange) zone. ${\beta }_5$ is the rate at which orange zone becoming green and ${\beta }_3$ is rate at which red zone become orange in 15 days. ${\beta }_4$ is the rate by which the orange zone is becoming a red zone in 14 days due to the high transmission rate of the infection. Since the model contains only the human population, initial conditions, and all the parameters used in the model are considered to be non-negative.

We know that $\left(G\left(t\right),O\left(t\right),R\left(t\right)\right)\ge 0$ if $\left(G\left(0\right),O\left(0\right),R\left(0\right)\right)\ge 0$ [SmithHL (2008)]. Based on system (1), we have $N=B-\mu$.

Where $N=G+O+R$, when $t\to \infty$, we have $N\le \frac{B}{\mu }$. Hence$N$is bounded and the feasible region$\left(\Lambda \right)$for system (1) is as follows,

$$\Lambda =\left\{(G,O,R)\in R_{+}^3:G+O+R\le \frac{B}{\mu }\right\}$$ (2)

The solutions of system (1) are called an equilibrium points of the model. The current model has two obvious equilibrium points;

• IDisease free equilibrium point, $E_0=(G_0,0,0)$, where $G_0=\frac{B}{\mu }$, and
• IIEndemic equilibrium point, $E^{*}=(G^{*},O^{*},R^{*})$

where, $G^{*}=r$, $O^{*}=-\frac{{\beta }_3(B-\mu r)}{\mu ({\beta }_{1}r-{\beta }_3-{\beta }_4-{\beta }_5-\mu )}$, $R^{*}=\frac{B{\beta }_2\left({\beta }_{1}r-{\beta }_4\right)-{\beta }_1\mu r\left({\beta }_3+\mu \right)+\left({\beta }_5+\mu \right)\left({\beta }_3\mu -B{\beta }_2\right)+{\mu }^2\left({\beta }_4+{\beta }_5+\mu \right)}{{\beta }_2\mu \left({\beta }_{1}r-{\beta }_3-{\beta }_4-{\beta }_5-\mu \right)}$

Here, $r$is a highest root of the polynomial, given by $p\left(z\right)=a_2{z}^2-a_{1}z+a_0$ and its coefficients are: $a_2={\beta }_1{\beta }_3$, $a_1={\beta }_1({\beta }_3+\mu )+{\beta }_2({\beta }_4+{\beta }_5+\mu )$, $a_0={\beta }_3({\beta }_5+\mu )+\mu ({\beta }_4+{\beta }_5+\mu )$.

3. The basic reproduction number

To get threshold value of zone wise transformation of the COVID-19, basic reproduction number is formulated using the next-generation matrix algorithm [16], [17]. The basic reproduction number for the system (1) is obtained as the spectral radius of the matrix $\left(F{V}^{-1}\right)$ around the disease-free equilibrium point.

The dynamical system (1) is split into two disjoint matrices $f$and $v$. $F$ and $V$are the formed Jacobian matrix of matrices $f$and $v$ respectively, where,

$$f=\left[\begin{array}{c}{\beta }_{1}GO\\ {\beta }_{2}GR\\ 0\end{array}\right]\text{and}v=\left[\begin{array}{c}{\beta }_{5}O-{\beta }_{3}R+{\beta }_{4}O+\mu O\\ {\beta }_{3}R-{\beta }_{4}O+\mu R\\ -B+{\beta }_{1}GO+{\beta }_{2}GR-{\beta }_{5}O+\mu G\end{array}\right]$$

The Jacobian matrices of $f$ and $v$ around disease free equilibrium point are:

$$F=\left[\begin{array}{ccc}\frac{{\beta }_{1}B}{\mu }& 0& 0\\ 0& \frac{{\beta }_{2}B}{\mu }& 0\\ 0& 0& 0\end{array}\right]\text{and}V=\left[\begin{array}{ccc}{\beta }_5+{\beta }_4+\mu & -{\beta }_3& 0\\ -{\beta }_4& {\beta }_3+\mu & 0\\ \frac{{\beta }_{1}B}{\mu }-{\beta }_5& \frac{{\beta }_{2}B}{\mu }& \frac{{\beta }_{2}B}{\mu }-\mu \end{array}\right]$$

The formulated basic reproduction number$R_0=\left(F{V}^{-1}\right)$is:

$$R_0=\frac{B\left(\begin{array}{c}{\beta }_1({\beta }_3+\mu )+{\beta }_2({\beta }_5+\mu )+{\beta }_2{\beta }_4\hfill \\ +{\left(\begin{array}{c}({\beta }_3+\mu )\left({\beta }_1^2({\beta }_3+\mu )-2{\beta }_1{\beta }_2({\beta }_5+\mu )+2{\beta }_1{\beta }_2{\beta }_4\right)\hfill \\ +{\beta }_2^2(2{\beta }_4+{\beta }_5+\mu )({\beta }_5+\mu )+{\beta }_2^2{\beta }_4\hfill \end{array}\right)}^{\frac{1}{2}}\hfill \end{array}\right)}{2\left(({\beta }_5+\mu )({\beta }_3+\mu )+{\beta }_4\mu \right)}$$ (3)

The value of the basic reproduction number$\left(R_0\right)$ also called the threshold value, depends on parameters based on various environmental factors that govern pathogen transmission. Hence, the magnitude of$R_0$varies by places and communities.

In particular, value of the transmission rates used in the model are based on the zone-wise distribution of districts in India from 15th April to 3rd May 2020 [4], [5]. The parameters are set as $B=0.2012$, $\mu =0.08$ (CIA World Factbook. Retrieved January 31, 2020. https://www.cia.gov/library/publications/the-world-factbook/fields/346.html#XX), ${\beta }_1=0.3123$, ${\beta }_2=0.0123$, ${\beta }_3=0.2533$, ${\beta }_4=0.5654$ and ${\beta }_5=0.6758$. The resulting basic reproduction number is $R_0=0.9307$. Note that $R_0<1$, this means that the epidemic outbreak in India can be controlled in the next upcoming months and it is very much possible that the disease dies out from the society and most of the infected district of India converted into the green zone in few months.

4. Stability analysis

In this section, local and global stability of disease-free and endemic equilibrium points is deliberated. The local stability conditions that emerged from the following theory suggest maximizing the transmission rates ${\beta }_3$ and${\beta }_5$. This simply means that to stabilise the model, we need to improve the rate at which the red (orange) zone area transformed into an orange (green) zone area.

4.1. Local stability

Theorem 1

The system (1) with disease free equilibrium point $E_0=(G_0,0,0)$ is locally asymptotically stable if and only if ${\beta }_3+{\beta }_4+{\beta }_5+2\mu >\frac{B}{\mu }\left({\beta }_1+{\beta }_2\right)$,${\beta }_5>B{\beta }_2$,${\beta }_4>B{\beta }_1>{\beta }_4\mu$,${\beta }_5\mu >B{\beta }_1$ and ${\mu }^3>B{\beta }_2{\beta }_5$.

Proof

The system (2) was being linearized around disease free equilibrium point to obtain the disease-free Jacobian matrix as follows,

$$J\left(E_0\right)=\left[\begin{array}{ccc}-\mu & -\frac{{\beta }_{1}B}{\mu }+{\beta }_5& -\frac{{\beta }_{2}B}{\mu }\\ 0& \frac{{\beta }_{1}B}{\mu }-{\beta }_4-{\beta }_5-\mu & {\beta }_3\\ 0& {\beta }_4& \frac{{\beta }_{2}B}{\mu }-{\beta }_3-\mu \end{array}\right]$$ (4)

The characteristic equation of (4) is:

$$P_0\left(\lambda \right)=\left(\lambda +\mu \right)\left({\lambda }^2-a_1\lambda +a_0\right)$$ (5)

with: $a_1=-\frac{1}{\mu }\left({\beta }_3\mu +{\beta }_4\mu +{\beta }_5\mu +2{\mu }^2-B{\beta }_1-B{\beta }_2\right)$

$$\begin{array}{ccc}\hfill a_0& =& \frac{1}{{\mu }^2}({\mu }^2({\beta }_4-B{\beta }_1)+{\mu }^2({\beta }_5-B{\beta }_2)+{\beta }_{2}B(B{\beta }_1-{\beta }_4\mu )+{\beta }_3\mu ({\beta }_5\mu -B{\beta }_1)\hfill \\ & & +\mu ({\mu }^3-B{\beta }_2{\beta }_5))\hfill \end{array}$$ (6)

One eigenvalue obtain from (5) is ${\lambda }_1^0=-\mu$and other two are${\lambda }_{2,3}^0=\frac{1}{2}\left(a_1\pm \sqrt{a_1^2-4a_0}\right)$.

Note that, is $a_0$ is negative then one of the eigenvalues, ${\lambda }_2^0$ or ${\lambda }_3^0$ is positive and others are negative.

• a)If $a_0>0$ and $a_1^2-4a_0\ge 0$, then the equilibrium point $E_0$ is in the form of nodes and if $a_1<0$ then $E_0$ is asymptotically stable.
• b)If $a_0>0$, $a_1^2-4a_0<0$ and $a_1\ne 0$, then the equilibrium point $E_0$ is in the form of focus and if $a_1<0$ then $E_0$ is asymptotically stable.
• c)If $a_1>0$ then $E_0$ is unstable.

Hence, the conditions ${\beta }_3+{\beta }_4+{\beta }_5+2\mu >\frac{B}{\mu }\left({\beta }_1+{\beta }_2\right)$,${\beta }_5>B{\beta }_2$,${\beta }_4>B{\beta }_1>{\beta }_4\mu$,${\beta }_5\mu >B{\beta }_1$ and ${\mu }^3>B{\beta }_2{\beta }_5$ implies $a_1<0$ and $a_0>0$ and which implies the disease free equilibrium point $E_0$is locally asymptotically stable.

Theorem 2

The system (1) with endemic equilibrium point $E^{*}$ is locally asymptotically stable if and only if ${\beta }_5{\beta }_3>{\beta }_{2}r({\beta }_4+{\beta }_5+\mu )+e_2\mu +\frac{e_1}{\mu }$, $r^2{\beta }_2{e}_2>{\beta }_3(B+e_{2}r)$, ${\beta }_3+{\beta }_4>{\beta }_{2}r$,$\mu r>B$ ${\beta }_3+{\beta }_4+{\beta }_5+2\mu >r\left({\beta }_1+{\beta }_2\right)$,$\mu r>B$, ${\beta }_{1}r>{\beta }_3+{\beta }_4+{\beta }_5+\mu$ and $\mu e_2+{\beta }_1{\beta }_{3}r<{\beta }_3{\beta }_5$.

Proof

The system (2) was being linearized around the endemic equilibrium point to obtain the Jacobian matrix as follows,

$$J\left(E^{*}\right)=\left[\begin{array}{ccc}\frac{{\beta }_1{\beta }_3(B-r\mu )}{\mu e_2}-\frac{e_1}{\mu e_2}-\mu & -{\beta }_{1}r+{\beta }_5& -{\beta }_{2}r\\ -\frac{{\beta }_1{\beta }_3(B-r\mu )}{\mu e_2}& {\beta }_{1}r-{\beta }_4-{\beta }_5-\mu & {\beta }_3\\ \frac{e_1}{\mu e_2}& {\beta }_4& {\beta }_{2}r-{\beta }_3-\mu \end{array}\right]$$ (6)

where $e_1=-{\mu }^2{e}_2-{\beta }_1{\beta }_3\mu r+{\beta }_3{\beta }_5\mu +B{\beta }_2({\beta }_{1}r-{\beta }_4-{\beta }_5-\mu )$ and $e_2={\beta }_{1}r-{\beta }_3-{\beta }_4-{\beta }_5-\mu$. Here, $e_1$ and $e_2$ are assumed positive with conditions${\beta }_{1}r>{\beta }_3+{\beta }_4+{\beta }_5+\mu$,$\mu e_2+{\beta }_1{\beta }_{3}r<{\beta }_3{\beta }_5$.

The characteristic equation of (5) is:

$$P^{*}\left(\lambda \right)=\left(\lambda +\mu \right)\left({\lambda }^2-b_1\lambda +b_0\right)$$ (7)

with: $b_1=-\frac{1}{\mu e_2}\left(e_2\mu ({\beta }_3+{\beta }_4+{\beta }_5+2\mu -r{\beta }_1-r{\beta }_2)+{\beta }_1{\beta }_3(\mu r-B)+e_1\right)$

$$b_0=\frac{1}{\mu e_2}\left(\begin{array}{c}{\beta }_1{\beta }_3(\mu r-B)({\beta }_3+{\beta }_4-{\beta }_{2}r)+{\beta }_1\mu \left(r^2{\beta }_2{e}_2-{\beta }_3(B+e_{2}r)\right)\hfill \\ +e_2\mu \left({\beta }_5{\beta }_3-{\beta }_{2}r({\beta }_4+{\beta }_5+\mu )-e_2\mu -\frac{e_1}{\mu }\right)\hfill \end{array}\right)$$

One eigenvalue obtain from (7) is ${\lambda }_1^{*}=-\mu$and other two are${\lambda }_{2,3}^{*}=\frac{1}{2}\left(b_1\pm \sqrt{b_1^2-4b_0}\right)$.

Note that, $b_1<0$ iff ${\beta }_3+{\beta }_4+{\beta }_5+2\mu >r\left({\beta }_1+{\beta }_2\right)$,$\mu r>B$ and $e_1>0$. And $b_0>0$ iff ${\beta }_5{\beta }_3>{\beta }_{2}r({\beta }_4+{\beta }_5+\mu )+e_2\mu +\frac{e_1}{\mu }$, $r^2{\beta }_2{e}_2>{\beta }_3(B+e_{2}r)$, ${\beta }_3+{\beta }_4>{\beta }_{2}r$ and $\mu r>B$. Using the same arguments used in the previous Theorem 1, we can say that, the endemic equilibrium point $E^{*}$is locally asymptotically stable when $b_1<0$ and $b_0>0$.

4.2. Global stability

The global stability of $E_0$ has been proved under the condition${\beta }_3\le \frac{B{\beta }_2}{\mu }$. It is noted that$G\le \frac{B}{\mu }$. Suppose the Lyapunov function as: $L=G+O$.

$$L^{\prime }=B-\mu G+R({\beta }_3-{\beta }_{2}G)-O({\beta }_4+\mu )\le R\left({\beta }_3-\frac{{\beta }_{2}B}{\mu }\right)-O({\beta }_4+\mu ).$$

We have $L^{\prime }\le 0$ for ${\beta }_3\le \frac{B{\beta }_2}{\mu }$, however, $L^{\prime }=0$ only if $G=\frac{B}{\mu }$ and $O=R=0$. Hence, by Laselle's Invariance Principle, all the roots to the equations of the system (1) having initial conditions in the feasible region (2) approaches $E_0$ as $t$ approaches to infinity. Hence, we have the result given below:

Theorem 3

The disease free equilibrium point$E_0$is globally asymptotically stable, provided${\beta }_3\le \frac{B{\beta }_2}{\mu }$.

Now, we prove the global stability of the endemic equilibrium point $\left(E^{*}\right)$ by applying the following theorem [18], [19] by proving that inside the invariant region, the GOR-model has no homoclinic loops, oriented phase polygons, and periodic solutions.

Theorem 4

Consider, $g(G,O,R)$ $=\left\{g_1(G,O,R),g_2(G,O,R),g_3(G,O,R)\right\}$ a piece-wise smooth vector field on ${\Lambda }^{*}$that satisfy the conditions $\left(curlg\right).\overrightarrow{n}<0$and $g.f=0$ inside ${\Lambda }^{*}$, where $f=(f_1,f_2,f_3)$ is a Lipchitz continuous field inside ${\Lambda }^{*}$, $\overrightarrow{n}$ is the normal vector to${\Lambda }^{*}$and $\left(curlg\right)=\left(\frac{\partial g_3}{\partial O}-\frac{\partial g_2}{\partial R}\right)\widehat{i}-\left(\frac{\partial g_3}{\partial G}-\frac{\partial g_1}{\partial R}\right)\widehat{j}+\left(\frac{\partial g_2}{\partial G}-\frac{\partial g_1}{\partial O}\right)\widehat{k}$. Then, the system of differential equations$G=f_1$, $O=f_2$, $R=f_3$ has no periodic solutions, homoclinic loops and oriented phase polygons inside ${\Lambda }^{*}$.

Proof

Let ${\Lambda }^{*}=\left\{(G,O,R):G+O+R=1,G>0,O\ge 0,R\ge 0\right\}$. Clearly${\Lambda }^{*}$is subset of $\Lambda$and it is positively invariant. Using $G+O+R=1$ we rewrite the system in equivalent form.

$$\begin{array}{c}{f}_1(G,O)=B-{\beta }_{1}GO-{\beta }_{2}G(1-G-O)+{\beta }_{5}O-\mu G\hfill \\ f_1(G,R)=B-{\beta }_{1}G(1-G-R)-{\beta }_{2}GR+{\beta }_5(1-G-R)-\mu G\hfill \\ f_2(G,O)={\beta }_{1}GO-{\beta }_{5}O+{\beta }_3(1-G-O)-{\beta }_{4}O-\mu O\hfill \\ f_2(R,O)={\beta }_1(1-R-O)O-{\beta }_{5}O+{\beta }_{3}R-{\beta }_{4}O-\mu O\hfill \\ f_3(G,R)={\beta }_{2}GR-{\beta }_{3}R+{\beta }_4(1-G-R)-\mu R\hfill \\ f_3(O,R)={\beta }_2(1-O-R)R-{\beta }_{3}R-{\beta }_{4}O-\mu R\hfill \end{array}$$ (8)

Let, $g=(g_1,g_2,g_3)$be a vector field, such that

$$\begin{array}{c}{g}_1=\frac{f_3(G,R)}{GR}-\frac{f_2(G,O)}{GO}=-{\beta }_1+{\beta }_2+\frac{{\beta }_3}{O}\left(1-\frac{1}{G}\right)+\frac{{\beta }_4}{R}\left(\frac{1}{G}-1\right)+\frac{{\beta }_5}{G}\hfill \\ g_2=\frac{f_1(G,O)}{GO}-\frac{f_3(O,R)}{OR}=\frac{B}{GO}-{\beta }_1+2{\beta }_2+\frac{{\beta }_2}{O}\left(R+G-2\right)+\frac{{\beta }_3}{O}+\frac{{\beta }_4}{R}+\frac{{\beta }_5}{G}\hfill \\ g_3=\frac{f_2(R,O)}{RO}-\frac{f_1(G,R)}{GR}=-\frac{B}{GR}-2{\beta }_1+\frac{{\beta }_1}{R}\left(2-O-G\right)+{\beta }_2+\frac{{\beta }_3}{O}-\frac{{\beta }_4}{R}+\frac{{\beta }_5}{G}\left(1-\frac{1}{R}\right)\hfill \end{array}$$

Note that, $g.f=0$and using normal vector $\overrightarrow{n}=(1,1,1)$, we get

$\left(curlg\right).\overrightarrow{n}=-\frac{BOR\left(O+R\right)+O^2\left(G{\beta }_4+R{\beta }_5\right)+G{R}^2{\beta }_3}{R^2{O}^2{G}^2}<0$. Hence, there are no solutions with homoclinic loops, periodicity and oriented phase polygons inside${\Lambda }^{*}$, hence$E^{*}$exist inside${\Lambda }^{*}$and it is globally asymptotically stable.

5. Multi-group model

Since the parametric values depend upon several social and environmental factors related to different geographical locations, the transmission rate of COVID-19 is different in every country. To analyse global transmission dynamics more accurately, we divide the globe into $n$-groups. Here each group is correlated to the other groups by parameter${\beta }_{kj}$, it shows the transmission of infection from $j^{th}$ group to $k^{th}$ group. Each $i^{th}-\text{group}$, $i=1,2,...,n$ contains three compartments; $G_i$, the compartment of green zone, $O_i$, the compartment of orange zone and $R_i$, the compartment of red zone. Meanwhile, due to heavy lockdown in most of the red zone around the world, the transmission of infection between two groups through red zone is not easily possible, hence it is not considered in this multi-group model. The transmission dynamics for the multi-group model is shown in the following Fig. 2 .

Fig. 2.

Transmission diagram of Multi-group dynamical model.

The formulation of the multi-group model is as below:

$$\begin{array}{c}{G^{\prime }}_k=B_k-\sum _{j=1}^n{\beta }_{kj}{G}_k{O}_j-{\beta }_2^k{G}_k{R}_k+{\beta }_5^k{O}_k-{\mu }_k{G}_k\hfill \\ {O^{\prime }}_k=\sum _{j=1}^n{\beta }_{kj}{G}_k{O}_j-{\beta }_5^k{O}_k+{\beta }_3^k{R}_k-{\beta }_4^k{O}_k-{\mu }_k{O}_k\hfill \\ {R^{\prime }}_k={\beta }_2^k{G}_k{R}_k-{\beta }_3^k{R}_k+{\beta }_4^k{O}_k-{\mu }_k{R}_k\hfill \end{array}$$ (9)

${\beta }_{kj}$ is the rate at which orange zone of $j^{th}-\text{group}$ transmit the infection to the green zone of $k^{th}-\text{group}$. Other parameters are considered same as model (1) for respective group.

For each $k$, adding the three equations in system (9) gives

$\left({G^{\prime }}_k+{O^{\prime }}_k+{R^{\prime }}_k\right)=B_k-{\mu }_k(G_k+O_k+R_k)$; hence, $\underset{t\to \infty }{\lim \sup }\left(G_k\left(t\right),O_k\left(t\right),R_k\left(t\right)\right)\le \frac{B_k}{{\mu }_k}$. Therefore$G_k\left(t\right)$, $O_k\left(t\right)$ and $R_k\left(t\right)$ are uniformly bounded in $R_{+}^{3n}$.

Hence, the feasible region for the system (9) is as follow:

$${\Lambda }_m=\left\{\left(G_1,O_1,R_1,...,G_n,O_n,R_n\right)\in R_{+}^{3n}/G_k+O_k+R_k\le \frac{B_k}{{\mu }_k},1\le k\le n\right\}$$ (10)

The system (9) always have the disease free equilibrium point, $E_0^m=\left(G_1^0,0,0,...,G_2^0,0,0\right)$ on the boundary of the feasible region (10) with $G_k^0=\frac{B_k}{{\mu }_k}$.

An equilibrium point $E_m^{*}=\left(G_m^{*},O_m^{*},R_m^{*}\right)$ is called the endemic equilibrium point, which satisfy the following equilibrium equations

$$B_k+{\beta }_5^k{O}_k^{*}=\sum _{j=1}^n{\beta }_{kj}{G}_k^{*}{O}_j^{*}+{\beta }_2^k{G}_k^{*}{R}_k^{*}+{\mu }_k{G}_k^{*}$$ (11)

$$\sum _{j=1}^n{\beta }_{kj}{G}_k^{*}{O}_k^{*}+{\beta }_3^k{R}_k^{*}=\left({\beta }_5^k+{\beta }_4^k+{\mu }_k\right)O_k^{*}$$ (12)

$$R_k^{*}\left({\beta }_2^k{G}_k^{*}-{\beta }_3^k-{\mu }_k\right)=-{\beta }_4^k{O}_k^{*}$$ (13)

and $\sum _{j=1}^n{\beta }_{kj}{G}_k^{*}{O}_j^{*}=O_k^{*}\left[{\beta }_5^k+{\beta }_4^k+{\mu }_k-\frac{{\beta }_3^k{\beta }_4^k}{{\beta }_3^k-{\beta }_2^k{G}_k^{*}+{\mu }_k}\right]$, which follows from (13) and (12). Let,

$$R_0^m=\rho \left(M_0\right)$$ (14)

where, $R_0^m$ is the basic reproduction number for the multi-group system (9), given by spectral radius of the matrix$M_0$.

Here, $M_0=M\left(G_1^0,G_2^0,...,G_n^0\right)={\left[\frac{{\beta }_{kj}{G}_k^0\left({\beta }_3^k-{\beta }_2^k{G}_k^0+{\mu }_k\right)}{\left({\beta }_5^k+{\beta }_4^k+{\mu }_k\right)\left({\beta }_3^k-{\beta }_2^k{G}_k^0+{\mu }_k\right)-{\beta }_3^k{\beta }_4^k}\right]}_{1\le k\le n}$Using this formula, one can easily find the threshold value for $n$ number of countries and analyse the effectiveness of the rate${\beta }_{kj}$, i.e. the rate at which one country transmits the virus to other countries.

6. Optimal control

In this section, we extend the system (1) by including time dependent control strategies,$u_1\left(t\right)$, $u_2\left(t\right)$, $u_3\left(t\right)$, $u_4\left(t\right)$ and $u_5\left(t\right)$ related to restriction in movement in a zone and essential medication facilities to fight against the COVID-19 pandemic outbreak. The role and place of each control variable are demonstrated in Table 1 and Fig. 3 respectively.

Table 1.

Control variables used in the optimal control theory.

Control variables
$u_1\left(t\right)$	This control variable suggests first stage lockdown, in which people are allowed to move only for a medical emergency and stock of food.
$u_2\left(t\right)$	This control variable applied when the transmission rate becomes extremely intense. In such a case, there is a possibility that the green zone turns out in red very quickly. In this situation, this control variable suggests complete lockdown, in which, almost all activities, excepting essential services related to medical emergencies, will be stopped in such immerging infective zones.
$u_3\left(t\right)$	This control strategy needs to control transmission between the primarily infected zone (orange zone) to the critically infected zone (red zone). Hence to control the situation we need to restrict movements in the zone, as well as we need to improve hospitalization facilities to cure infected individuals in the orange zone.
$u_4\left(t\right)$	This control variable has remarkable importance; it suggests the improved medication strategy to support the rate at which red zone converting to an orange zone.
$u_5\left(t\right)$	This control strategy helps the orange zone to become green zone by curing infected cases in the orange zone. Which needs better medication to improve the immunity of infected individuals.

Fig. 3.

COVID-19 model with control variables.

According to the above assumptions, the COVID-19 model (1) is modified as follow:

$$\begin{array}{c}{G}^{\prime }=B-{\beta }_{1}GO-{\beta }_{2}GR+{\beta }_{5}O-\mu G+u_{1}G+u_{2}G+u_{5}O\hfill \\ O^{\prime }={\beta }_{1}GO-{\beta }_{5}O+{\beta }_{3}R-{\beta }_{4}O-\mu O-u_{1}G-u_{5}O+u_{3}O+u_{4}R\hfill \\ R^{\prime }={\beta }_{2}GR-{\beta }_{3}R+{\beta }_{4}O-\mu R-u_{2}G-u_{3}O-u_{4}R\hfill \end{array}$$ (15)

According to this extended model, the optimal control situation with the objective function is formulated by

$$\begin{array}{ccc}\hfill \text{Optimise}J(u_1,u_2,u_3,u_4,u_5)& =& \underset{0}{\stackrel{T}{\int }}(A_{1}G\left(t\right)+A_{2}O\left(t\right)+A_{3}R\left(t\right)+w_1{u}_1^2+w_2{u}_2^2\hfill \\ & & +w_3{u}_3^2+w_4{u}_4^2+w_5{u}_5^2)\mathrm{d}t\hfill \end{array}$$ (16)

In above objective function (16), $A_i,i=1,2,3$, are weight constants of the respective compartments and $w_j,j=1,2,...,5$ are weight constant of respective control variables. The objective is to minimize the spread of COVID-19 through zones by determining optimal control functions $(u_1^{*},u_2^{*},u_3^{*},u_4^{*},u_5^{*})$, subject to the system (15), such that

$$J\left(u_1^{*},u_2^{*},u_3^{*},u_4^{*},u_5^{*}\right)=optimise\left(J(u_1,u_2,u_3,u_4,u_5)/(u_1,u_2,u_3,u_4,u_5)\in \varphi \right)$$ (17)

where, the control strategy set is:

$$\varphi =\left\{(u_1,u_2,u_3,u_4,u_5)/u_i\left(t\right)\text{is}\text{Lebesgue}\text{measurable}\text{on}[0,T],0\le u_i\left(t\right)\le 1,i=1,2,...,5\right\}$$

The integrand, $A_{1}G\left(t\right)+A_{2}O\left(t\right)+A_{3}R\left(t\right)+w_1{u}_1^2+w_2{u}_2^2+w_3{u}_3^2+w_4{u}_4^2+w_5{u}_5^2$ of the objective function (16) is convex in the set $\varphi$. The control strategy set $\varphi$ is also close and convex by definition. Since the model (15) is bounded and linear in the control variables, the conditions for the existence of optimal control are satisfied [Fleming and Rishel, (1975)]. Hence, here exist $(u_1^{*},u_2^{*},u_3^{*},u_4^{*},u_5^{*})\in \varphi$ such that (17) exist.

Let we convert this optimality problem into a problem of maximizing a Langrangian function$L$, with respect to all control variables. We use Pontryagins maximum principle for necessary condition of an optimal control problem (Pontryagin 1987).

$$\begin{array}{c}L=A_1{G}^2\left(t\right)+A_2{O}^2\left(t\right)+A_3{R}^2\left(t\right)+w_1{u}_1^2+w_2{u}_2^2+w_3{u}_3^2+w_4{u}_4^2+w_5{u}_5^2\hfill \\ +{\lambda }_1\left(B-{\beta }_{1}GO-{\beta }_{2}GR+{\beta }_{5}O-\mu G+u_{1}G+u_{2}G+u_{5}O\right)\hfill \\ +{\lambda }_2\left({\beta }_{1}GO-{\beta }_{5}O+{\beta }_{3}R-{\beta }_{4}O-\mu O-u_{1}G-u_{5}O+u_{3}O+u_{4}R\right)\hfill \\ +{\lambda }_3\left({\beta }_{2}GR-{\beta }_{3}R+{\beta }_{4}O-\mu R-u_{2}G-u_{3}O-u_{4}R\right)\hfill \end{array}$$

For given optimal control $u^{*}=(u_1^{*},u_2^{*},u_3^{*},u_4^{*},u_5^{*})$ and corresponding state solutions of the corresponding system (5), there exist adjoint functions,${\lambda }_i$, $i=1,2,3$, as follow:

$${\lambda }_1^{\prime }=\frac{-\partial L}{\partial G}=-2A_{1}G+\left({\lambda }_1-{\lambda }_2\right)\left({\beta }_{1}O-u_1\right)+\left({\lambda }_1-{\lambda }_3\right)\left({\beta }_{2}R-u_2\right)+{\lambda }_1\mu$$

$${\lambda }_2^{\prime }=\frac{-\partial L}{\partial O}=-2A_{2}O+\left({\lambda }_1-{\lambda }_2\right)\left({\beta }_{1}G-{\beta }_5-u_5\right)+\left({\lambda }_2-{\lambda }_3\right)\left({\beta }_4-u_3\right)+{\lambda }_2\mu$$

$${\lambda }_3^{\prime }=\frac{-\partial L}{\partial R}=-2A_{3}R+\left({\lambda }_1-{\lambda }_3\right){\beta }_{2}G+\left({\lambda }_3-{\lambda }_2\right)\left({\beta }_3-u_4\right)+{\lambda }_3\mu$$

Using terminal condition${\lambda }_i\left(T\right)=0$, for $i=1,2,3$ and optimality condition, $-\frac{\partial L}{\partial u_i}=0$, for $i=1,2,...,5$, the optimal control variables $u_1^{*}$, $u_2^{*}$, $u_3^{*}$, $u_4^{*}$ and $u_5^{*}$ is solved.

$$u_1=\frac{\left({\lambda }_2-{\lambda }_1\right)G}{2w_1},u_2=\frac{\left({\lambda }_3-{\lambda }_1\right)G}{2w_2},u_3=\frac{\left({\lambda }_3-{\lambda }_2\right)O}{2w_3},u_4=\frac{\left({\lambda }_3-{\lambda }_2\right)R}{2w_4}\text{and}{u}_5=\frac{\left({\lambda }_2-{\lambda }_1\right)O}{2w_5}$$ (18)

Moreover, optimal control strategies $u_1^{*}$, $u_2^{*}$, $u_3^{*}$, $u_4^{*}$ and $u_5^{*}$ are given by:

$$u_1^{*}=\max \left\{0,\min \left\{1,\frac{\left({\lambda }_2-{\lambda }_1\right)G}{2w_1}\right\}\right\}$$ (19)

$$u_2^{*}=\max \left\{0,\min \left\{1,\frac{\left({\lambda }_3-{\lambda }_1\right)G}{2w_2}\right\}\right\}$$ (20)

$$u_3^{*}=\max \left\{0,\min \left\{1,\frac{\left({\lambda }_3-{\lambda }_2\right)O}{2w_3}\right\}\right\}$$ (21)

$$u_4^{*}=\max \left\{0,\min \left\{1,\frac{\left({\lambda }_3-{\lambda }_2\right)R}{2w_4}\right\}\right\}$$ (22)

$$u_5^{*}=\max \left\{0,\min \left\{1,\frac{\left({\lambda }_2-{\lambda }_1\right)O}{2w_5}\right\}\right\}$$ (23)

7. Numerical simulation

This section attempts to describe the graphical representation of variations in the model, which helps to visualize flow of the COVID-19 outbreak and the influence of optimal control strategies on it.

Fig. 4 shows movement of zones from one stage to other stage. The initialisation of the compartment of green, orange, and red zone is given by $G\left(0\right)=0.66$, $O\left(0\right)=2.84$, and $R\left(0\right)=1.70$ respectively. It can be seen that compartments of both the infected zones, red and green zone, are decreases with time and become a very low in number after twenty weeks of the outbreak. Moreover, a gradual improvement is observed in the compartment of green zone.

Fig. 4.

Variation in compartment with time.

Fig. 5 shows the transmission between all three compartments with time. It shows the high intensity of transmission of COVID-19 during the initial period of the outbreak.

Fig. 5.

Scatter diagram of COVID-19 model.

Fig. 6 shows a bifurcation diagram which helps to validate the qualitative information about the basic reproduction number. Here, the blue vertical line indicates value of the basic reproduction number, $R_0=0.93$ and the red vertical line indicates value of the critical point$R_C$, which is 1.52. This higher value of critical point indicates that this is the point from which the system's stability switches from unstable to stable state. To effectively control the spread of COVID-19, the critical point, $R_C$ should be brought below 1.

Fig. 6.

Bifurcation diagram for the COVID-19 model.

Fig. 7 shows the variation in each compartment under influence of with and without control strategies. It is observed that the COVID-19 outbreak can be controlled up to a significant level in twenty weeks after applying all the control strategies together. Notable progress in the compartment of green zone and degradation in the compartment of orange and red zone is observed after twenty weeks after applying optimal control theory to the GOR-model.

Fig. 7.

Change in each compartment with and without control strategies.

Deviation in the intensity of control variables with time is shown in Fig. 8 . Furthermore, the figure also suggests the time and amount of intensity of the mixed control strategies should be applied to control the COVID-19 outbreak in around 50 days. It can be observed that the requirement of the first two control strategies $u_1\left(t\right)$ and $u_2\left(t\right)$, consist of lockdown strategies are present throughout the outbreak. The requirement of the fourth control strategy which helps the red zone turned out to the green zone by improving recovery rate, is very essential in between the outbreak.

Fig. 8.

Change in control variables with time.

8. Conclusion

To control the increasing number of infected cases of COVID-19 in around the world, we need to control the geographic transmission of the virus. It is easy to reduce impact of the virus in one zone compared to the whole group of the zone, hence once the geographic transmission or multi-zonal transmission controlled, the transmission of COVID-19 can be controlled in a short time duration. Hence, to explore and analyse the geographic transmission of the virus, the GOR-model is constructed. The basic reproduction number (3) is formulated which shows the transmission rate of infected zone through the compartment of green zone which is 0.9307. This transmission rate of the infected zone is in one group. In the present work, GOR-model is expanded to study multi-group transmission dynamics. The basic reproduction number (14) for the multi-group model is expressed. Local and global stability of the GOR-model is proved and stability conditions also demand to reduce the number of infected zones. The study of parametric inequality used in the conditions formulated in the process of stabilising the GOR-model helps to develop certain strategies to stabilise the current epidemic situation. Using optimal control theory, several effective control strategies are applied to the model and their effect on disease spread is also visualized in the simulation section.

The model is applied to the released data based on zone wise transmission of COVID-19 through all 276 districts of India from 1^st May to 15^th May 2020. The model suggests that strategic partially lockdown according to intensity of infection should be kept throughout this pandemic outbreak of COVID-19. The multi-group model can also be useful to predict the variation in threshold value with deviation in the rate at which the virus transmits from one group to another. Undeniably, this investigation has one limitation, due to the limited data accessible for zone-wise distribution of other countries than India, the global geographic transmission using the constructed multi-group model cannot calculate easily. In the future, one can develop further this model.

Declaration of Competing Interest

The authors declare no conflict of interest.