key: cord-1001419-nwtwymdr authors: Shah, N. H.; Sheoran, N.; Jayswal, E. N.; Shukla, D.; Shukla, N.; Shukla, J.; Shah, Y. title: Modelling COVID-19 Transmission in the United States through Interstate and Foreign Travels and Evaluating Impact of Governmental Public Health Interventions date: 2020-05-26 journal: nan DOI: 10.1101/2020.05.23.20110999 sha: 2c85077387febd1aa1829dca1f3a8a881567d447 doc_id: 1001419 cord_uid: nwtwymdr 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 labeled as a pandemic by WHO. Modelling, evaluating, and predicting the rate of disease transmission is crucial for epidemic 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 is developed to study the transmission dynamics of the novel coronavirus. A system of non-linear differential equations was formulated and the basic reproduction number R0 was computed. The 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 increase while placing interstates individuals and foreigners under quarantine, decreases the chances of exposure and subsequent infection, leading to an increase in the recovery rate. Conclusion: Interstate and foreign travel restrictions, in addition to quarantine, help in effectively controlling the epidemic. While the United States has implemented numerous public health interventions, it has not implemented a ban on interstate travel. According to World Health Organization (WHO), New cases of COVID-19 have emerged in 210 countries with 1,733, 945 confirmed cases and 106, 518 confirmed deaths globally as of 10 th 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 (Castillo-Chavez et al. 2003; Tang et al. 2020) . In order to prevent the transmission of Covid-19 within the US, the mode of transmission must first be modeled and understood. Mathematical modeling 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. (Wu et al. 2020) . 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). Mathematical modeling plays vital role in determining dynamics of diseases. In this paper we consider disjoint mutually exclusive compartmental model with compartments as follows: Death rate due to Covid-19 0.0027 Table 2 : Parametric definitions and its values This model considers new recruitment in the exposed class at the rate B and all the compartments have mortality rate μ . Here interstate population (it is the population within the country) is exposed to Covid-19 at the rate 1 β and Foreigner population (it is population arriving in a particular country through air ways) is exposed to Covid-19 at the rate 2 β . Next, foreigners F come in contact with interstate population at the rate 3 β joining S I . Interstate population and foreigners quarantine themselves at the rate 4 β and 6 β respectively. Similarly, after getting exposed to Covid-19, interstate and foreigner population gets the infection joining infectious class I with the rate 5 β and 7 β respectively. Quarantined humans also get the infection at the rate 8 β . Next, we assume infected population gets hospitalized at the rate 9 β joining H .We also assume population gets admitted to the hospital at the initial exposure of the disease at the rate 10 β . Hospitalized patients after undergoing treatment gets recover joining R at the rate 11 β We take into the consideration death due to Covid-19 when the individual is hospitalized. Fig.1 gives rise to the following set of non-linear ordinary differential equations . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 26, 2020. Adding all the differential equations of model (1), we get, Then, feasible region for the system is defined as ⎭ 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 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 (Diekmann et al. 2010) . We also compute the value of reproduction number F R when there are no foreigners present in the total population. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 26, 2020. . https://doi.org/10.1101/2020.05.23.20110999 doi: medRxiv preprint 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(Routh). Theorem 1: The foreigner free equilibrium point is locally asymptotically stable if Proof: The Jacobian of system (1) at Foreigner free equilibrium is as follows . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 26, 2020. . https://doi.org/10.1101/2020.05.23.20110999 doi: medRxiv preprint The eigen values of the Jacobian F J are ( ) ( ) 1 2 2 8 3 9 4 5 1 1 2 2 6,7 1 10 10 10 1 . Hence the Foreigner free equilibrium point is locally asymptotically stable. Proof: The Jacobian matrix of system (1) for endemic equilibrium is given by where, * * * * 11 1 2 10 22 1 4 5 33 2 3 6 7 44 8 55 9 66 11 ,a ,a , , , The characteristic polynomial for Jacobian * J is . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 26, 2020. ) ( a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a . Then by Routh-Hurwitz criterion we say the endemic equilibrium point is locally asymptotically stable. 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: 1 u : To prevent exposed foreign individuals in the interstate 2 u : Exposed interstate individuals should be quarantined 3 u : Exposed foreign individuals should be quarantined 4 u : Infected individuals should be quarantined The objective function is, . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 26, 2020. Based on analytical results, numerical simulation is given in next section. In this section we discuss the simulation performed for the system (1). 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. 3 . Scatter plotting Scatter plotting is shown in figure 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 figure 3(c), one can say that infected foreigner would be quarantined at higher rate. All quarantined individuals infected more than, they get hospitalised which is observed in figure 3(d). Figure 3 (e) describes that how interstate infected individuals are quarantined. Fig. 4 . Phase plot of interstate class when exposed to Covid-19 (a) (b) 1 1 (c) (d) (e) Phase plot of foreigner when exposed to covid-19 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 26, 2020. . https://doi.org/10.1101/2020.05.23.20110999 doi: medRxiv preprint 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. 6-1 . Behaviour of interstate class with quarantine class Fig. 6-2 . Phase plot of quaratine with foreingners Figs. 6-1, 6-2 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. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 26, 2020 . . https://doi.org/10.1101 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 26, 2020 . . https://doi.org/10.1101 (c) (d) (e) (f) (g) Fig. 11 . Oscillations Effect Figures 11 (a)-(g) show the oscillating behaviour of each compartment. As epidemic nature of disease increases, this can oscillate whole situation of model. In some interval 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 has 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) ). . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 26, 2020 . . https://doi.org/10.1101 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 26, 2020. . The above oscillating nature of the model is controlled by the fig.12 (a)-(g). All the four controls are effective to this model. Control applied decreases the number of exposed individuals. Quarantining interstate and foreign individuals reduce the infection leading to increase in recovery rate. 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%. 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 make contact with 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 (Tatem et al. 2006) . Second, the guidelines and strategies for addressing the epidemic differ among countries. For example, while India has enforced total lockdown, the US government have not mandated enforced lockdown (Chatterjee et al. 2020) . 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(Aaron Miller, Mac Josh Reandelar, Kimberly Fasciglione, Violeta Roumenova, Yan Li and Otazu 2013). 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 CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 26, 2020. . https://doi.org/10.1101/2020.05.23.20110999 doi: medRxiv preprint recommend significant public health interventions at both international and interstate levels otherwise large cities with close inter-transport systems could become outbreak epicenters. 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. 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World Health Organization The first three authors thank DST-FIST file # MSI-097 for technical support to the department. Third author (ENJ) is funded by UGC granted National Fellowship for Other Backward Classes (NFO-2018-19-OBC-GUJ-71790). 2 2 2 2 2 2 2 2 1 2 3 4 5 6 7 11 2 2 0 2 2 3 3 4