Modelling COVID-19 transmission in the United States through interstate and foreign travels and evaluating impact of governmental public health interventions

Abstract

Background: The first case of COVID-19 was reported in Wuhan, China in December 2019. The disease has spread to 210 countries and has been labelled as a pandemic by the World Health Organization (WHO). Modelling, evaluating, and predicting the rate of disease transmission is crucial in understanding optimal methods for prevention and control. Our aim is to assess the impact of interstate and foreign travel and public health interventions implemented by the United States government in response to the COVID-19 pandemic. Methods: A disjoint mutually exclusive compartmental model was developed to study transmission dynamics of the novel coronavirus. A system of nonlinear differential equations was formulated and the basic reproduction number $R_0$ was computed. Stability of the model was evaluated at the equilibrium points. Optimal controls were applied in the form of travel restrictions and quarantine. Numerical simulations were conducted. Results: Analysis shows that the model is locally asymptomatically stable, at endemic and foreigners free equilibrium points. Without any mitigation measures, infectivity and subsequent hospitalization of the population increased. When interstate and foreign travel was restricted and the population placed under quarantine, the probability of exposure and subsequent infection decreased significantly; furthermore, the recovery rate increased substantially. Conclusion: Interstate and foreign travel restrictions, in addition to quarantine, are necessary in effectively controlling the pandemic. The United States has controlled COVID-19 spread by implementing quarantine and restricting foreign travel. The government can further strengthen restrictions and reduce spread within the nation more effectively by implementing restrictions on interstate travel.

1. Introduction

In December 2019, an unidentified pneumonia was found in Wuhan, Hubei province, China. The responsible virus was later identified as the novel coronavirus 2019, and the disease as coronavirus disease 2019 (COVID-19) [4]. COVID-19 is transmitted via direct contact with an infected person through respiratory droplets when a person coughs or sneezes, or by indirect contact with contaminated surfaces with respiratory droplets from infected person and then touching their eyes, nose or mouth [6]. Furthermore, the virus remains on surfaces from a few hours to several days and has an incubation period between 1-14 days. Consequently, the disease spread rapidly from Wuhan to all parts of the country and overseas.

Introduction and spread of COVID-19 within the United States is a direct result of transmission through foreign and interstate travel. The first known case of COVID-19 in the U.S.A. was confirmed on 20th January 2020 in a 35-year-old individual who had travelled from Wuhan to Washington state [6]. Soon, cases started appearing and rising in many other states of the U.S.A. including New York, New Jersey, Illinois, Florida, Georgia, Texas, Pennsylvania due to interstate travel. The CDC alarmed that hospitals may get overwhelmed by a large number of people seeking care at the same time due to widespread transmission of disease which may lead to otherwise preventable deaths (2020) [2]. In response to the oncoming epidemic the US government implemented the following regulations (Table 1 ).

Table 1.

Timeline of public health intervention implemented by the United States government.

Date	Action
17th Jan	Public health entry screening at 3 U.S. Airport.
31st Jan	Coronavirus declared Public health emergency, Chinese travel restrictions, restricted entry into U.S.A for foreign nationals who pose a risk of transmission, Funnel all flights from China to just 7 U.S. domestic airports.
29th Feb	Barred all travel to Iran, level 4 travel advisory to areas of Italy and South Korea.
11th March	Travel restriction for foreigners who visited Europe in the last 14 days.
14th March	Europe travel ban extended to UK and Ireland. Implement Social Distancing and Closure of teaching institutes in many states.
18th March	Temporary closure of U.S. Canada border for non-essential traffic.
19th March	Americans to avoid all international travel.
20th March	The U.S. and Mexico agreed to restrict non-essential cross border traffic. Closure of non-essential businesses and shelter in place order in NY.
24th March	Self-quarantine for 14 days for individuals who recently visited New York.
28th March	For residents in NY, NJ, CT, Avoid non-essential domestic travel for 2 weeks.
29th March	Social distancing extended through 30th April.
3rd April	All American wear non-medical, fabric or cloth masks to prevent asymptomatic spread of coronavirus.

While the United States has implemented numerous public health interventions, it has not implemented a ban on interstate travel. According to the World Health Organization (WHO) [12]. New cases of COVID-19 have emerged in 210 countries with 1,733, 945 confirmed cases and 106, 518 confirmed deaths globally as of 10th April 2020. The United States has implemented quarantine measures, close contact tracing, early testing for individuals with symptoms, hospitalization if needed, and closing of teaching institutes and non-essential businesses. Studies have shown that in other countries, the complete lockdown of travel has decreased the spread of the disease in the surrounding states ([1]; [10]). In order to prevent the transmission of COVID-19 within the US, the mode of transmission must first be modelled and understood. Mathematical modelling is ideal for evaluating and predicting the rate of disease transmission. Data-driven mathematical modelling plays an important role in epidemic mitigation, in preparedness for future epidemic and in the evaluation of control effectiveness [13]. In this study, we adopted a disjoint mutually exclusive compartmental model to shed light on the transmission dynamics from foreign and interstate travel of the novel coronavirus and our aim is to assess the impact of public health interventions on infection by measuring basic reproduction number, contact rate, newly confirmed cases, total confirmed cases, total death. Our estimated parameters are largely in line with World Health Organization estimates and previous studies (2019).

2. Mathematical modelling

Mathematical modelling plays a vital role in determining dynamics of diseases. In this paper we consider a disjoint mutually exclusive compartmental model with compartments as follows: Exposed to COVID-19 E i.e. this compartment consists of individuals (Both foreign population and interstate population) who are exposed to COVID-19, next compartment is $I_S$ i.e. transmission of COVID-19 through interstate travel, COVID-19 transmission through foreigners F - this compartment includes U.S. population which is exposed to COVID-19, Quarantined class Q, COVID-19 Infected I - this class includes infected population as well infectious population, hospitalized H - this class includes hospitalization of both COVID-19 infectives and also those who are exposed to COVID-19 and last compartment includes recovered population from hospitalized population denoted by R. Notations and parametric values used in the formulation of dynamical system model are given in the following Table 2 .

Table 2.

Model parameters and their interpretation.

Notation	Parameter description	Parametric values
B	Birth rate	0.0181
β1	Rate at which U.S. population gets exposed to COVID-19 via interstate travel	0.0011
β2	Rate at which U.S. population gets exposed to COVID-19 through contact with foreigners	0.0003
β3	Rate of COVID-19 exposure through foreigners engaged in interstate travel	0.0012
β4	Rate at which Interstate population goes for quarantine	0.0018
β5	Rate at interstate population gets infected	0.0035
β6	Rate at which foreigner quarantine themselves	0.000015
β7	Rate at which foreigner gets infected	0.000001
β8	Rate at which quarantine humans gets infected by COVID-19	0.0025
β9	Rate at which infected humans gets hospitalized	0.0037
β10	Rate at which exposed humans gets hospitalized	0.000002
β11	Rate at which hospitalized humans gets recovered	0.0000003
μ	Death rate	0.000119
μc	Death rate due to COVID-19	0.0027

This model considers new recruitment in the exposed class at the rate B and all the compartments have mortality rate μ. Here ${\beta }_1$ is the US population exposed to COVID-19 via interstate travel, and ${\beta }_2$ is the rate at which the US population gets exposed to COVID-19 through contact with foreigners. Next, ${\beta }_3$ is the rate of COVID-19 exposure through foreigners engaged in interstate travel. The US population engaging in interstate travel and foreigners quarantine themselves at the rate ${\beta }_4$ and ${\beta }_6$ respectively. Similarly, after getting exposed to COVID-19, US population engaging in interstate travel and foreigner population gets the infection joining infectious class I with the rate ${\beta }_5$ and ${\beta }_7$ respectively. Quarantined humans also get the infection at the rate ${\beta }_8$. Next, we assume infected population gets hospitalized at the rate ${\beta }_9$ joining H. We also assume population gets admitted to the hospital at the initial exposure of the disease at the rate ${\beta }_{10}$. Hospitalized patients after undergoing treatment gets recover joining R at the rate ${\beta }_{11}$. We take into consideration death due to COVID-19 ${\mu }_c$, when the individual is hospitalized.

The Fig. 1 gives rise to the following set of non-linear ordinary differential equations

$$\begin{gathered} \frac{dE}{dt}=B-{\beta }_{1}E{I}_S-{\beta }_{2}EF-{\beta }_{10}E-\mu E \\ \frac{d{I}_S}{dt}={\beta }_{1}E{I}_S+{\beta }_{3}F-({\beta }_4+{\beta }_5+\mu )I_S \\ \frac{dF}{dt}={\beta }_{2}EF-({\beta }_3+{\beta }_6+{\beta }_7+\mu )F \\ \frac{dQ}{dt}={\beta }_4{I}_S+{\beta }_{6}F-({\beta }_8+\mu )Q \\ \frac{dI}{dt}={\beta }_5{I}_S+{\beta }_{7}F+{\beta }_{8}Q-({\beta }_9+\mu )I_M \\ \frac{dH}{dt}={\beta }_{9}I+{\beta }_{10}E-({\beta }_{11}+{\mu }_C+\mu )I_M \\ \frac{dR}{dt}={\beta }_{11}H-\mu R \end{gathered}$$ (1)

where, $N(t)=E(t)+I_S(t)+F(t)+Q(t)+I(t)+H(t)+R(t)$.

Fig. 1.

Compartmental diagram showing movement of individuals from one compartment to another compartment.

Adding all the differential equations of model, we get,

$$\frac{dN}{dt}\le B-\mu (E+I_S+F+Q+I+H+R)\ge 0$$

Hence, $\frac{dN}{dt}\le B-\mu N$.

So that $\underset{t\to \infty }{\lim }\mathrm{supN}\le \frac{B}{\mu }$.

Then, Feasible Region for the system is defined as

$$\mathrm{\Lambda }=\{(E,I_S,F,Q,I,H,R);E+I_S+F+Q+I+H+R\le \frac{B}{\mu },\}$$ (2)

with $E>0,I_S>0,F>0,Q>0,I>0,H>0,R>0$.

This system has following equilibrium points

i. Foreigner free equilibrium point

ii. Endemic equilibrium point

3. Reproduction number

Basic reproduction number is defined as the total number of secondary infections in a total susceptible population. Here, we calculate the reproduction number using Diekmann et al., when the disease in its endemic stage i.e. for this model it is defined as percentage of population infected by a single infection in a totally exposure situation [5]. We also compute the value of reproduction number $R_F$ when there are no foreigners present in the total population.

$$F_1=\left[\begin{array}{ccccccc}\hfill {\beta }_{1}E\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\beta }_1{I}_S\hfill \\ \hfill 0\hfill & \hfill {\beta }_{2}E\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\beta }_{2}F\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \hfill \end{array}\right]$$

$$V_1=\left[\begin{array}{ccccccc}\hfill {\beta }_4+{\beta }_5+\mu \hfill & \hfill -{\beta }_3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill l1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -{\beta }_4\hfill & \hfill -{\beta }_6\hfill & \hfill {\beta }_8+\mu \hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -{\beta }_5\hfill & \hfill -{\beta }_7\hfill & \hfill -{\beta }_8\hfill & \hfill {\beta }_9+\mu \hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\beta }_9\hfill & \hfill {\beta }_{11}+\mu +{\mu }_C\hfill & \hfill 0\hfill & \hfill -{\beta }_{10}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\beta }_{11}\hfill & \hfill \mu \hfill & \hfill 0\hfill \\ \hfill {\beta }_{1}E\hfill & \hfill {\beta }_{2}E\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill l2\hfill \\ \hfill \hfill \end{array}\right]$$

where, $l1={\beta }_3+{\beta }_6+{\beta }_7+\mu$, $l2={\beta }_1{I}_S+{\beta }_{2}F+{\beta }_{10}+\mu$.

The reproduction number $R_{E^{⁎}},R_F$ is the spectral radius of $F_1{V_1}^{-1}(E^{⁎})$ and $F_1{V_1}^{-1}(E^F)$ respectively. The value of $R_{E^{⁎}}=81\text{\%}$ and $R_F=1.10$.

4. Stability analysis

In this section we study the stability analysis of the model. Here we study Local stability of all the equilibrium points using Routh-Hurwitz criterion by Routh 1877 [9].

Theorem 1

The foreigner free equilibrium point is locally asymptotically stable if $({\beta }_4+{\beta }_5+\mu )<\max \{\frac{B{\beta }_1}{{\beta }_{10}+\mu },\frac{{\beta }_1({\beta }_3+{\beta }_6+{\beta }_7+\mu )}{{\beta }_2}\}$ .

Proof

The Jacobian of system (1) at Foreigner free equilibrium is as follows

$$J^F=\left[\begin{array}{ccccccc}\hfill -t_1-{\beta }_{10}-\mu \hfill & \hfill -({\beta }_4+{\beta }_5+\mu )\hfill & \hfill \frac{-{\beta }_2({\beta }_4+{\beta }_5+\mu )}{{\beta }_1}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill t_1\hfill & \hfill 0\hfill & \hfill {\beta }_3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill t_2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\beta }_4\hfill & \hfill {\beta }_6\hfill & \hfill -t_3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\beta }_5\hfill & \hfill {\beta }_7\hfill & \hfill {\beta }_8\hfill & \hfill -t_4\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill {\beta }_{10}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\beta }_9\hfill & \hfill -t_5\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\beta }_{11}\hfill & \hfill -\mu \hfill \\ \hfill \hfill \end{array}\right]$$

$$t_1=\frac{B{\beta }_1-({\beta }_{10}+\mu )({\beta }_4+{\beta }_5+\mu )}{({\beta }_4+{\beta }_5+\mu )},t_2=\frac{{\beta }_2({\beta }_4+{\beta }_5+\mu )}{{\beta }_1}-({\beta }_3+{\beta }_6+{\beta }_7+\mu ),t_3=({\beta }_8+\mu ),t_5=({\beta }_{11}+\mu +{\mu }_C).$$

The eigen values of the Jacobian $J^F$ are

$${\lambda }_1=t_2,{\lambda }_2=-({\beta }_8+\mu ),{\lambda }_3=-({\beta }_9+\mu ),{\lambda }_4=-\mu ,{\lambda }_5=-({\beta }_{11}+\mu +{\mu }_C),{\lambda }_{6,7}=-\frac{1}{2}(t_1+{\beta }_{10}+\mu \pm \sqrt{\xi }),\xi ={({\beta }_{10}+\mu )}^2+2{\beta }_{10}{t}_1-4t_1({\beta }_4+{\beta }_5)-2\mu t_1+t_1^2$$

If it has imaginary roots i.e. $\xi <0$. Then we have negative real part. Hence the theorem. But if $\xi \ge 0$, then eigen values are negative if $({\beta }_4+{\beta }_5+\mu )<\max \{\frac{B{\beta }_1}{{\beta }_{10}+\mu },\frac{{\beta }_1({\beta }_3+{\beta }_6+{\beta }_7+\mu )}{{\beta }_2}\}$. Hence the Foreigner free equilibrium point is locally asymptotically stable.

Theorem 2

The endemic equilibrium point is locally asymptotically stable if $E^{⁎}<\max \{\frac{{\beta }_4+{\beta }_5+\mu }{{\beta }_1},\frac{{\beta }_3+{\beta }_6+{\beta }_7+\mu }{{\beta }_2}\}$ .

Proof

The Jacobian matrix of system (1) for endemic equilibrium is given by

$$J^{⁎}=\left[\begin{array}{ccccccc}\hfill -a_{11}\hfill & \hfill -{\beta }_1{E}^{⁎}\hfill & \hfill -{\beta }_2{E}^{⁎}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill {\beta }_1{I}_S^{⁎}\hfill & \hfill -a_{22}\hfill & \hfill {\beta }_3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill {\beta }_2{F}^{⁎}\hfill & \hfill 0\hfill & \hfill -a_{33}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\beta }_4\hfill & \hfill {\beta }_6\hfill & \hfill -a_{44}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\beta }_5\hfill & \hfill {\beta }_7\hfill & \hfill {\beta }_8\hfill & \hfill -a_{55}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill {\beta }_{10}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\beta }_9\hfill & \hfill -a_{66}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\beta }_{11}\hfill & \hfill -\mu \hfill \\ \hfill \hfill \end{array}\right]$$

where, $a_{11}={\beta }_1{I}_S^{⁎}+{\beta }_2{F}^{⁎}+{\beta }_{10}+\mu$, $a_{22}=-{\beta }_1{E}^{⁎}+{\beta }_4+{\beta }_5+\mu$, $a_{33}=-{\beta }_2{E}^{⁎}+{\beta }_3+{\beta }_6+{\beta }_7+\mu$, $a_{44}={\beta }_8+\mu$, $a_{55}={\beta }_9+\mu$, $a_{66}={\beta }_{11}+{\mu }_C+\mu$.

The characteristic polynomial for Jacobian $J^{⁎}$ is

$${\lambda }^7+b_6{\lambda }^6+b_5{\lambda }^5+b_4{\lambda }^4+b_3{\lambda }^3+b_2{\lambda }^2+b_1\lambda +b_0$$

where,

$$\begin{gathered} b_0=a_{44}{a}_{55}{a}_{66}\mu (E^{⁎}(F^{⁎}{\beta }_2(a_{22}{\beta }_2+{\beta }_1{\beta }_3)+)+a_{11}{a}_{22}{a}_{33})b_1=E^{⁎}{F}^{⁎}{\beta }_2^2{a}_{66}\mu (a_{22}{a}_{44}+(a_{22}+a_{44})a_{55})+E^{⁎}{F}^{⁎}{\beta }_1{\beta }_2{\beta }_3\mu (a_{44}{a}_{55}{a}_{66}+a_{44}{a}_{55}+a_{44}{a}_{66}+a_{55}{a}_{66})+E^{⁎}{I}_S^{⁎}{\beta }_1^2\mu (a_{33}(a_{44}+a_{55})+a_{44}{a}_{55})+a_{11}{a}_{22}((a_{33}{a}_{44}{a}_{55}(a_{66}+\mu ))+a_{66}\mu (a_{33}(a_{44}+a_{55})+a_{44}{a}_{55}))+a_{33}{a}_{44}{a}_{55}{a}_{66}\mu (a_{11}+a_{22})+(a_{66}+\mu )E^{⁎}{a}_{44}{a}_{55}(F^{⁎}{\beta }_2^2{a}_{22}+I_S{\beta }_1^2{a}_{33})b_2=E^{⁎}{F}^{⁎}{\beta }_2^2((a_{66}+\mu )(a_{22}(a_{44}+a_{55})+a_{44}{a}_{55})+a_{66}\mu (a_{22}+a_{44}+a_{55})+a_{22}{a}_{44}{a}_{55})+E^{⁎}{F}^{⁎}{\beta }_1{\beta }_2{\beta }_3((a_{66}+\mu )(a_{44}+a_{55})+a_{66}\mu +a_{44}{a}_{55})+E^{⁎}{I}_S^{⁎}{\beta }_1^2((a_{66}+\mu )(a_{33}(a_{44}+a_{55})+a_{44}{a}_{55})+a_{66}\mu (a_{33}+a_{44}+a_{55})+a_{33}{a}_{44}{a}_{55})+(a_{66}+\mu )(a_{11}{a}_{22}(a_{44}+a_{33}(a_{44}+a_{55}))+a_{33}{a}_{44}{a}_{55}(a_{11}+a_{22}))+(a_{44}+a_{55})(a_{11}{a}_{66}\mu (a_{22}+a_{33})+a_{22}{a}_{33}{a}_{66}\mu )+a_{44}{a}_{55}{a}_{66}\mu (a_{22}+a_{33})+a_{11}{a}_{66}\mu (a_{44}{a}_{55}+a_{22}{a}_{33})+a_{11}{a}_{22}{a}_{33}{a}_{44}{a}_{55})b_3=E^{⁎}{F}^{⁎}{\beta }_2^2((a_{66}+\mu )(a_{22}+a_{44}+a_{55})+a_{66}\mu +a_{22}(a_{44}+a_{55}))+E^{⁎}{F}^{⁎}{\beta }_1{\beta }_2{\beta }_3(a_{44}+a_{55}+a_{66}+\mu )+E^{⁎}{I}_S^{⁎}{\beta }_1^2((a_{66}+\mu )(a_{33}+a_{44}+a_{55})+a_{33}(a_{44}+a_{55})+a_{66}\mu )+(a_{66}+\mu )((a_{44}+a_{55})(a_{11}(a_{22}+a_{33})+a_{22}{a}_{33})+a_{44}{a}_{55}(a_{11}+a_{22}+a_{33})+a_{11}{a}_{22}{a}_{33})+(a_{44}+a_{55})(a_{66}\mu (a_{11}+a_{22}+a_{33})+a_{11}{a}_{22}{a}_{33})+(a_{44}{a}_{55}+a_{66}\mu )(a_{11}(a_{22}+a_{33})+a_{22}{a}_{33})+a_{44}{a}_{55}{a}_{66}\mu \\ {b}_4=E^{⁎}(F^{⁎}{\beta }_2^2{a}_{22}+I_S^{⁎}{\beta }_1^2{a}_{33})+E^{⁎}{F}^{⁎}{\beta }_1{\beta }_2{\beta }_3+a_{11}{a}_{22}{a}_{33})+a_{66}\mu (a_{22}+a_{33}+a_{44}+a_{55})+a_{55}(a_{66}+\mu )(a_{11}+a_{22}+a_{33}+a_{44})+a_{44}(a_{55}+a_{66}+\mu )(a_{11}+a_{22}+a_{33})+(a_{44}+a_{55}+a_{66}+\mu )(E^{⁎}(F^{⁎}{\beta }_2^2+I_S^{⁎}{\beta }_1^2)+a_{11}(a_{22}+a_{33})+a_{22}{a}_{33})b_5=E^{⁎}(F^{⁎}{\beta }_2^2+I_S^{⁎}{\beta }_1^2)+(a_{11}+\mu )(a_{22}+a_{33}+a_{44}+a_{55}+a_{66})+a_{55}{a}_{66}+(a_{55}+a_{66})(a_{22}+a_{33}+a_{44})+a_{44}(a_{22}+a_{33})+a_{22}{a}_{33} \\ {b}_6=a_{11}+a_{22}+a_{33}+a_{44}+a_{55}+a_{66}+\mu \end{gathered}$$

Here, all the eigen values are negative if $a_{11}>0,a_{22}>0,a_{33}>0,a_{44}>0,a_{55}>0,a_{66}>0$ i.e. $E^{⁎}<\max \{\frac{{\beta }_4+{\beta }_5+\mu }{{\beta }_1},\frac{{\beta }_3+{\beta }_6+{\beta }_7+\mu }{{\beta }_2}\}$. Then by Routh-Hurwitz criterion we say the endemic equilibrium point is locally asymptotically stable.

5. Optimal control

The novel corona virus is spread through human contact with infected individuals. Therefore, one can put control on respective situation to prevent its spreading.

Control description:

$u_1$: To prevent exposed foreign individuals in the interstate

$u_2$: Exposed interstate individuals should be quarantined

$u_3$: Exposed foreign individuals should be quarantined

$u_4$: Infected individuals should be quarantined

The objective function is,

$$J(c_i,\mathrm{\Lambda })=\underset{0}{\overset{T}{\int }}(A_1{E}^2+A_2{I_S}^2+A_3{F}^2+A_4{Q}^2+A_5{I}^2+A_6{H}^2+A_7{R}^2+w_1{u_1}^2+w_2{u_2}^2+w_3{u_3}^2+w_4{u_4}^2)dt$$

where, Λ denotes set of all compartmental variables, $A_1,A_2,A_3,A_4,A_5,A_6,A_7$ denote non-negative weight constants for compartments $E,I_S,F,Q,I,H,R$ respectively. $w_1,w_2,w_3$ and $w_4$ are the weight constants for each control $u_i$ where $i=1,2,3,4$ respectively.

Now, calculate every values of control variables from $t=0$ to $t=T$ such that, $J(u_i(t))=min\{J({u_i}^{⁎},\mathrm{\Lambda })/(u_i)\in \varphi \}$, $i=1,2,3,4$ where, ϕ is a smooth function on the interval $[0,1]$.

Related Langrangian function is given by,

$$\begin{gathered} L(\mathrm{\Lambda },A_i)=A_1{E}^2+A_2{I_S}^2+A_3{F}^2+A_4{Q}^2+A_5{I}^2+A_6{H}^2+A_7{R}^2+w_1{u_1}^2+w_2{u_2}^2+w_3{u_3}^2+w_4{u_4}^2 \\ +{\lambda }_1(B-{\beta }_1{I}_{S}E-{\beta }_{2}EF-{\beta }_{10}E-\mu E)+{\lambda }_2({\beta }_1{I}_{S}E-{\beta }_4{I}_S-{\beta }_5{I}_S+{\beta }_{3}F-\mu S-(u_1+u_2)I_S)+{\lambda }_3({\beta }_{2}EF-({\beta }_3+{\beta }_6+{\beta }_7+\mu )F-\mu F+u_1{I}_S-u_{3}F)+{\lambda }_4({\beta }_4{I}_S+{\beta }_{6}F-{\beta }_{8}Q-\mu Q+u_2{I}_S \\ +u_{4}I+u_{3}F)+{\lambda }_5({\beta }_{7}F+{\beta }_5{I}_s+{\beta }_{8}Q-{\beta }_{9}I-\mu I-u_{4}I)+{\lambda }_6({\beta }_{9}I+{\beta }_{10}E-{\beta }_{11}H \\ -(\mu +{\mu }_C)H)+{\lambda }_7({\beta }_{11}H-\mu R) \end{gathered}$$

The adjoint equation variables, ${\lambda }_i=({\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4,{\lambda }_5,{\lambda }_6,{\lambda }_7)$ for the system is calculated by taking partial derivatives of the Langrangian function with respect to each compartment variable.

$$\stackrel{•}{{\lambda }_1}=-\frac{\partial L}{\partial E}=-2A_{1}E+({\lambda }_1-{\lambda }_2){\beta }_1{I}_S+({\lambda }_1-{\lambda }_3){\beta }_{2}F+({\lambda }_1-{\lambda }_6){\beta }_{10}+{\lambda }_1\mu ,\stackrel{•}{{\lambda }_2}=-\frac{\partial L}{\partial I_S}=-2A_2{I}_S+({\lambda }_1-{\lambda }_2){\beta }_{1}E+({\lambda }_2-{\lambda }_4)({\beta }_4+u_2)+({\lambda }_2-{\lambda }_3)u_1+({\lambda }_1-{\lambda }_2){\beta }_5+{\lambda }_2\mu ,\stackrel{•}{{\lambda }_3}=-\frac{\partial L}{\partial F}=-2A_{3}F+({\lambda }_1-{\lambda }_3){\beta }_{2}E+({\lambda }_3-{\lambda }_4)({\beta }_6+u_3)+({\lambda }_3-{\lambda }_2){\beta }_3+({\lambda }_3-{\lambda }_5){\beta }_7+{\lambda }_3\mu ,\stackrel{•}{{\lambda }_4}=-\frac{\partial L}{\partial Q}=-2A_{4}Q+({\lambda }_4-{\lambda }_5){\beta }_8+{\lambda }_4\mu ,\stackrel{•}{{\lambda }_5}=-\frac{\partial L}{\partial I}=-2A_{5}I+({\lambda }_5-{\lambda }_4)u_4+({\lambda }_5-{\lambda }_6){\beta }_9+{\lambda }_5\mu ,\stackrel{•}{{\lambda }_6}=-\frac{\partial L}{\partial H}=-2A_{6}H+({\lambda }_6-{\lambda }_7){\beta }_{11}+(\mu +{\mu }_C){\lambda }_6,\stackrel{•}{{\lambda }_7}=-\frac{\partial L}{\partial R}=-2A_{7}R+{\lambda }_7\mu .$$

This calculation leads with resulting conditions as (Pontryagin, 1986) [8],

$${u_1}^{⁎}=max(a_1,min(b_1,\frac{I_S({\lambda }_2-{\lambda }_3)}{2w_1})),{u_2}^{⁎}=max(a_2,min(b_2,\frac{I_S({\lambda }_2-{\lambda }_4)}{2w_2})),{u_3}^{⁎}=max(a_3,min(b_3,\frac{F({\lambda }_3-{\lambda }_4)}{2w_3})),$$

and

$${u_4}^{⁎}=max(a_4,min(b_4,\frac{I({\lambda }_5-{\lambda }_4)}{2w_4})).$$

Based on analytical results, numerical simulation is given in next section.

6. Numerical simulation

In this section we discuss the simulation performed for the system (1)

From Fig. 2 , we observe 30% of interstate population is exposed to COVID-19 in 17.2 days. Whereas 24.55% of foreigner's population is exposed to COVID-19 in 21.8 days. 21% of foreigners come in contact with Interstate individuals in 7.1 days which increases the infectives of interstate to 22.78% in 9.2 days. Also 27.59% of interstate population gets hospitalized in 14.5 days.

Fig. 2.

Trajectories of each compartment showing flow of individuals in respective compartment.

Scatter plotting is shown in Fig. 3 . Combined effect of group of three compartments is revealed in each plot. Fig. 3(a) depicts that; more infected interstate and foreign individuals will be hospitalised at higher rate of level. Fig. 3(b) shows that, individuals who travelled more will be quarantined. From Fig. 3(c), one can say that infected foreigner would be quarantined at higher rate. Infectedness in quarantined individuals increases which leads to the hospitalization of individuals as observed in Fig. 3(d). Fig. 3(e) describes that how interstate infected individuals are quarantined.

Fig. 3.

Scatter plotting between different compartment combination is observed showing the behaviour of respective combinations.

Fig. 4 shows the periodic nature of the interstate class exposed to the virus COVID-19. It indicates that interstate population is exposed again and again to the disease. It happens if the lockdown, social distancing is not followed as per government system. Which shows the importance of the government action taken against COVID-19 to protect the population. Fig. 5 shows the stability of the respective compartments at endemic equilibrium point. Since the government has decided to quarantine foreigners as soon as they arrive in their countries this makes the system stable as they are not exposed much to the COVID-19.

Fig. 4.

We plot phase diagram of interstate class when exposed to COVID-19 observing again and again exposure of interstate population to COVID-19.

Fig. 5.

This indicates phase plot of foreigner class when exposed to COVID-19. Here we observe convergent behaviour of respective classes making it stable.

Fig. 6a, 6b shows the trajectory at the endemic equilibrium point for the system (1). Here we observe the importance of quarantine as the system is stable when interstate and foreigners are quarantined.

Fig. 6.

Behaviour of interstate class with quarantine class (a) and Phase plot of quarantine with foreigners (b).

From Fig. 7 , we observe foreigner are moving towards interstate population.

Fig. 7.

Transition diagram of interstate and foreigner population. Here the movement of individuals between respective classes is observed.

Fig. 8 and Fig. 9 illustrates the flow of interstate population and foreigners with the COVDI-19 infection. It shows that the interstate population gets the infection at a slower rate as compared to foreigners.

Fig. 8.

Shows directional plot of Interstate population infected with COVID-19 indicating the infectiousness of interstate population.

Fig. 9.

Depicts behaviour of Foreigner population infected with COVID-19. Here we observe that foreign travellers are getting infection at large.

Fig. 10 shows the flow of interstate and foreigners towards hospitalization. Foreigners gets hospitalized at faster rate than interstate population.

Fig. 10.

Directional plot of hospitalization of interstate population (a) and foreigner (b).

Fig. 11 (a)-(g) show the oscillating behaviour of each compartment. As the epidemic nature of disease increases, this can oscillate the whole situation. In some intervals of data, exposed individuals increase (Fig. 11(a)) who are either interstate (Fig. 11(b)) or foreigner (Fig. 11(c)). If quarantined individuals do not follow quarantines rules which have been observed in Fig. 11(d). This leads to a greater number of infected individuals (Fig. 11(e)) hence they should be hospitalised (Fig. 11(f)) which effects on recovery rate (Fig. 11(f)).

Fig. 11.

Here we observe continuous fluctuation in all the compartments which very well depicts the scenario of COVID-19 among interstate and foreign travellers.

The above oscillating nature of the model is controlled by the Fig. 12 (a)-(g). All the four controls are effective to our system (1). In the presence of all the controls we observe decrease in the number of exposed individuals. Quarantining interstate and foreign individuals also reduce the infection when controls are applied.

Fig. 12.

Effect of controls applied to the system (1) is observed on each compartment. Here it can be seen that after control is applied, population of each compartment decreases.

The Fig. 13 , shows intensity of mortality due to COVID-19 among interstate and foreign travellers.

Fig. 13.

Represents chaotic diagram showing mortality rate of 2019-nCoV.

The Fig. 14 clearly shows the infected population of foreigner is more than that of interstate population which shows the importance of complete ban on air arrivals.

Fig. 14.

Percentage wise distribution of interstate (a) and foreigner population in COVID-19 scenario (b). In Fig. 14 a, we observe out of 26% of interstate population 17% is infected and from Fig. 14 b, among 19% of foreigners we have 18% infected population.

From the Fig. 15 it can be observed that 7% of population is exposed to COVID-19. Interstate population share the largest percentage. 14% of the population is quarantined including foreigners and interstate population. Similarly, the infection is 13%. The hospitalization is done at 15%. Of the total population recovery is 16%.

Fig. 15.

Figure indicates percentage wise distribution of all the compartments of the system (1). Here we have 21% of interstate population, 14% of foreign population out of which 13% gets the infection with 15% getting hospitalised.

7. Discussion

Our model indicates that foreigners exhibit a larger infected population, hospitalization rate, and infection rate when compared to the interstate population. Moreover, as foreign individuals contact interstate individuals, the rate of infection within the interstate population increases significantly. To the best of our knowledge, our study is the first to create a disjoint mutually exclusive compartmental model. Our model suggests that both foreign and interstate travel lead to increased risk of infection within the United States population. Consequently, we validate the effectiveness of quarantine as a public health intervention model by the US government and encourage implementation of efforts to mitigate interstate travel.

There are multiple reasons that foreigners have increased risk of obtaining COVID-19 when compared to interstate population. First, as people travel, they risk exposing themselves to a greater number of other individuals. The WHO indicates that transmission of COVID-19 occurs primarily through droplet transmission (2019). Most methods of international travel, including airways, railways, and waterways, crowd individuals in compact and enclosed spaces. Being in close contact with individuals with respiratory symptoms in an enclosed environment increases the risk of being exposed to infected mucosae [11]. Second, the guidelines and strategies for addressing the epidemic differ among countries. For example, while India has enforced total lockdown, the US government has not mandated enforced lockdown [3]. Consequently, when individuals from countries with different regulations arrive, they may be infected and increase the incidence of COVID-19. Finally, the vaccination standards differ among countries. In particular, BCG vaccine, believed to confer protective effects against COVID-19 is recommended in some countries, but not the US [7]. As a result, future research and modelling is necessary to determine the protective effects of the BCG vaccine, and its potential to reduce the incidence of COVID-19 within the United States.

Given that 2019-nCoV is no longer contained within Wuhan, we recommend the United States government close their borders to both foreign and interstate travel. We recommend significant public health interventions at both international and interstate levels otherwise large cities with close inter-transport systems could become outbreak epicentres. Finally, we recommend preparedness plans and mitigation interventions be readied for quick deployment on both a state and federal level. Based on our model, compliance with these recommendations will effectively reduce the transmission of COVID-19 as a result of foreign and interstate travel.

Submitted by S.G. Krantz