Abstract
This study employs a mathematical model to analyze and forecast the severe outbreak of SARS-CoV-2 (Severe Acute Respiratory Syndrome Coronavirus 2), focusing on the socio-economic ramifications within the Thai population and among foreign tourists. Specifically, the model examines the impact of the disease on various population groups, including susceptible (S), exposed (E), infected (I), quarantined (Q), and recovered (R) individuals among tourists visiting the country. The stability theory of differential equations is utilized to validate the mathematical model. This involves assessing the stability of both the disease-free equilibrium and the endemic equilibrium using the basic reproduction number. Emphasis is placed on local stability, the positivity of solutions, and the invariant regions of solutions. Additionally, a sensitivity analysis of the model is conducted. The computation of the basic reproduction number (R0) reveals that the disease-free equilibrium is locally asymptotically stable when R0 is less than 1, whereas the endemic equilibrium is locally asymptotically stable when R0 exceeds 1. Notably, both equilibriums are globally asymptotically stable under the same conditions. Through numerical simulations, the study concludes that the outcome of COVID-19 is most sensitive to reductions in transmission rates. Furthermore, the sensitivity of the model to all parameters is thoroughly considered, informing strategies for disease control through various intervention measures.
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Introduction
Coronavirus disease (COVID-19) is an infectious disease caused by the SARS-COV-2 virus which has spread throughout the world. The World Health Organization (WHO) has declared it a serious epidemic1,2,3,4,5. The World Health Organization (WHO) has coordinated and asked for international cooperation to stop the spread of the coronavirus -19, which the epidemic is continuously spreading. From reports around the world starting with H1N1 influenza infection, there was a clear outbreak in 2009, with a new outbreak starting on December 31, 2019. A group of cases of pneumonia of unknown ethology in Wuhan, Hubei Province in China. The outbreak was later reported to WHO in January 2022. An outbreak of a new virus1,2,3,4 and6,7,8,9 was identified, and the new virus was later named the 2019 novel coronavirus. By analyzing the genetics of viruses from personal illnesses, including Coronavirus Disease 2019 by WHO in February 2020 on behalf of the virus. This virus is called SARS-CoV-2 and a disease in the same family is COVID-193,4,5,6,7,8,9,10,11.
Coronaviruses are a set of viruses that cause sicknesses such as respiratory diseases or gastrointestinal diseases. Respiratory diseases can extend from the common cold to the more serious diseases e.g. Middle East Respiratory Syndrome (MERS-COV), Severe Acute Respiratory Syndrome (SARS-COV). The novel coronavirus (nCOV) is a new strain that has not been identified in humans. New diseases caused by viruses are named according to where they were first discovered, such as the Spanish flu and the Hong Kong flu. West Nile Flu, etc. The official name of the disease in this article is COVID-19, not Wuhan Flu (or Chinese flu) Coronaviruses are zoonotic1,2,3 and6,7,8,9 and14,15,16,17,18, which means they are transmitted between animals and humans meaning that they are transmitted between animals and humans. It has been definite that MERS-COV was transmitted from dromedary camels to humans and SARS-COV from civet cats to humans5,6,7,8. While the original source of the COVID-19 virus has not been precisely determined, ongoing investigations point it to be zoonotic15,16.
In a person infected with the COVID-19 virus, respiratory symptoms can appear almost immediately. In most cases, the person can exhibit no symptoms or mild symptoms. The symptoms of this disease are very similar to those of seasonal flu17 and20,21,22,23. Laboratory and clinical signs of the COVID-19 infection can appear 2–14 days after exposure. The period between the initial exposure to the disease and the time when the symptoms first appear is called the incubation period. During the incubation of the disease, there is a probability of transmitting or spreading the COVID-19 virus. The clinical signs of COVID-19 infections are a fever, a cough, and a general tiredness. Other early symptoms of COVID-19 of the slight loss of taste or smell, shortness of breath or difficult breathing, muscle aches, chills, sore throat, runny nose, headache, chest pain, pink eye (conjunctivitis), nausea. While many of the other illnesses are caused by other viruses, the main cause at the early stages of the current pandemic was the COVID-19 and it seems to have targeted older people21,22,23,24,25. Older people (people over 70 years of age) often suffer from other serious chronic illnesses, such as diabetes, cardiovascular disease, chronic respiratory disease, cancer, hypertension, chronic liver disease and people who are physically inactive1,2,3,4 and13,14,15,16 have weaker immune symptoms and may succumb to the disease (COVID-19). The WHO reported cases of COVID-19 from January 2020 to the present. The number of cases and the number of deaths in Thailand are shown in the following in Fig. 1. The reasons for separating the populations into Thais and foreigners are that there is shortage of season labor (needed for the farming industry) and tourism is one of the top industries in Thailand. The spread of this disease to become a pandemic is due to the ease of moving from one country to another. The slowness of the great Spanish Flu was the difficulty of traveling from country to country or continent to continent.
WHO has issued guidelines for the treatment of COVID-19 in the high-risk groups (older people and people with serious chronic illness). These are the people who are the most susceptible to infection by the virus and who are in most danger of dying. The World Health Organization has issued guidelines for preventing COVID-19 infection. Not separated from each other, but able to live together with groups of people at risk. They will take care of their treatment and social care. In Thailand from January 2020 to October 2022, there were 4,689,897 confirmed cases of COVID-19 and 32,922 deaths, according to the WHO 2022 September report. A total of 142,635,014 vaccine doses have been administered.
To understand the nature and dynamics of the COVID-19 of epidemic (a pandemic in the larger scheme), mathematical modeling is used to forecast the transmission dynamics needed for controlling and planning strategies. Most epidemiological modelling studies of COVID-19 are based on WHO data. The studies on COVID-19 modelling done in Thailand16 and26. The authors considered a mathematical model for the transmission dynamics of COVID-19. The data from Thailand, which considers the special features pertaining to Thailand and other neighboring countries4,5,6 and16,17,18, and24. From the information obtained, we estimate the values of unknown parameters by statistical and mathematical methods. It should be noted that the effective parameters for the spread of the virus differ from country to country and that the effective control over the rate of virus transmission from country to country will be different. In necessary to stop the spread of the virus. It was found in other studies, that the spread of COVID-19 be managed by minimizing the contact rate of infected and increasing the quarantine of exposed individuals17,18,19,20,21,22,23,24,25. This study examined a mathematical model of the COVID-19 transmission dynamics by dividing it into two groups of coronavirus transmission. The research was organized as follows: explanation of the mathematical models, formulation of the differential equations, mathematical analysis of models, followed by numerical solutions of the differential equations, summarization and discussion.
Materials and methods
In this study, a deterministic mathematical model was created. It covers the well-known SEIR epidemic model20,21,22,23,24,25,26. By adding people who are in quarantine and do not have symptoms of the disease. Symptomatic and asymptomatic infected people will be collected. The SEIQR epidemic model was thus obtained, which evolved with the following subpopulations: Susceptible (S), Exposed (E- (people not yet infectious)), Infectious (I), Quarantined (Q- (setting aside individuals who are exposed), and Recovered (R). This is because people in the Q (Quarantined) group, which represents people who are required to stay in the hospital and at home for a period of time due to the disease, are concerned about their illness. The COVID-19 pandemic in Thailand, we used a ten-dimensional SEIQR (Susceptible, Exposed, Infected, Quarantined and Recovered) containing two populations \(S_{1, } E_{1, } I_{1, } Q_{1, } R_{1 }\) are Thais population respectively and \(S_{2, } E_{2, } I_{2, } Q_{2, } R_{2 }\) are Foreign (tourist) or migrant workers) population respectively of COVID-19 transmission model21,22,23,24,25,26.
The Recruitment term of the susceptible population in Thais and the rest if the Foreign (tourist) are given as \(\mu \) and \(C\) respectively. Only exposed and infectious are considered, it is assumed that those infected show symptoms of Thais and Foreign (tourist). The natural death rate of Thais population and the natural death rate of Foreign (tourist) population is assumed to be the same across the world are given as \(\delta_{1}\) and \(\delta_{2}\). The force of infections in Thais population \(\varphi_{1 } S_{1 } \left( {E_{1} + I_{1} } \right)\) (Transmission rate of virus between population from Thais population to Foreign (tourist) population (in Thais) and \(\varphi_{12 } S_{1 } \left( {E_{2} + I_{2} } \right)\) (When Foreign (tourist) are present, a susceptible Thais can also be infected by an infected or exposed Foreign (tourist) (in Thais)) are the new infections caused by other infected individuals in Thais. The force of infection in rest of Foreign (tourist) population \(\varphi_{2 } S_{2 } \left( {E_{2} + I_{2} } \right)\)(Transmission rate of virus between population from Foreign (tourist) population to Thais population (in Foreign (tourist)) and \(\varphi_{21 } S_{2 } \left( {E_{1} + I_{1} } \right)\) (When Thais are present, the susceptible Thais can also be infected by an infected or exposed Thais (in Foreign (tourist)) are the new infections caused by other infected individuals in Foreign (tourist). Taking into consider the above discretion, the schematic flow diagram for COVID-19 model is appeared in Fig. 2.
The flowchart illustration the dynamics of the model.
The host population was divided into five compartments: \(S_{1}\) number of Thais susceptible to COVID-19 infection at time \(t\), \(E_{1}\) number of Thais exposed to COVID-19 infection at time \(t\), \(I_{1}\) number of infectious Thais at time \(t\), \(Q_{1}\) number of Thais quarantined for COVID-19 at time \(t\), \(R_{1}\) number of recovered Thais at time \(t\), \(S_{2}\) number of Foreign (tourist) susceptible at time \(t\), \(E_{2}\) number of Foreign (tourist) exposed at time \(t\), \(I_{2}\) number of Foreign (tourist) infected at time \(t\), \(Q_{2}\) number of Foreign (tourist) quarantined at time \(t\), \(R_{2}\) number of Foreign (tourist) recovered at time \(t\).
A system of ordinary differential equations can be used to model the influence of two populations on each other as a set nonlinear differential equation20 and24,25 as follows:
with initial densities: \(S_{1} \ge 0, E_{1} \ge 0, I_{1} \ge 0, Q_{1} \ge 0, R_{1} \ge 0\) in Thais population and \(S_{2} \ge 0, E_{2} \ge 0, I_{2} \ge 0, Q_{2} \ge 0, R_{2} \ge 0\) in the Foreign (tourist) population.
All the parameters and corresponding biological meaning are defined in Table 1.
The total Thais population \(N_{h}\) is \(S_{1} + E_{1} + I_{1} + Q_{1} + R_{1}\). The equations for the human compartment are the following Eq. (1) and the total Foreign (tourist) population is \(N_{T} = S_{2 } + E_{2} + I_{2} + Q_{2} + R_{2 }\). We assume that there are constant total number of human Thais population and of Foreign (tourist) population. Therefore the rate of change for total number of human Thais population and of Foreign (tourist) population are equivalent to zero. Thus, the Recruitment term of human and death rate are equivalent. We defined the new state variables as follows: \(\frac{{S_{1} }}{{N_{h} }} = {\mathop {S_{1} }\limits^{\prime } } , \frac{{E_{1} }}{{N_{h} }} = {\mathop {E_{1} }\limits^{\prime } } , \frac{{I_{1} }}{{N_{h} }} = {\mathop {I_{1} }\limits^{\prime } } , \frac{{Q_{1} }}{{N_{h} }} = {\mathop {Q_{1} }\limits^{\prime } } , \frac{{R_{1} }}{{N_{h} }} = R_{1}^{\prime } , \) \(\frac{{S_{2} }}{{N_{T} }} = S_{2}^{\prime } , \frac{{E_{2} }}{{N_{T} }} = E_{2}^{\prime } , \frac{{I_{2} }}{{N_{T} }} = I_{2}^{\prime } , \frac{{Q_{2} }}{{N_{T} }} = Q_{2}^{\prime } , \frac{{R_{2} }}{{N_{T} }} = R_{2}^{\prime }\) Renormalizing model (1) we obtain the following:
Positivity of solution
Model (2) must been found to be biologically and epidemiologically meaningful and well positioned. To do this, we needed to show that the solutions of all state variables were non-negative all the time. The following theorem24,25,26 were required.
Theorem 1: The given solution \(\left\{ {S_{1} ,E_{1} ,I_{1} ,Q_{1} ,R_{1,} S_{2 } ,E_{2} ,I_{2} ,Q_{2} ,R_{2 } } \right\}\) of the epidemiological systems (2) with non-negative initial data when \(S_{1} \ge 0, E_{1} \ge 0, I_{1} \ge 0, Q_{1} \ge 0, R_{1} \ge 0\) and \(S_{2} \ge 0, E_{2} \ge 0, I_{2} \ge 0, Q_{2} \ge 0,\)
\( R_{2} \ge 0\) stills non-negative for all time non-negative \(t > 0\).
Proof of Theorem 1
Given the initial data \(S_{1} \left( 0 \right),E_{1} \left( 0 \right),I_{1} \left( 0 \right),Q_{1} \left( 0 \right),R_{1} \left( 0 \right)\) and \( S_{2 } \left( 0 \right),E_{2} \left( 0 \right),I_{2} \left( 0 \right),Q_{2} \left( 0 \right),R_{2 } \left( 0 \right)\) are non—negative. It is clear from the first sub-equation of the model (2) that
\(\frac{{d{\mathop {S_{1} }\limits^{\prime } } }}{dt} = \left[ {\varphi_{1} {\mathop {S_{1} }\limits^{\prime } } \left( {{\mathop {E_{1} }\limits^{\prime } } + {\mathop {I_{1} }\limits^{\prime } } } \right) + \delta_{1} {\mathop {S_{1} }\limits^{\prime } } + \varphi_{12} {\mathop {S_{1} }\limits^{\prime } } \left( {E_{2}^{\prime } + I_{2}^{\prime } } \right)} \right] \ge 0\) so that
Integrating (3) gives
Further, one sees from the second sub-equation of the model (2) that
\(\frac{{d{\mathop {E_{1} }\limits^{\prime } } }}{dt}\left[ { \delta_{1} {\mathop {E_{1} }\limits^{\prime } } + \frac{1}{{IIP_{1} }}{\mathop {E_{1} }\limits^{\prime } } } \right] \ge 0\) implies \(\frac{d}{dt}\left[ { {\mathop {E_{1} }\limits^{\prime } } {\text{exp }}(\delta_{1} + \frac{1}{{IIP_{1} }} \mathop \smallint \limits_{0}^{t} {\mathop {E_{1} }\limits^{\prime } } (\zeta_{1} ))d\zeta_{1} } \right] \ge 0\) which on integration yields
Further, one sees from the third sub-equation of the model (2) that
which upon integration yields
Further, one sees from the fourth sub-equation of the model (2) that
\(\frac{{d\mathop Q\limits^{\prime }_{1} }}{dt} = \left[ {\gamma_{1} + \delta_{1} } \right)\mathop Q\limits^{\prime }_{1} ] \ge 0\) implies \(\frac{{d\mathop Q\limits^{\prime }_{1} }}{dt}\left[ {\mathop Q\limits^{\prime }_{1} \exp (\gamma_{1} + \delta_{1} \mathop \smallint \limits_{0}^{t} \mathop Q\limits^{\prime }_{1} (\zeta_{1} } \right))d\zeta_{1} ] \ge 0\) which upon integration yields
Further, one sees from the fifth sub-equation of the model (2) that \(\frac{{d\mathop R\limits^{\prime }_{1} }}{dt} = \gamma_{1} \mathop Q\limits^{\prime }_{1} - (\delta_{1} + \alpha_{1} )\mathop R\limits^{\prime }_{1}\), \(\frac{{d\mathop R\limits^{\prime }_{1} }}{dt}\left[ {(\delta_{1} + \alpha_{1} )\mathop R\limits^{\prime }_{1} } \right] \ge 0\) implies \(\frac{{d\mathop R\limits^{\prime }_{1} }}{dt}\left[ {\mathop R\limits^{\prime }_{1} \exp \left( {\delta_{1} + \alpha_{1} \mathop \smallint \limits_{0}^{t} \mathop R\limits^{\prime }_{1} \left( {\zeta_{1} } \right)} \right)d\zeta_{1} } \right] \ge 0\) which upon integration yields.
In a similar model, it can be shown that \( \mathop S\limits^{\prime }_{2} \left( t \right) \ge 0,\mathop E\limits^{\prime }_{2} \left( t \right) \ge 0,\mathop I\limits^{\prime }_{2} \ge 0,\mathop Q\limits^{\prime }_{2} \ge 0\) and \(\mathop R\limits^{\prime }_{2} \left( t \right) \ge 0\) for all time \(t > 0\). This completes the proof. It is important to note that the model (2) has be analyzed in the region \(\beta\) given by
Divided into two groups \( \mathop S\limits_{1} + \mathop E\limits_{1} + \mathop I\limits_{1} + \mathop Q\limits_{1} + \mathop R\limits_{1} = 1\) and \( \mathop S\limits_{2} + \mathop E\limits_{2} + \mathop I\limits_{2} + \mathop Q\limits_{2} + \mathop R\limits_{2} = 1\) which can easily be shown to be positively univariate according to model (2). In the following, model (2) is epidemiologically and mathematically well-positioned in \( \beta\).
Theorem 2:
The solution of system (1) is possible for all if entering an invariant region. \({\Omega } = {\Omega }_{1} \times {\Omega }_{2}\), where \({\Omega }_{1} = \{ S_{1} ,E_{1} ,I_{1} ,Q_{1} ,R_{1} \in R_{ + }^{5} :0 < N_{h} \left( t \right) \le \frac{\mu }{{\delta_{1} }}\}\) as \(t \to \infty\), when \(\theta = {\text{min}}\left\{ {\delta_{1} , \delta_{1} + \rho_{1} } \right\}\) and \({\Omega }_{2} = \left\{ { S_{2 } ,E_{2} ,I_{2} ,Q_{2} ,R_{2 } \in R_{ + }^{5} :0 < N_{T} \left( t \right) \le \frac{C}{{\delta_{2} }}} \right\}\) as \( t \to \infty\), when \(\theta_{2} = \min \left\{ {\delta_{2} + \vartheta , \delta_{2} + \rho_{2} + \vartheta } \right\}.\)
Proof of Theorem 2
The invariant region is received from the bounded situation of the system. Here, \(N_{h} \left( t \right) = S_{1} \left( t \right) + E_{1} \left( t \right) + I_{1} \left( t \right) + Q_{1} \left( t \right) + R_{1} \left( t \right)\) and \(N_{T} \left( t \right) = S_{2} \left( t \right) + E_{2} \left( t \right) + I_{2} \left( t \right) + Q_{2} \left( t \right) + R_{2} \left( t \right)\). It following that,
\(\begin{aligned} \frac{{dN_{h} }}{dt} & = \frac{{dS_{1} }}{dt} + \frac{{dE_{1} }}{dt} + \frac{{dI_{1} }}{dt} + \frac{{dQ_{1} }}{dt} + \frac{{dR_{1} }}{dt} \\ & = \mu - \left( {\rho_{1} + g_{1} } \right)I_{1} - g_{1} E_{1} - \delta_{1} N_{h} \\ & \le \mu - \delta_{1} N_{h} \\ \end{aligned}\)
This inequality can be expressed in a general solutions as
where \(N_{h} \left( 0 \right)\) is the initial values, i.e., \(N_{h} \left( t \right) = N_{h} \left( 0 \right)\) at \(t = 0\).
In a similar model, it can be shown that \(N_{T} \left( t \right) = S_{2} \left( t \right) + E_{2} \left( t \right) + I_{2} \left( t \right) + Q_{2} \left( t \right) + R_{2} \left( t \right)\) for the bounded situation of the system all time \( t > 0\). Moreover, every solution for systems (1) with initial conditions in \({{ \Omega }}\) remains in \({{ \Omega }}\) for all \(t > 0\). Therefore, the dynamics of our model will be poised in \({\Omega }\).
Analysis of the model
Basic reproduction number
The next generation matrix method is used to calculate the basic reproduction number, \(R_{0}\)27,28,29,30,31,32,33,34,35,36 the number of secondary infections caused by a single infected individual in a completely susceptible population (including of the local Thais and the Foreign (tourist)). The behavior of the disease in the total system defined by Eq. (2)15,16,17,18,19 will be determined by \(R_{0}\) which has the form
where \(R_{0T} = \frac{{\left( {q_{2T} + \gamma_{1} + \delta_{1} } \right) \mu \left( {1 + IIP_{1} \left( {\delta_{1} + \rho_{1} } \right)} \right)\varphi_{1} }}{{\delta_{1} \left( {1 + IIP_{1} \delta_{1} } \right)\left( {g_{1} q_{2T} + \left( {q_{2T} + \gamma_{1} + \delta_{1} } \right)\left( {\delta_{1} + \rho_{1} } \right)} \right)}}\) is the basic reproduction number for Thais and \(R_{0F} = \frac{{\alpha_{3} C\left( {1 + \alpha_{2} IIP_{2} } \right)\varphi_{2} }}{{\alpha_{1} \left( {\alpha_{2} \alpha_{3} + g_{2} q_{3T} } \right)IIP_{2} \left( {\delta_{2} + \vartheta } \right)}}\) is the basic reproduction number for the Foreign (tourist) population only with \(\alpha_{1} = \delta_{2} + \vartheta + \frac{1}{{IIP_{2} }} , \alpha_{2} = \delta_{2} + \vartheta + \rho_{2} , \alpha_{3} = \delta_{2} + \vartheta + \gamma_{2} + q_{3T}\) and \(\alpha_{4} = \delta_{2} + \vartheta + \alpha_{2}\), we have
Theorem 3:
To find the basic reproduction number of our proposed differential Eq. (2), using help of the next generation matrix formulas17,18,19,20,21,22,23,24,25. We initially define \( K = (\mathop E\limits_{1} ,\mathop I\limits_{1} ,\mathop Q\limits_{1} )^{T}\) and \( K_{1} = (\mathop E\limits_{2} ,\mathop I\limits_{2} ,\mathop Q\limits_{2} )^{T}\). The model (2) is rewritten in the following form \(\frac{dy}{{dt}} = F\left( y \right) - V\left( y \right)\), where \( F\left( y \right)\) is the non-negative matrix of the newly infected (Thais and Foreign (tourist) populations) and \( V\left( y \right)\) is the non-singular matrix for the transfers between the parts in the infective equations (Thais and Foreign (tourist) populations) (when \(y\) represents Thais populations and Foreign (tourist) populations) as follows:
for the Thais population.
for the Foreign (tourist) population.
The basic reproductive number \(\left( {R_{0} } \right)\) is the threshold for the stability of the disease-free equilibrium B0. It can be calculated by \(R_{0} = \rho \left( {FV^{ - 1} } \right)\) where, \( FV^{ - 1}\) is called the next generation matrix and \(\rho \left( {FV^{ - 1} } \right)\) is the spectral radius of the matrix \(FV^{ - 1}\). Then we get reproduction number \(\left( {R_{0} } \right)\) where,
Finally, the Routh–Hurwitz criteria is used for determining the stabilities of the model. If \(R_{0} > 1\), then the endemic equilibrium is local asymptotically stable, but if \(R_{0} < 1\), then the disease free equilibrium point is local asymptotically stable.
Equilibrium point
The standard method is used to analyze the model. The equilibrium points are found by setting the right-hand side of Eq. (2) to zero. By doing this, the equilibrium points are determined as follows24,25,26,27,28,29,30,31,32,33,34,35,36,37.
-
(A)
The COVID-19 free equilibrium of the Eq. (2) exists and then given by
$$ B_{0} = \left( {\mathop S\limits_{1} ,\mathop E\limits_{1} ,\mathop I\limits_{1} ,\mathop Q\limits_{1} ,\mathop R\limits_{1} ,\mathop S\limits_{2} ,\mathop E\limits_{2} ,\mathop I\limits_{2} ,\mathop Q\limits_{2} ,\mathop R\limits_{2} } \right),B_{0} = \left( {\frac{\mu }{{\delta_{1} }},0,0,0,0,\frac{C}{{\delta_{2} }},0,0,0,0} \right) $$ -
(B)
The COVID-19 endemic equilibrium of the Eq. (2) exists with infection and then given by
$$ \begin{gathered} B_{1} = \left( {\mathop S\limits_{1}^{*} ,\mathop E\limits_{1}^{*} ,\mathop I\limits_{1}^{*} ,\mathop Q\limits_{1}^{*} ,\mathop R\limits_{1}^{*} ,\mathop S\limits_{2}^{*} ,\mathop E\limits_{2}^{*} ,\mathop I\limits_{2}^{*} ,\mathop Q\limits_{2}^{*} ,\mathop R\limits_{2}^{*} } \right) \hfill \\ \mathop S\limits_{1}^{*} = \frac{{\mathop R\limits_{1}^{*} \alpha_{1} + \mu }}{{\varphi_{1} \left( {\mathop E\limits_{1}^{*} + \mathop I\limits_{1}^{*} } \right) + \delta_{1} + \varphi_{12} \left( {\mathop E\limits_{2}^{*} + \mathop I\limits_{2}^{*} } \right)}}, \hfill \\ \mathop E\limits_{1}^{*} = \frac{{IIP_{1} \mathop S\limits_{1}^{*} \left( {\varphi_{1} \mathop I\limits_{1}^{*} + \varphi_{12} \left( {\mathop E\limits_{2}^{*} + \mathop I\limits_{2}^{*} } \right)} \right)}}{{1 + IIP_{1} \left( {\delta_{1} - \mathop S\limits_{1}^{*} \varphi_{1} } \right)}}, \hfill \\ \mathop I\limits_{1}^{*} = \frac{{\mathop E\limits_{1}^{*} }}{{IIP_{1} \left( { - q_{2T} + \delta_{1} + \rho_{1} } \right)}}, \hfill \\ \mathop Q\limits_{1}^{*} = \frac{{g_{1} \left( {\mathop E\limits_{1}^{*} + \mathop I\limits_{1}^{*} } \right) - \mathop I\limits_{1}^{*} q_{2T} }}{{\gamma_{1} + \delta_{1} }}, \hfill \\ \mathop R\limits_{1}^{*} = \frac{{\gamma_{1} \mathop Q\limits_{1}^{*} }}{{\delta_{1} + \alpha_{1} }}, \hfill \\ \mathop S\limits_{2}^{*} = \frac{{C + \mathop R\limits_{2}^{*} \alpha_{2} }}{{\varphi_{2} \left( {\mathop E\limits_{2}^{*} + \mathop I\limits_{2}^{*} } \right) - \varphi_{21} \left( {\mathop E\limits_{1}^{*} + \mathop I\limits_{1}^{*} } \right) + (\delta_{2} + \vartheta )}}, \hfill \\ \mathop E\limits_{2}^{*} = \frac{{IIP_{2} \mathop S\limits_{2}^{*} \left( {\varphi_{2} \mathop I\limits_{2}^{*} + \varphi_{21} \left( {\mathop E\limits_{1}^{*} + \mathop I\limits_{1}^{*} } \right)} \right)}}{{1 + IIP_{2} \left( {\delta_{2} + \vartheta - \mathop S\limits_{2}^{*} \varphi_{2} } \right)}}, \hfill \\ \mathop I\limits_{2}^{*} = \frac{{\mathop E\limits_{2}^{*} }}{{IIp_{2} \left( { - q_{3T} + \delta_{2} + \vartheta + \rho_{2} } \right)}}, \hfill \\ \mathop Q\limits_{2}^{*} = \frac{{g_{2} \left( {\mathop E\limits_{2}^{*} + \mathop I\limits_{2}^{*} } \right) - \mathop I\limits_{2}^{*} q_{3T} }}{{\gamma_{2} + \delta_{2} + \vartheta + \rho_{2} }}, \hfill \\ \mathop R\limits_{2}^{*} = \frac{{\gamma_{2} \mathop Q\limits_{2}^{*} }}{{\delta_{2} + \vartheta + \alpha_{2} }}. \hfill \\ \end{gathered} $$(13)
Local asymptotically stability of disease—free equilibrium point
Lemma 1:
(The Generalized Routh–Hurwitz Criterion). Given the charactistic equation
Define \(k\) matrices as follows:
where the \(\left( {l,m} \right)\) term in the matrix \(H_{j}\) is \(a_{2l - m}\) for \(0 < 2l - m < k\), \(1\) for \(2l = m\).
\(0\) for \(0 < 2l\) or \(2l < k + m.\)
Then all eigenvalues have negative real parts; that is, the steady-state \(\overline{N}\) is stable if and only if the determinants of all Hurwitz are positive:
When is \(\overline{N} = \overline{N}_{i} + a_{1} e^{{\lambda_{1} t}} + a_{2} e^{{\lambda_{2} t}} + \cdots + a_{k} e^{{\lambda_{k} t}}\).
Theorem 4:
The local stability of disease-free equilibrium point is determined from the Jacobian matrix of the model of Eq. (2) evaluated at the equilibrium points. If \(R_{0} > 1\), the point is stable and unstable otherwise.12,13,14,15,16,17,18, and24,25,26,27,28,29, and31.
Proof of Theorem 4
To determine the local stability of \(J_{0}\), we evaluate the Jacobian matrix at the disease-free state to be
where \( \theta_{1} = \varphi_{1} \mathop S\limits_{1} - \left( { \delta_{1} + \frac{1}{{IIP_{1} }}} \right)\), \( \theta_{2} = \varphi_{2} \mathop S\limits_{2} - \left( {\delta_{2} + \vartheta + \frac{1}{{IIP_{2} }}} \right)\), \(\theta_{3} = - (\delta_{2} + \vartheta + q_{3T} + \rho_{2}\)), \(\theta_{4} = (\delta_{2} + \vartheta + \gamma_{2} + \rho_{2}\)) and \(\theta_{5} = - \left( {\delta_{2} + \vartheta + \alpha_{2} } \right)\).
The eigenvalues of the \(J_{0} \) are obtained by solving \(Det \left( {J_{0} - \lambda I } \right) = 0\). We obtain the characteristic equation, where \(\lambda\) is an eigenvalue of the matrix \( J_{0}\). The, root of the model (2) i.e., eigenvalue of the matrix \(J_{0}\) are
The three eigenvalues from Eq. (15) were \(\lambda_{1} = - \delta_{2} - \vartheta - \alpha_{2}\), \(\lambda_{2} = - \delta_{1} - \alpha_{1}\) and \(\lambda_{3} = - \delta_{1}\) and all of them must have negative real parts. For the other seven eigenvalues, we examine the stability of disease-free equilibrium state by using the Routh Hurwitz principle (R-H criterion) to show that all eigenvalues given by Eq. (14) has a negative real part, i.e., coefficients of the seventh order the polynomial appearing in Eq. (14) satisfies all R-H conditions when \(A_{1} ,A_{2} ,A_{3} ,A_{4} ,A_{5} ,A_{6} ,A_{7 } > 0 \)(The coefficients appearing in Eq. 15 from the Routh Hurwitz condition are plotted on the graph by the x axis being the coefficient \(A_{2}\). and the Y-axis is the coefficient of \(A_{1} ,A_{2} ,A_{3} ,A_{4} ,A_{5} ,A_{6}\) and \(A_{7 }\) obtained by finding determinants from size nxn, parameter values from Table 1 by the use the Mathematica program.) This is displayed for \(R_{0} < 1\), disease-free equilibrium point will be stable as showed in Fig. 3.
The parameter areas for disease free equilibrium state which satisfies the Routh-Hurwitz criteria with the value of parameters: respectively, for with \((\lambda^{7} + A_{1} \lambda^{6} + A_{2} \lambda^{5} + A_{3} \lambda^{4} + A_{4} \lambda^{3} + A_{5} \lambda^{2} + A_{6} \lambda^{{}} + A_{7} = 0\).
When
and \({\beta }_{7}= {A}_{8}({A}_{7}\left({A}_{6}\left(-{A}_{5}\left({A}_{3}^{2}{A}_{4}-{A}_{2}{A}_{3}{A}_{5}+{A}_{5}^{2}+{A}_{3}^{3}{A}_{6}\right)+\left({A}_{4}\left({A}_{3}^{2}{A}_{4}-{A}_{2}{A}_{3}{A}_{5}+{A}_{5}^{2}\right)+{A}_{3}\left(-2{A}_{2}{A}_{3}+ 3{A}_{5}\right){A}_{6}\right){A}_{7}+\left({A}_{2}^{2}{A}_{3}-2{A}_{3}{A}_{4}-{A}_{2}{A}_{5}\right){A}_{7}^{2}+{A}_{7}^{3}\right)+\left({A}_{5}^{4}-{A}_{3}{A}_{5}^{2}\left({A}_{2}{A}_{5}+4{A}_{7}\right)-\left({A}_{3}^{3}\left({A}_{5}{A}_{6}+2{A}_{4}{A}_{7}\right)+{ A}_{3}^{2}{A}_{4}{A}_{5}^{2}+3{A}_{2}{A}_{5}{A}_{7}+2{A}_{7}^{2}\right)\right){A}_{8}+{A}_{3}^{4}{A}_{8}^{2}+{A}_{1}\left({A}_{7}\left(-{A}_{6}\left({A}_{5}\left(-{A}_{2}{A}_{3}{A}_{4}+{A}_{2}^{2}-2{A}_{4}{A}_{5}\right)\right){A}_{7}- \left({A}_{3}^{2}-{3A}_{2}{A}_{4}+3{A}_{6}\right){A}_{7}^{2}\right)+\left(-2{A}_{4}{A}_{5}^{3}+3{A}_{3}{A}_{5}^{2}{A}_{6}+4{A}_{3}{A}_{4}{A}_{5}{A}_{7}+{A}_{3}^{2}{A}_{6}{A}_{7}+4{A}_{5}{A}_{7}^{2}+{ A}_{2}^{2}\left({A}_{5}^{3}-3{A}_{3}{A}_{5}{A}_{7}\right)+{A}_{2}\left({A}_{3}{A}_{5}\left(-{A}_{4}{A}_{5}+{A}_{3}{A}_{6}\right)+\left(2{A}_{3}^{2}{A}_{4}+{A}_{5}^{2}\right){A}_{7}-5{A}_{3}{A}_{7}^{2}\right)\right){A}_{8}-{A}_{3}^{2}\left({A}_{2}{A}_{3}+ 4{A}_{5}\right){A}_{8}^{2}\right)+\left({A}_{1}^{2}\left({A}_{7}\left(-{A}_{4}{A}_{5}{A}_{6}+{A}_{4}^{3}{A}_{7}+{A}_{6}^{2}\left(2{A}_{2}{A}_{5}+3{A}_{7}\right)+{A}_{4}{A}_{6}\left({A}_{3}{A}_{6}-3{A}_{2}{A}_{7}\right)\right)- \left({A}_{5}\left(-{A}_{4}^{2}{A}_{5}+{A}_{3}{A}_{4}{A}_{6}+2{A}_{2}{A}_{5}{A}_{6}\right)+\left(2{A}_{3}{A}_{4}^{2}-{A}_{2}{A}_{4}{A}_{6}\right)+{A}_{2}{A}_{3}{A}_{6}+5{A}_{5}{A}_{6}\right){A}_{7}+3\left(-{A}_{2}+{A}_{4}\right){A}_{7}^{2}\right){A}_{8}+{ A}_{3}^{2}{A}_{4}+3{A}_{2}{A}_{3}{A}_{5}+2{A}_{5}^{2}+4{A}_{3}{A}_{7}\right){A}_{8}^{2}\right)\).
Local asymptotically stability of disease endemic equilibrium point
Theorem 5:
The disease endemic equilibrium point is set from the Jacobian matrix of the system of Eq. (2) evaluated at every equilibrium point. If \({R}_{0} <1\) the state is stable and unstable otherwise12,13,14,15,16,17,18, and24,25,26, and30,31.
Proof of Theorem 5
The Jacobian matrix of the model (2) at pandemic equilibrium point is
Where \( \eta_{1} = - \varphi_{1} \left( {\mathop E\limits_{1} + \mathop I\limits_{1} } \right) - \varphi_{12} \left( {\mathop E\limits_{2} + \mathop I\limits_{2} } \right) - \delta_{1}\) , \( \eta_{2} = \varphi_{1} \left( {\mathop E\limits_{1} + \mathop I\limits_{1} } \right) + \varphi_{12} \left( {\mathop E\limits_{2} + \mathop I\limits_{2} } \right)\) \( \eta_{3} = \varphi_{1} \mathop S\limits_{1} - \left( {\delta_{1} + \frac{1}{{IIP_{1} }}} \right)\), \( \eta_{4} = - \varphi_{2} \left( {\mathop E\limits_{2} + \mathop I\limits_{2} } \right) - \varphi_{21} \left( {\mathop E\limits_{1} + \mathop I\limits_{1} } \right) - (\delta_{2} + \vartheta )\) , \( \eta_{5} = - \varphi_{2} \left( {\mathop E\limits_{2} + \mathop I\limits_{2} } \right) + \varphi_{21} \mathop S\limits_{2} \left( {\mathop E\limits_{1} + \mathop I\limits_{1} } \right)\), \( \eta_{6} = - \varphi_{2} \mathop S\limits_{2} - (\delta_{2} + \vartheta + \frac{1}{{IIP_{2} }}\), \( \eta_{7} = - \left( {q_{3T} + \delta_{2} + \rho_{2} + \vartheta } \right),\) \(\eta_{8} = - (\gamma_{2} + \delta_{2} + \rho_{2} + \vartheta ) \) and \(\eta_{9} = \left( {\delta_{2} + \vartheta + \alpha_{2} } \right)\).
The endemic equilibrium point (B1) exists and is positive if \(R_{0} > 1\). The eigenvalues of \(J_{1} \) are obtained by solving \(Det \left( {J_{1} - \lambda I} \right) = 0\). The characteristic equation is as follows; we obtain the characteristic equation \(\lambda^{10} + W_{1} \lambda^{9} + W_{2} \lambda^{8} + W_{3} \lambda^{7} + W_{4} \lambda^{6} + W_{5} \lambda^{5} + W_{6} \lambda^{4} + W_{7} \lambda^{3} + W_{8} \lambda^{2} + W_{9} \lambda^{{}} + W_{10} = 0\) where \(\lambda\) are eigenvalues of the matrix \(J_{1}\). To consider the local stability of the endemic equilibrium state, we check the stability of endemic equilibrium state by using the Routh-Hurwitz criteria required for all the eigenvalues defined by Eq. (16) to have negative real parts. We find that the Routh-Hurwitz conditions for the above the all the eigenvalues of the above 10th order polynomial to have negative real parts when \(W_{1} ,W_{2} ,W_{3} ,W_{4} ,W_{5} ,W_{6} ,W_{7} ,W_{8} ,W_{9} ,W_{10} > 0\) (The coefficients appearing in Eq. (15) from the Routh Hurwitz condition are plotted on a graph by the x axis being the coefficient \(W_{2}\) and the Y-axis is the coefficient of \( W_{1} ,W_{2} ,W_{3} ,W_{4} ,W_{5} ,W_{6} ,W_{7} ,W_{8} ,W_{9} ,W_{10}\), obtained by finding determinants from size nxn, parameter values from Table 1 by the use the Mathematica program.) This is displayed for \({R}_{0}>1\), endemic equilibrium point will be stable as showed in Fig. 4.
The parameter areas for endemic equilibrium point which satisfies the Routh-Hurwitz criteria with the value of parameters: respectively, for with.
\(\lambda^{10} + W_{1} \lambda^{9} + W_{2} \lambda^{8} + W_{3} \lambda^{7} + W_{4} \lambda^{6} + W_{5} \lambda^{5} + W_{6} \lambda^{4} + W_{7} \lambda^{3} + W_{8} \lambda^{2} + W_{9} \lambda^{{}} + W_{10} = 0\). When
, and
Numerical results
Numerical simulations of the impact of the strategies to control the spread of coronavirus disease 19 (COVID-19) in the Thais population when there are Foreign (tourist) also present. Numerical values of various parameters and data points needed for the numerical calculations in Table 2. The Data collected were from the official website of the Ministry of Public Health and World Health Organization (WHO)1,2,3,4,5,6 and24,25,26,27,28,29,30. Using the numerical values in Table 2, we obtained the time evolutions of a susceptible Thais individual, an exposed Thais, an infectious Thais, a quarantined Thais, a recovered Thais, a susceptible Foreigner (tourist), an exposed Foreigner (tourist), an infectious Foreigner (tourist), a quarantined Foreigner (tourist), and a recovered Foreigner (tourist). The values of the parameters were first chosen to lead to \(R_{0}\) to be less than one so the equilibrium state will be disease free State (0.80893). The time evolutions of the ten states were plotted in Figs. 5. Next, we change the values of the parameters so that the value of \(R_{0}\) will be greater than one, meaning that the equilibrium state will be the endemic state (9.4175). In Fig. 6, we see the evolution of the ten categories of individuals (susceptible Thais, exposed Thais, infectious Thais, quarantined Thais, recovered Thais, susceptible Foreign (tourist), exposed Foreign (tourist), infectious Foreign (tourist), quarantined Foreign (tourist), recovered Foreign (tourist)) converge to their epidemic equilibrium values (0.002951, 0.0000217, 0.0001608, 0.0000236, 0.000125, 0.0000001974, 0.00000134, 0.000001218).
Numerical simulations of each population for the disease free state. We will see that the solutions converge to the disease free state when it satisfy.
Numerical simulations of each population for the disease state.
The behavior’s of the endemic, we has plotted the 2-D trajectories of the following thirteen pairs (Thais susceptible-Thais exposed), (Thais susceptible-Thais infectious), (Thais susceptible-Thais quarantined), (Thais exposed-Thais infectious), (Thais exposed-Thais quarantined), (Thais infectious-Thais quarantined), (susceptible Foreign (tourist) -exposed Foreigner), (susceptible Foreign (tourist) -infectious Foreign (tourist)), (exposed Foreign (tourist) -quarantined Foreign (tourist)), (infectious Foreign (tourist) -quarantined Foreign (tourist)), (infectious Thais-exposed Foreign (tourist)), (infectious Thais-infectious Foreign (tourist)) and (quarantined Thais-quarantined Foreign (tourist). These 2D trajectories are shown in Fig. 7. We can see that all the trajectories converge to a central point (the equilibrium pot).
The trajectories of the numerical projected onto the 2D (a)\((S_{1} ,E_{1} )\), (b) \( (S_{1} ,I_{1} )\), (c) \( (S_{1} ,Q_{1} )\), (d) \( (E_{1} ,I_{1} )\), (e) \( (I_{1} ,Q_{1} )\), (f) \( (S_{2} ,E_{2} )\), (g) \( (S_{2} ,Q_{2} )\), (h) \( (I_{2} ,Q_{2} )\), (i) \( (E_{1} ,E_{2} )\), (j) \( (I_{1} ,I_{2} )\) and (k) \( (Q_{1} ,Q_{2} )\). planes when there was no vertical transmission and equilibrium state the endemic state.
Global Stability of disease free equilibrium for model
The solutions to Eq. (2) were asymptotically stable locally in section “Analysis of the model”. We have now proved that the two equilibrium points are asymptotically stable globally through the following theorem.
Theorem 6:
If \(R_{0}^{{}} \le 1\), then the disease—free equilibrium \(E^{*}\) is globally asymptotically stable, by
Proof of Theorem 6
The Lyapunov function may be constructed for the model (1) through the use of the function
Differentiating with respect to time yields.
Using the condition (17), Eq. (17a) may be rewrite as,
Substitute with Eq. (17) we obtain
Note that on Ω, we have \(S_{1}^{*} = \frac{{\mu N_{h} }}{{\delta_{1} }}\) and \(S_{2}^{*} = \frac{{CN_{T} }}{{\delta_{2} + \vartheta }}\) with this in mind, Eq. (17) becomes
Hence, \(\dot{P}\left( t \right) \le 0\). By using LaSalle’ s (1976)31,32,33,34,35,36,37 extension to Lyapunov method, the limit of each solution is contained in the largest invariant set for which \(S_{1} = S_{1}^{*} , E_{1} = 0,I_{1} = 0, Q_{1} = 0, R_{1} = 0, S_{2} = S_{2}^{*} , E_{2} = 0,I_{2} = 0, Q_{2} = 0\) and \(R_{2} = 0\) which is the singleton \(\left\{ {E_{0} } \right\}\). This means that the disease—free equilibrium \( E^{*} = \left\{ {S_{1}^{*} ,\mathop E\limits_{1}^{*} ,\mathop I\limits_{1}^{*} ,\mathop Q\limits_{1}^{*} ,\mathop R\limits_{1}^{*} ,\mathop S\limits_{2}^{*} ,\mathop E\limits_{2}^{*} ,\mathop I\limits_{2}^{*} ,\mathop Q\limits_{2}^{*} ,\mathop R\limits_{2}^{*} } \right\}\) is globally asymptotically stable on \(\Omega \). This achieves the proof of the theorem.
Theorem 7:
If \(R_{0} > 1\), then the positive endemic equilibrium state of system (1) exists and is globally asymptotically stable on \(\Omega\), by assuming that \(\varphi_{1} = \frac{{\delta_{1} + \rho_{1} }}{{S_{1}^{*} }} \), \(\rho_{1} = \gamma_{1}\), \(\varphi_{2} = \frac{{\delta_{2} + \rho_{2} + \vartheta }}{{S_{2}^{*} }}\),
Proof of Theorem 7
We construct the Lypunov function from the model as follows
Since
Substituting the relations in Eqs. (18), we have
Substituting the relations in Eq. (18a), we have
Hence, the condition (16) show that \(\dot{\omega }\left( t \right) \le 0\) of all terms. Then the equilibrium steady state \( B_{1} = \left( {S_{1}^{*} ,\mathop E\limits_{1}^{*} ,\mathop I\limits_{1}^{*} ,\mathop Q\limits_{1}^{*} ,\mathop R\limits_{1}^{*} ,\mathop S\limits_{2}^{*} ,\mathop E\limits_{2}^{*} ,\mathop I\limits_{2}^{*} ,\mathop Q\limits_{2}^{*} ,\mathop R\limits_{2}^{*} } \right)\) is the globally asymptotically stable in the \(\Omega\).
Sensitivity analysis
The model of the parameters will affect the spread and spread of COVID-19, the results of insertion into model (2) will be subjected to a sensitivity analysis. We begin by first introducing the following definitions of30,31,32,33,34,35,36.
Definition 1:
The normalized forward sensitivity index of the variable (\(R_{0}\)), depending on the parameter difference, is given as: \(E_{\zeta }^{\emptyset } = \frac{\partial \emptyset }{{\partial \zeta }} \times \frac{\zeta }{\emptyset }\). A new expression for \(R_{0}\) is introduced as:
Then the sensitivity indices of the basic reproduction number (\({R}_{0}\)), with respect to the system model depends on the nineteenth parameter are computed as below.
We can estimate the sensitivity indices (S.I) of the basic reproduction number \({(R}_{0})\), taking into account the parameter of the model (2). The signs of sensitivity indices (S.I) are shown in the Table 3 and bar chart Fig. 8.
A bar chart showing the measurement of the sensitivity indices with various parameters of model (2) and the reference values as indicated in Table 3.
The effects of changing parameter values on the functional value of the reproduction number \({R}_{0}\) are obtainable in this section. The necessary parameters must be found, which may be important criteria in disease management. The desirable changes of the occurred when their changes produce a positive effect, i.e., when their sensitivity indices have positive sign, i.e.\(\alpha_{1} ,\alpha_{2} ,IIP_{2} ,g_{1} ,g_{2} ,q_{3T} ,\mu_{{}} ,C_{{}} ,\varphi_{1} ,\varphi_{12} ,\varphi_{2}\) and \(\varphi_{21}\) have a positive effect on \(R_{0}\). The determine that the increase in the number of two exposed population \(\left( {E_{1} ,E_{2} } \right)\) and two infectious host population \(\left( {I_{1} ,I_{2} } \right)\) with the value \(IIP_{1} , g_{1} , g_{2} ,IIP_{2}\) may lead to an outbreak. On the other hand, the negative sign of the sensitivity indices (S.I) in the \(R_{0}\) i.e. \(\delta_{1} , \delta_{2} , IIP_{1} ,q_{2T} ,\rho_{1} ,\gamma_{1} ,\vartheta_{{}} ,\gamma_{2}\) and \(\rho_{2}\) has a negative effect on the spread of disease according to the system (2). Thus, the sensitivity indices (S.I) of the Covid-19 (2) shows that there will been appreciable change at the beginning of the transmission the disease. This would help the public health official to plan on how best to develop a reasonable interference strategy to prevent and manage the spread of the disease.
Conclusions and discussion
Tourism has become an important source of foreign currency for many countries. This is especially true for Thailand. It is the second major source of currency. This means that tourists are coming to Thailand every year. When combined with the need for temporary, seasonal farmer workers to support the main source of income in Thailand, that is the agricultural industry, foreigners (tourists), and these foreign workers diseases can be brought into Thailand. Thailand must always be aware of the arrival of new infectious diseases. Most recently, the novel coronavirus COVID-19 appeared in China. From a few hundred infections in Wuhan, China, it is quickly evolved into a pandemic, which spread to five continents public health authorities in Thailand have initiated public health measures to control the spread and stop the spread of this virus in the United States. More than one million people have died from the disease. In this article a standard SEIQR model has been introduced for the transmission dynamics of COVID-19 infection in Thailand and for Foreign (tourists) entering the Thai population. Affecting the change of COVID-19 among Thais people, they took the SEIQR model for each population and linked them together, allowing members of each population to cross-infection with each other. The impact of factors causing changes in the spread of COVID-19 is examined. After that, we performed a basic reproductive number analysis and saw how homeostasis changes. Taking cross-infection (mixed) into account, we find that our model achieves an infection-free equilibrium. When the basic reproductive number is less than one. This model achieves local equilibrium at multiple points when the number is greater than one. Our analysis shows that the rate of recovery rate of both Thais and tourists would be affected by decreases in the recruitment rates and death rates. This result shows that the recovery rate for both Thais and foreigners has increased. This is because changes in recruitment and death rates will result in a decrease in the basic reproductive number. However, changes \(IIP_{1}\) (capita rate of progression of Thais population from the exposed state to the infectious state), \(IIP_{2}\) (capita rate of progression of foreign human from the exposed state to the infectious state), \(q_{2T}\) ( the number of infected Thais that leave the quarantine period with the virus intact) and \(q_{3T}\) (the number of infected Foreign that leave the quarantine period with the virus intact), \(g_{1}\) (the rate at which the exposed Thais are put into quarantine from the exposed and infected Thais) and \(g_{2}\) (the rate at which the exposed Foreign (tourists) are put into quarantine from the exposed and infected Foreign (tourists)), \(\varphi_{1}\)( transmission rate of virus between population in Thais population), \(\varphi_{2}\) (transmission rate of virus between population in Foreign (tourists) population), \(\mu\) (recruitment term of the susceptible population in Thais) and \(C\)(recruitment term of the susceptible population in Foreign (tourists)) would cause the basic reproductive number to increase meaning increases in the severity of the pandemic, more people being infected by the COVID-19 coronavirus. Therefore, we controlled the number of new confirmed cases or new infections significantly by introducing a positive change in the parameter memory value in the sensitivity analysis.
In summary, from the study of the spread of the COVID-19 virus, which is an infectious disease. This disease is a health problem, leading to a rapid decline in the impact on the economy. Although governments and the World Health Organization have implemented international control measures and prevented interference. By creating a mathematical model that uses data from disease outbreaks between Thais population and Foreign (tourist) entering Thailand. To determine some parameters affecting the outbreak under proper control of the disease. By relying on the strategies of the government and the World Health Organization, including controlling the spread of infection, incubation, treatment and prevention of fever. It was found that controlling the disease transmission will be a guideline for reducing the spread and reducing the number of cases between Thai people and foreigners.
Data availability
The data in the analysis is taken from Bureau of Epidemiology Ministry of Public Health, Thailand (https://ddc.moph.go.th/viralpneumonia/eng/index.php).
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Acknowledgements
The authors thank the handling editor and anonymous referees for their valuable comments and suggestions which led to an improvement of our original paper. R.S would like to thank Research and Development Institute and Faculty of Science and Technology, Phuket Rajabhat University. P.P would like to thank School of Science, King Mongkut’s Institute of Technology Ladkrabang.
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Research and Development Institute and Faculty of Science and Technology, Phuket Rajabhat University.
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R.S.: Conceptualization (equal); Funding acquisition (equal); Formal analysis (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal);. I.M.: Conceptualization (equal); Supervision (equal); Writing – review & editing (equal); Investigation (equal); Validation (equal). P.P.: Conceptualization (equal); Formal analysis (equal); Supervision (equal);Methodology (equal);Writing – original draft (equal); Writing – review & editing (equal);Investigation (equal); Validation (equal).
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Sungchasit, R., Tang, IM. & Pongsumpun, P. Sensitivity analysis and global stability of epidemic between Thais and tourists for Covid -19. Sci Rep 14, 21569 (2024). https://doi.org/10.1038/s41598-024-71009-x
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DOI: https://doi.org/10.1038/s41598-024-71009-x
