
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

71009
10.1038/s41598-024-71009-x
Article
Sensitivity analysis and global stability of epidemic between Thais and tourists for Covid -19
Sungchasit Rattiya 1
Tang I.-Ming 2
Pongsumpun Puntani puntani.po@kmitl.ac.th

3
1 https://ror.org/03mh64g46 grid.444142.3 0000 0000 9555 1411 Department of Mathematics, Faculty of Science, Phuket Rajabhat University, Phuket, Thailand
2 https://ror.org/01znkr924 grid.10223.32 0000 0004 1937 0490 Department of Physics, Faculty of Science, Mahidol University, Bangkok, Thailand
3 https://ror.org/055mf0v62 grid.419784.7 0000 0001 0816 7508 Department of Mathematics, School of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok, Thailand
16 9 2024
16 9 2024
2024
14 2156912 12 2023
23 8 2024
© The Author(s) 2024
2024
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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.

Subject terms

Applied mathematics
Computational science
Diseases
issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

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 epidemic1–5. 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 virus1–4 and6–9 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-193–11.

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 zoonotic1–3 and6–9 and14–18, 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 humans5–8. 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 and20–23. 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 people21–25. 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 inactive1–4 and13–16 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.Fig. 1 Number of patients and deaths of COVID-19 cases per month around the world and in Thailand1–4.

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 countries4–6 and16–18, 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 individuals17–25. 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 model20–26. 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 S1,E1,I1,Q1,R1 are Thais population respectively and S2,E2,I2,Q2,R2 are Foreign (tourist) or migrant workers) population respectively of COVID-19 transmission model21–26.

The Recruitment term of the susceptible population in Thais and the rest if the Foreign (tourist) are given as μ 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 δ1 and δ2. The force of infections in Thais population φ1S1E1+I1 (Transmission rate of virus between population from Thais population to Foreign (tourist) population (in Thais) and φ12S1E2+I2 (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 φ2S2E2+I2(Transmission rate of virus between population from Foreign (tourist) population to Thais population (in Foreign (tourist)) and φ21S2E1+I1 (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.Fig. 2 The flowchart illustration the dynamics of the model.

The host population was divided into five compartments: S1 number of Thais susceptible to COVID-19 infection at time t, E1 number of Thais exposed to COVID-19 infection at time t, I1 number of infectious Thais at time t, Q1 number of Thais quarantined for COVID-19 at time t, R1 number of recovered Thais at time t, S2 number of Foreign (tourist) susceptible at time t, E2 number of Foreign (tourist) exposed at time t, I2 number of Foreign (tourist) infected at time t, Q2 number of Foreign (tourist) quarantined at time t, R2 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:1 dS1dt=μNh-φ1S1E1+I1-δ1S1+α1R1-φ12S1E2+I2,dE1dt=φ1S1E1+I1+φ12S1E2+I2-δ1E1-1IIP1E1,dI1dt=1IIP1E1-q2TI1-δ1I1+ρ1I1,dQ1dt=q2TI1-γ1Q1-δ1Q1+g1E1+I1,dR1dt=γ1Q1-(δ1+α1)R1,dS2dt=CNT-φ2S2E2+I2-φ21S2E1+I1-δ2S2-ϑS2+α2R2,dE2dt=φ2S2E2+I2+φ21S2E1+I1-δ2E2-ϑE2-1IIP2E2,dI2dt=1IIP2E2-q3TI2-(δ2+ρ2+ϑ)I2,dQ2dt=q3TI2-γ2Q2-(δ2+ρ2+ϑ)Q2+g2(E2+I2),dR2dt=γ2Q2-(δ2+ϑ+α2)R2.

with initial densities: S1≥0,E1≥0,I1≥0,Q1≥0,R1≥0 in Thais population and S2≥0,E2≥0,I2≥0,Q2≥0,R2≥0 in the Foreign (tourist) population.

All the parameters and corresponding biological meaning are defined in Table 1.Table 1 The description of the state variables and parameters of the model.

Description	Symbol	
Recruitment term of the susceptible population in Thais	μ	
Total Thais population	Nh	
Recruitment term of the susceptible population in Foreign (tourist)	C	
Total Foreign (tourist) population	NT	
Transmission rate of virus between population from Thais population to Foreign (tourist) population (in Thai) φ1S1E1+I1	φ1	
When Foreign (tourist) are present, a susceptible Thais can also be infected by an infected or exposed Foreign (tourist)(in Thais) φ12S1E2+I2	φ12	
Transmission rate of virus between population from Foreign (tourist) population to Thais population (in Foreign (tourist)) φ2S2E2+I2	φ2	
When Thais are present, a susceptible Thais can also be infected by an infected or exposed Thais (in Foreign (tourist))φ21S2E1+I1	φ21	
Per capita rate of progression of Thais population from the exposed state to the infectious state	IIP1	
Per capita rate of progression of Foreign (tourist) population from the exposed state to the infectious state	IIP2	
The rate at which the exposed Thais are put into quarantine from the exposed and infected Thais	g1	
The rate at which the exposed Foreign (tourist) are put into quarantine from the exposed and infected Foreign (tourist)	g2	
The number of infected Thais that leave the quarantine period with the virus intact	q2T	
The number of infected Foreign (tourist) that leave the quarantine period with the virus intact	q3T	
Per capita recovery rate for population in Thais from the infectious state to the recovered state	γ1	
Per capita recovery rate for population in Foreign (tourist) from the infectious state to the recovered state	γ2	
Natural death rate of Thais population	δ1	
Natural death rate of Foreign (tourist) population	δ2	
Per capita rate of loss of immunity in Thais population	α1	
Per capita rate of loss of immunity in Foreign (tourist) population	α2	
Rate at which Foreign (tourist) population move out the country	ϑ	
Death rate due to COVID-19 of Thais population	ρ1	
Death rate due to COVID-19 of Foreign (tourist) population	ρ2	

The total Thais population Nh is S1+E1+I1+Q1+R1. The equations for the human compartment are the following Eq. (1) and the total Foreign (tourist) population is NT=S2+E2+I2+Q2+R2. 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: S1Nh=S1′,E1Nh=E1′,I1Nh=I1′,Q1Nh=Q1′,R1Nh=R1′, S2NT=S2′,E2NT=E2′,I2NT=I2′,Q2NT=Q2′,R2NT=R2′ Renormalizing model (1) we obtain the following:2 dS1′dt=μ-φ1S1′E1′+I1′-δ1S1′+α1R1′-φ12S1′E2′+I2′,dE1′dt=φ1S1′E1′+I1′+φ12S1′E2′+I2′-δ1E1′-1IIP1E1′,dI1′dt=1IIP1E1′-q2TI1′-δ1I1′+ρ1I1′,dQ1dt=q2TI1′-γ1Q1′-δ1Q1′+g1E1′+I1′,dR1′dt=γ1Q1′-(δ1+α1)R1′,dS2′dt=C-φ2S2′E2′+I2′-φ21S2′E1′+I1′-(δ2+ϑ)S2′+α2R2′,dE2′dt=φ2S2′E2′+I2′+φ21S2′E1′+I1′-(δ2+ϑ)E2′-1IIP2E2′dI2′dt=1IIP2E2′-q3TI2′-(δ2+ρ2+ϑ)I2′,dQ2dt=q3TI2′-γ2Q2′-(δ2+ρ2+ϑ)Q2′+g2(E2′+I2′),dR2′dt=γ2Q2′-(δ2+ϑ+α2)R2′.

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 theorem24–26 were required.

Theorem 1: The given solution S1,E1,I1,Q1,R1,S2,E2,I2,Q2,R2 of the epidemiological systems (2) with non-negative initial data when S1≥0,E1≥0,I1≥0,Q1≥0,R1≥0 and S2≥0,E2≥0,I2≥0,Q2≥0,

R2≥0 stills non-negative for all time non-negative t>0.

Proof of Theorem 1

Given the initial data S10,E10,I10,Q10,R10 and S20,E20,I20,Q20,R20 are non—negative. It is clear from the first sub-equation of the model (2) that

dS1′dt=φ1S1′E1′+I1′+δ1S1′+φ12S1′E2′+I2′≥0 so that3 ddtS1′exp(δ1+φ1+φ12∫0tE1′ζ1+I1′ζ1+E2′ζ1+I2′ζ1)dζ1]≥0

Integrating (3) gives4 S1′t≥S1′0exp-δ1+φ1+φ12∫0tE1′ζ1+I1′ζ1+E2′ζ1+I2′ζ1dζ1≥0,∀t>0

Further, one sees from the second sub-equation of the model (2) that5 dE1′dt=φ1S1′E1′+I1′+φ12S1′E2′+I2′-δ1E1′-1IIP1E1′

dE1′dtδ1E1′+1IIP1E1′≥0 implies ddtE1′exp(δ1+1IIP1∫0tE1′(ζ1))dζ1≥0 which on integration yields6 E1′t≥E1′0exp-δ1+1IIP1∫0tE1′(ζ1))dζ1>0,∀t>0.

Further, one sees from the third sub-equation of the model (2) thatdI1′dt=1IIP1E1′-q2TI1′-δ1I1′+ρ1I1′

7 dI1′dtq2TI1′+δ1I1′+ρ1I1′≥0sothatdI1′dtI1′expq2T+δ1+ρ1∫0tI1′ζ1)dζ1≥0

which upon integration yields8 I1′t≥I1′0exp-q2T+δ1+ρ1∫0tI1′ζ1dζ1>0,∀t>0

Further, one sees from the fourth sub-equation of the model (2) thatdQ1′dt=q2TI1′-γ1Q1′-δ1Q1′+g1E1′+I1′,

dQ1′dt=γ1+δ1Q1′]≥0 implies dQ1′dtQ1′exp(γ1+δ1∫0tQ1′(ζ1)dζ1]≥0 which upon integration yields9 Q1′t≥Q1′0exp-(γ1+δ1∫0tQ1′(ζ1))dζ1≥0,∀t>0.

Further, one sees from the fifth sub-equation of the model (2) that dR1′dt=γ1Q1′-(δ1+α1)R1′, dR1′dt(δ1+α1)R1′≥0 implies dR1′dtR1′expδ1+α1∫0tR1′ζ1dζ1≥0 which upon integration yields.10 R1′t≥R1′0exp-δ1+α1∫0tR1′ζ1dζ1≥0,∀t>0

In a similar model, it can be shown that S2′t≥0,E2′t≥0,I2′≥0,Q2′≥0 and R2′t≥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 β given byβ=S1,E1,I1,Q1,R1,S2,E2,I2,Q2,R2∈R+10:S1+E1+I1+Q1+R1+S2+E2+I2+Q2+R2=1

Divided into two groups S1+E1+I1+Q1+R1=1 and S2+E2+I2+Q2+R2=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 β.

Theorem 2:

The solution of system (1) is possible for all if entering an invariant region. Ω=Ω1×Ω2, where Ω1={S1,E1,I1,Q1,R1∈R+5:0<Nht≤μδ1} as t→∞, when θ=minδ1,δ1+ρ1 and Ω2=S2,E2,I2,Q2,R2∈R+5:0<NTt≤Cδ2 as t→∞, when θ2=minδ2+ϑ,δ2+ρ2+ϑ.

Proof of Theorem 2

The invariant region is received from the bounded situation of the system. Here, Nht=S1t+E1t+I1t+Q1t+R1t and NTt=S2t+E2t+I2t+Q2t+R2t. It following that,

dNhdt=dS1dt+dE1dt+dI1dt+dQ1dt+dR1dt=μ-ρ1+g1I1-g1E1-δ1Nh≤μ-δ1Nh

This inequality can be expressed in a general solutions asNht≤μδ1+Nh0-μδ1e-δ1t,

where Nh0 is the initial values, i.e., Nht=Nh0 at t=0.

In a similar model, it can be shown that NTt=S2t+E2t+I2t+Q2t+R2t for the bounded situation of the system all time t>0. Moreover, every solution for systems (1) with initial conditions in Ω remains in Ω for all t>0. Therefore, the dynamics of our model will be poised in Ω.

Analysis of the model

Basic reproduction number

The next generation matrix method is used to calculate the basic reproduction number, R027–36 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)15–19 will be determined by R0 which has the form11 R0=R0TR0F

where R0T=q2T+γ1+δ1μ1+IIP1δ1+ρ1φ1δ11+IIP1δ1g1q2T+q2T+γ1+δ1δ1+ρ1 is the basic reproduction number for Thais and R0F=α3C1+α2IIP2φ2α1α2α3+g2q3TIIP2δ2+ϑ is the basic reproduction number for the Foreign (tourist) population only with α1=δ2+ϑ+1IIP2,α2=δ2+ϑ+ρ2,α3=δ2+ϑ+γ2+q3T and α4=δ2+ϑ+α2, we have

Theorem 3:

To find the basic reproduction number of our proposed differential Eq. (2), using help of the next generation matrix formulas17–25. We initially define K=(E1,I1,Q1)T and K1=(E2,I2,Q2)T. The model (2) is rewritten in the following form dydt=Fy-Vy, where Fy is the non-negative matrix of the newly infected (Thais and Foreign (tourist) populations) and Vy 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:Fy=φ1S1E1+I1+φ12S1E2+I200andVy=δ1E1+1IIP1E1-1IIP1E1+δ1I1+ρ1I1+q2TI1-g1E1+I1+γ1+δ1Q1-q2TI1

for the Thais population.F1y=φ2S2E2+I2+φ21S2E1+I100andV1y=δ2+ϑE2+1IIP2E2-1IIP2E2+δ2+ϑ+ρ2I2+q3TI2-g2E2+I2+γ2+δ2+ϑ+ρ2Q2-q3TI2

for the Foreign (tourist) population.

The basic reproductive number R0 is the threshold for the stability of the disease-free equilibrium B0. It can be calculated by R0=ρFV-1 where, FV-1 is called the next generation matrix and ρFV-1 is the spectral radius of the matrix FV-1. Then we get reproduction number R0 where,12 R0=α3C1+α2IIP2φ2q2T+γ1+δ1μ1+IIP1δ1+ρ1φ1α1α2α3+g2q3TIIP2δ2+ϑδ11+IIP1δ1g1q2T+γ1+δ1δ1+ρ1).

Finally, the Routh–Hurwitz criteria is used for determining the stabilities of the model. If R0>1, then the endemic equilibrium is local asymptotically stable, but if R0<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 follows24–37.(A) The COVID-19 free equilibrium of the Eq. (2) exists and then given byB0=S1,E1,I1,Q1,R1,S2,E2,I2,Q2,R2,B0=μδ1,0,0,0,0,Cδ2,0,0,0,0

(B) The COVID-19 endemic equilibrium of the Eq. (2) exists with infection and then given by13 B1=S1∗,E1∗,I1∗,Q1∗,R1∗,S2∗,E2∗,I2∗,Q2∗,R2∗S1∗=R1∗α1+μφ1E1∗+I1∗+δ1+φ12E2∗+I2∗,E1∗=IIP1S1∗φ1I1∗+φ12E2∗+I2∗1+IIP1δ1-S1∗φ1,I1∗=E1∗IIP1-q2T+δ1+ρ1,Q1∗=g1E1∗+I1∗-I1∗q2Tγ1+δ1,R1∗=γ1Q1∗δ1+α1,S2∗=C+R2∗α2φ2E2∗+I2∗-φ21E1∗+I1∗+(δ2+ϑ),E2∗=IIP2S2∗φ2I2∗+φ21E1∗+I1∗1+IIP2δ2+ϑ-S2∗φ2,I2∗=E2∗IIp2-q3T+δ2+ϑ+ρ2,Q2∗=g2E2∗+I2∗-I2∗q3Tγ2+δ2+ϑ+ρ2,R2∗=γ2Q2∗δ2+ϑ+α2.

Local asymptotically stability of disease—free equilibrium point

Lemma 1:

(The Generalized Routh–Hurwitz Criterion). Given the charactistic equationλk+a1λk-1+a2λk-2+⋯+ak=0

Define k matrices as follows:H1=a1,H2=a11a3a2,H3=a110a3a2a1a5a4a3,…

HJ=a1100…0a3a2a11…0a5a4a3a2…0a2j-1a2j-2a2j-3a2j-4…aj⋯Hk=a110…0a3a2a1…0⋮⋮⋮⋮00…ak.

where the l,m term in the matrix Hj is a2l-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 N¯ is stable if and only if the determinants of all Hurwitz are positive:detHj>0j=1,2,…,k.

When is N¯=N¯i+a1eλ1t+a2eλ2t+⋯+akeλkt.

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 R0>1, the point is stable and unstable otherwise.12–18, and24–29, and31.

Proof of Theorem 4

To determine the local stability of J0, we evaluate the Jacobian matrix at the disease-free state to be14 J0=δ1-φ1S1′-φ1S1′0α10-φ12S1′-φ12S1′000θ1φ1S1′000φ12S1′φ12S1′0001IIP1-δ1+ρ1+q2T00000000g1q2T+g1-γ1+δ1000000000γ1-(δ1+α1)000000-φ21S2′-φ21S2′00-(δ2+ϑ)-φ2S2′-φ2S2′0α20φ21S2′φ21S2′000θ2φ2S2′000000001IIP2θ300000000g2q3T+g2θ4000000000γ2θ5=0

where θ1=φ1S1-δ1+1IIP1, θ2=φ2S2-δ2+ϑ+1IIP2, θ3=-(δ2+ϑ+q3T+ρ2), θ4=(δ2+ϑ+γ2+ρ2) and θ5=-δ2+ϑ+α2.

The eigenvalues of the J0 are obtained by solving DetJ0-λI=0. We obtain the characteristic equation, where λ is an eigenvalue of the matrix J0. The, root of the model (2) i.e., eigenvalue of the matrix J0 are15 λ+δ2+ϑ+α2λ+δ1+α1λ+δ1(λ7+A1λ6+A2λ5+A3λ4+A4λ3+A5λ2+A6λ+A7=0

The three eigenvalues from Eq. (15) were λ1=-δ2-ϑ-α2, λ2=-δ1-α1 and λ3=-δ1 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 A1,A2,A3,A4,A5,A6,A7>0(The coefficients appearing in Eq. 15 from the Routh Hurwitz condition are plotted on the graph by the x axis being the coefficient A2. and the Y-axis is the coefficient of A1,A2,A3,A4,A5,A6 and A7 obtained by finding determinants from size nxn, parameter values from Table 1 by the use the Mathematica program.) This is displayed for R0<1, disease-free equilibrium point will be stable as showed in Fig. 3.Fig. 3 The parameter areas for disease free equilibrium state which satisfies the Routh-Hurwitz criteria with the value of parameters: respectively, for with (λ7+A1λ6+A2λ5+A3λ4+A4λ3+A5λ2+A6λ+A7=0.

Whenβ1=A1A2-A3,β2=-A32-A12A4+A1A2A3+A5,

β3=-A4-A1A2A3+A32+A12A4+-A1A23+A2A3+2A1A4A5-A52,

β4=-A5A4-A1A2A3+A32+A12A4+-A2A3+A1A23-2A4A5+A52+A32-A1A3A2A3+2A5+A12A3A4+A2A5A6,

β5=A6(-A5A4-A1A2A3+A32+A12A4+-A2A3+A1A23-2A4A5+A52+A33-A1A3A2A3+2A5+A12A3A4+A2A5A6,

β6=A7(A6-A5A4-A1A2A3+A32+A12A4+-A2A3+A1A23-2A4A5+A52+A33-A1A3A2A3+3A5+A12A3A4+2A2A5A6-A13A62+A1A4-A2A3A4+A23A5-2A4A5+A4A32A4-A2A3A5+A52+A3-A2A3+3A5A6+A12A23A3+A3A4-A2A5A6+A12A43-3A2A4A6+3A62A7--A22A3+2A3A4+A2A5+A1A23-A2A4+3A6A72+A73),

and β7=A8(A7A6-A5A32A4-A2A3A5+A52+A33A6+A4A32A4-A2A3A5+A52+A3-2A2A3+3A5A6A7+A22A3-2A3A4-A2A5A72+A73+A54-A3A52A2A5+4A7-A33A5A6+2A4A7+A32A4A52+3A2A5A7+2A72A8+A34A82+A1A7-A6A5-A2A3A4+A22-2A4A5A7-A32-3A2A4+3A6A72+-2A4A53+3A3A52A6+4A3A4A5A7+A32A6A7+4A5A72+A22A53-3A3A5A7+A2A3A5-A4A5+A3A6+2A32A4+A52A7-5A3A72A8-A32A2A3+4A5A82+A12A7-A4A5A6+A43A7+A622A2A5+3A7+A4A6A3A6-3A2A7-A5-A42A5+A3A4A6+2A2A5A6+2A3A42-A2A4A6+A2A3A6+5A5A6A7+3-A2+A4A72A8+A32A4+3A2A3A5+2A52+4A3A7A82.

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 R0<1 the state is stable and unstable otherwise12–18, and24–26, and30,31.

Proof of Theorem 5

The Jacobian matrix of the model (2) at pandemic equilibrium point is16 J1=η1-φ1S1′-φ1S1′0α10-φ12S1′-φ12S1′00η2η3φ1S1′000φ12S1′φ12S1′0001IIP1-(q2T+δ1+ρ1)00000000g1g1+q2T-(γ1+δ1)000000000γ1-(α1+δ1)000000-φ21S2′-φ21S2′00η4-φ2S2′-φ2S2′0α20φ21S2′φ21S2′00η5η6φ2S2′000000001IIP2η700000000g2g2+q3Tη8000000000γ2η9=0

Where η1=-φ1E1+I1-φ12E2+I2-δ1 , η2=φ1E1+I1+φ12E2+I2 η3=φ1S1-δ1+1IIP1, η4=-φ2E2+I2-φ21E1+I1-(δ2+ϑ) , η5=-φ2E2+I2+φ21S2E1+I1, η6=-φ2S2-(δ2+ϑ+1IIP2, η7=-q3T+δ2+ρ2+ϑ, η8=-(γ2+δ2+ρ2+ϑ) and η9=δ2+ϑ+α2.

The endemic equilibrium point (B1) exists and is positive if R0>1. The eigenvalues of J1 are obtained by solving DetJ1-λI=0. The characteristic equation is as follows; we obtain the characteristic equation λ10+W1λ9+W2λ8+W3λ7+W4λ6+W5λ5+W6λ4+W7λ3+W8λ2+W9λ+W10=0 where λ are eigenvalues of the matrix J1. 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 W1,W2,W3,W4,W5,W6,W7,W8,W9,W10>0 (The coefficients appearing in Eq. (15) from the Routh Hurwitz condition are plotted on a graph by the x axis being the coefficient W2 and the Y-axis is the coefficient of W1,W2,W3,W4,W5,W6,W7,W8,W9,W10, obtained by finding determinants from size nxn, parameter values from Table 1 by the use the Mathematica program.) This is displayed for R0>1, endemic equilibrium point will be stable as showed in Fig. 4.Fig. 4 The parameter areas for endemic equilibrium point which satisfies the Routh-Hurwitz criteria with the value of parameters: respectively, for with.

λ10+W1λ9+W2λ8+W3λ7+W4λ6+W5λ5+W6λ4+W7λ3+W8λ2+W9λ+W10=0. Whenκ1=W1W2-W3,κ2=-W32-W12W4+W1(W3W2+W5),κ3=-W4-W1W2W3+W32+W12W4+-W1W23+W2W3+2W1W4W5-W52,

κ4=-W5W4-W1W2W3+W32+W12W4+-W2W3+W1W23-2W4W5+W52+(W32-W1W3W2W3+2W5+W12W3W4+W2W5)W6,

κ5=W6(-W5W4-W1W2W3+W32-W12W4+-W2W3+W1W23-2W4W5+W52+W33-W1W3W2W3+2W5+W12W3W4+W2W5W6),

κ6=W7(W6-W5W4-W1W2W3+W32+W12W4+-W2W3+W1W23-2W4W5+W52+W33-W1W3W2W3+3W5+W12W3W4+2W2W5W6-W13W62+W1W4-W2W3W4+W23W5-W2W5+W4W32W4-W2W3W5+W52+W3-W2W3+3W5W6+W12W23W3+W3W4-W2W5W6+W12W43-3W2W4W6+3W62W7--W22W3+2W3W4+W2W5+W1W23-W2W4+3W6W72+W73),

κ7=W8W7W6-W5W32W4-W2W3W5+W52+W33W6+W4W32W4-W2W3W5+W52+W3-2W2W3+3W5W6W7+W22W3-2W3W4-W2W5W72+W73+W54-W3W52W2W5+4W7-W33W5W6+2W4W7+W32W4W52+3W2W5W7+2W72W8+W34W82+W14W83+W1W7-W6W5-W2W3W4+W22-2W4W5W7-W23-3W2W4+3W6W72+-2W4W53+3W3W52W6+4W3W4W5W7+W32W6W7+4W5W72+W22W53-3W3W5W7+W2W3W5-W4W5+W3W6+2W32W4+W52W7-5W3W72W8-W32W2W3+4W5W82+(W12W7-W4W5W6+W43W7+W622W2W5+3W7+W4W6W3W6-3W2W7-W5-W42W5+W3W4W6+2W2W5W6+2W3W42-W2W4W6+W2W3W6+5W5W6W7+3-W2+W4W72W8+W32W4+3W2W3W5+2W52+4W3W7)W82)),

κ8=W15W104-W103W35+W102W55-W3W53(W2W5+5W7+W342W6W7+W5W8+3W4W9+W32W5W4W52+4W2W5W7+5W72+5W5W9-W33(W52W6+3W4W5W7+2W2W72+5W7W9))-W10(W75-W5W73W2W7+5W9+W54(W7W8+2W3W9+W34W8W7W8+3W6W9+W52W7W4W72+4W2W7W9+5W92-W53W6W72+3W4W7W9+2W2W92+W33W7W62W7-W5W6W8-2W4W7W8-2W5W62-W4W6W7+W4W5W8+5W2W7W8W9-3-W42+W2W6+W8W9+W32W7W42W72+W4W5-W6W7+W5W8+W7-2W2W6W7+3W2W5W8+2W7W8+-3W42W5W7+W64W2W5-3W7+W42W52W6+W2W72+W5W2W5+7W7W8W9+-5W2W4W5+7W5W6+4W32W7-7W4W7W9+3W2W92+W3(W7(W7W2W52W6+W5-2W53W6+3W4W52W7+W72-W56W5W6W7+6W4W72+W52W8W9+2W23+3W4W5-7W2W5W7+5W72W9-5W5W93))+W9(W34W83+W8-W53W6W7+W4W52W72-W2W5W73+W74+W54W8+(-W6W73+W5W72W6W2-4W8+W52W7-W4W6+3W2W8+W53W62-2W4W8)W9))

, and κ9=W14(W84W9-W103(2W5W6+3W4W7+W3W8+4W2W9)-W10W82(W7W8+4W6W9)+W102(3W6W7W8+2W62W9+W8(W5W8+4W4W9)))+W13(W9(W8(-W63W7+W6(W5W6+3W4W7)W8-(2W4W5+W3W6+3W2W7W82)+(W64-4W4W62W8+2(W42+2W2W6)W82-4W83)W9)+W10(W7(W63W7-W6(W5W6+3W4W7)W8+(2W4W5+W3W6+3W2W7)W82)+(W62-(-2W5W6+W4W7)+(4W4W5W6-5W42W7+W6(3W3W6+2W2W7))W8+(W3W4+2W2W5+7W7)W82)W9+4w(W6(W42-W2W6)+2(-W2W4+W6)W8)W92)+W103(W32W6+W5(2W2W5+5W7)+W3(3W4W5+4W2W7+5W9))-W102(3W73(-W42+W2W6+W8)-W52(W62-2W4W8)+(-5W2W4+7W6)W7W9+2(-3W22+2W4)W92+W5(-W4W6W7+W3W6W8+5W2W7W8+3W42W9-2W2W6W9+6W8W9)+W3(2W62W7+W4W7W8+5W4W6W9+W2W8W9)))+W1(W103W33(W2W3+5W5)+W10(W7(W7(2W42W5W7-W4W6(2W52+W3W7)+3W6(W3W5W6+W72))-(2W4W52+3W3W52W6+4W3W4W5W7+W32W6W7+4W5W72)W8+4W32W5W82)+(-6W42W52W7-3W5W6(2W3W5W6+3W72)+(11W32W5W6+3W52W7+13W3W72)W8+W32W82+W4(4W53W6+5W3W5W6W7-2W73-W3(2W52+5W3W7)W8))W9+(W32W4W6+4W5(2W5W6+W4W7)+W3(6W42W5+W6W7-2W5W8))W92+(-7W3W4+5W7)W93+W23(W74-4W5W72W9+2(W52+2W3W7)W92)+W22(-W7(W52W6W7+2W3W52W7+4W3W5W6W7+W3W4W72+W73+W3(W52-5W3W7)W8)W9-(5W3W4W5+3W32W6+3W5W7)W92+7W3W93)+W2(W7(W7(W32W62+W7(W5W6-3W4W7)+W3W4(-W5W6+W4W7))-(W3W5(-W4W5+W3W6)+(2W32W4+W52)W7-5W3W72)W8+W33W82)+(W5W7(-2W5W6+11W4W7)+W3(2W4W52W6-3W42W5W7-5W4W7)+W32(3W43-8W8))W92-6W5W93))+W9(W8(-2W43W5W73+W32W8(W6W7-4W5W8)+3W3W6W5(-W6W7+W5W8)+W72(-3W6W7+4W5W8)+W4(2W52W6W7+W3W6W72-2W53W8+4W3W5W7W8))+(W6(2W42W5W7-W4W6(2W52+W3W7)+3W6(W3W5W6+W72))+(4W42W52+W6(-W32W6+2W5W7)+2W4(-4W3W5W6+W72))W8)))).

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)1–6 and24–30. 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 R0 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 R0 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).Table 2 Values of the parameter of the model (2) on COVID-19 transmissions.

Parameter	Description	Value/range (Units)	
δ1	Natural death rate of Thais population	0.0000365	
δ2	Natural death rate of Foreign (tourist) population	0.000033	
α1	Per capita rate of loss of immunity in Thais population	0.045	
α2	Per capita rate of loss of immunity in Foreign (tourist) population	0.067	
IIP1	Per capita rate of progression of Thais population from the exposed state to the infectious state	0.13–0.785

Estimation26–32

	
IIP2	Per capita rate of progression of Foreign (tourist) population from the exposed state to the infectious state	0.13–0.785

Estimation27–30

	
g1	The rate at which the exposed Thais are put into quarantine from the exposed and infected Thais	0.341–0.854

Estimation24–30

	
g2	The rate at which the exposed Foreign (tourist) are put into quarantine from the exposed and infected Foreign (tourist)	0.341–0.854

Estimation23–29

	
q2T	The number of infected Thais that leave the quarantine period with the virus intact	0.08–0.099

Estimation22–27

	
q3T	The number of infected Foreign (tourist) that leave the quarantine period with the virus intact	0.08–0.099

Estimation26–32

	
γ1	Per capita recovery rate for population in Thais from the infectious state to the recovered state	0.0035–0.097	
γ2	Per capita recovery rate for population in Foreign (tourist) from the infectious state to the recovered state	0.0035–0.097	
ρ1	Death rate due to COVID-19 of Thais population	0.000067–0.003	
ρ2	Death rate due to COVID-19 of Foreign (tourist) population	0.000067–0.003	
φ1	Transmission rate of virus between population in Thais population	0.03125	
φ12	When Foreign are present, a susceptible Thais can also be infected by an infected or exposed Foreign (in Thais)	0.239–0.988	
φ2	Transmission rate of virus between population in Foreign (tourist) population	0.04167	
φ21	When Thais are present, a susceptible Thais can also be infected by an infected or exposed Thais (in Foreign (tourist)	0.239–0.988	
ϑ	Rate at which Foreign (tourist) population move out the country	0.0078–0.5	

Fig. 5 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.

Fig. 6 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).Fig. 7 The trajectories of the numerical projected onto the 2D (a)(S1,E1), (b) (S1,I1), (c) (S1,Q1), (d) (E1,I1), (e) (I1,Q1), (f) (S2,E2), (g) (S2,Q2), (h) (I2,Q2), (i) (E1,E2), (j) (I1,I2) and (k) (Q1,Q2). 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 R0≤1, then the disease—free equilibrium E∗ is globally asymptotically stable, byφ1=δ1+ρ1S1∗andφ2=δ2+ρ2+ϑS2∗(∗)

Proof of Theorem 6

The Lyapunov function may be constructed for the model (1) through the use of the function17 Pt=(S1-S1∗lnS1)+E1+I1+Q1+R1+(S2-S2∗lnS2)+E2+I2+Q2+R2

Differentiating with respect to time yields.P˙t=S1-S1∗S1++E˙1+I˙1+Q˙1+R˙1+S2-S2∗S2+E˙2+I˙2+Q˙2+R˙2

=μNh-φ1S1E1+I1-δ1S1+α1R1-φ12S1E2+I2S1-S1∗S1+(φ1S1E1+I1+φ12S1E2+I2-δ1E1-1IIP1E1)+(1IIP1E1-q2TI1-(δ1I1+ρ1I1))+(q2TI1-γ1Q1-δ1Q1+g1(E1+I1))+(γ1Q1-(δ1+α1)R1)+((CNT-φ2S2E2+I2-φ21S2E1+I1-δ2S2-ϑS2+α2R2)S2-S2∗S2)+(φ2S2E2+I2+φ21S2E1+I1-δ2E2-ϑE2-1IIP2E2)+(1IIP2E2-q3TI2-(δ2+ρ2+ϑ)I2)+(q3TI2-γ2Q2-(δ2+ρ2+ϑ)Q2+g2(E2+I2))+(γ2Q2-(δ2+ϑ+α2)R2)

=μNh1-S1∗S1+CNT1-S2∗S2-δ1S1∗1-S1S1∗-δ2S2∗1-S2∗S2-ϑS2∗1-S2S2∗+φ1I1S1∗+φ1S1∗Q1-α1R1S1∗S1-δ1R1-δ1E1-δ1+ρ1I1-δ1Q1+φ2I2S2∗+φ2Q2S2∗-α2R2S2∗S2-(δ2+ϑ)R2-(δ2+ϑ)E2-(δ2+ρ2+ϑ)I2-(δ2+ϑ)Q2

17a =μNh1-S1∗S1+CNT1-S2∗S2-δ1S1∗1-S1S1∗-δ2S2∗1-S2∗S2-ϑS2∗1-S2S2∗-δ1E1-δ1+ρ1-φ1S1∗I1-δ1-φ1S1∗)Q1-R1(α1R1S1∗S1+δ1-(δ2+ϑ)E2-(δ2+ρ2+ϑ-φ2S2∗)I2-(δ2+ϑ-φ2S2∗)Q2-(α2S2∗S2+(δ2+ϑ))R2

Using the condition (17), Eq. (17a) may be rewrite as,P˙t=μNh1-S1∗S1+CNT1-S2∗S2-δ1S1∗1-S1S1∗-δ2S2∗1-S2∗S2-ϑS2∗1-S2S2∗-δ1E1-δ1+ρ1-φ1S1∗I1-δ1-φ1S1∗)Q1-R1(α1R1S1∗S1+δ1-(δ2+ϑ)E2-(δ2+ρ2+ϑ-φ2S2∗)I2-(δ2+ϑ-φ2S2∗)Q2-(α2S2∗S2+(δ2+ϑ))R2

Substitute with Eq. (17) we obtainP˙t=μNh1-S1∗S1+δ1S1∗1-S1S1∗+CNT1-S2∗S2+δ2S2∗1-S2∗S2+ϑS2∗1-S2S2∗-δ1E1-(δ2+ϑ)E2-δ1R1-(δ2+ϑ)R2

Note that on Ω, we have S1∗=μNhδ1 and S2∗=CNTδ2+ϑ with this in mind, Eq. (17) becomesP˙t=μNh1-S1∗S1+δ1μNhδ11-S1S1∗+CNT1-S2∗S2+(δ2+ϑ)CNTδ2+ϑ1-S2∗S2-δ1E1-(δ2+ϑ)E2-δ1R1-(δ2+ϑ)R2

=μNh2-S1∗S1-S1S1∗+CNT2-S2∗S2-S2S2∗-δ1E1-(δ2+ϑ)E2-δ1R1-(δ2+ϑ)R2

P˙t=-μNhS1∗-S12S1∗S1-CNTS2∗-S22S2∗S2-δ1E1-(δ2+ϑ)E2-δ1R1-(δ2+ϑ)R2≤0

Hence, P˙t≤0. By using LaSalle’ s (1976)31–37 extension to Lyapunov method, the limit of each solution is contained in the largest invariant set for which S1=S1∗,E1=0,I1=0,Q1=0,R1=0,S2=S2∗,E2=0,I2=0,Q2=0 and R2=0 which is the singleton E0. This means that the disease—free equilibrium E∗=S1∗,E1∗,I1∗,Q1∗,R1∗,S2∗,E2∗,I2∗,Q2∗,R2∗ is globally asymptotically stable on Ω. This achieves the proof of the theorem.

Theorem 7:

If R0>1, then the positive endemic equilibrium state of system (1) exists and is globally asymptotically stable on Ω, by assuming that φ1=δ1+ρ1S1∗, ρ1=γ1, φ2=δ2+ρ2+ϑS2∗, 18 ρ2=γ2,μNh=S1∗∗δ1andCNT=S2∗∗δ2+ϑ.

Proof of Theorem 7

We construct the Lypunov function from the model as followsω˙t=(S1-S1∗∗lnS1)+E1+I1+Q1+R1+(S2-S2∗∗lnS2)+E2+I2+Q2+R2,

ω˙t=S˙11-S1∗∗S1+E˙1+I˙1+Q˙1+R˙1+S˙21-S2∗∗S2+E˙2+I˙2+Q˙2+R˙2,

=μNh-φ1S1E1+I1-δ1S1-φ12S1E2+I2+α1(S1+E1+I1+Q1)1-S1∗∗S1+(φ1S1E1+I1+φ12S1E2+I2-δ1E1-1IIP1E1)+(1IIP1E1-q2TI1-δ1I1+ρ1I1+q2TI1)+(q2TI1-(γ1+δ1)Q1+g1(E1+I1))+)+((CNT-φ2S2E2+I2-φ21S2E1+I1-(δ2+ϑ)S2+α2((S2+E2+I2+Q2))1-S2∗∗S2+(φ2S2E2+I2+φ21S2E1+I1-(δ2+ϑ)E2-1IIP2E2)+(1IIP2E2+q3TI2-(δ2+ρ2+ϑ)I2)+(q3TI2-(δ2+ρ2+ϑ+γ2)Q2+g2(E2+I2)),

=μNh1-S1∗∗S1+φ1E1+I1S1∗∗+φ12E2+I2S1∗∗-δ1S1+δ1S1∗∗+α1-α1S1∗∗S1-α1S1+α1S1∗∗-α1E1+α1E1S1∗∗S1-α1I1+α1I1S1∗∗S1-α1Q1+α1Q1S1∗∗S1-δ1+1IIP1E1-δ1I1-ρ1I1-γ1Q1-δ1Q1-q2TI1+CNT1-S2∗∗S2+φ2E2+I2S2∗∗+φ21E1+I1S2∗∗-δ2S2+δ2S2∗∗-ϑS2+ϑS2∗∗+α2-α2S2∗∗S2-α2S2+α2S2∗∗-α2E2+α2E2S2∗∗S2-α2I2+α2I2S2∗∗S2-α2Q2+α2Q2S2∗∗S2-δ2+ϑ+1IIP2E2-(δ2+ρ2+ϑ)I2+q3TI2-(δ2+ρ2+ϑ+γ2)Q2,

18a =μNh1-S1∗∗S1+δ1S1∗∗1-S1S1∗∗+α11-S1-E1-I1-Q11-S1∗∗S1-δ1E1-1IIP1E1-δ1+ρ1-φ1S1∗∗I1-γ1+δ1-φ1S1∗∗Q1+CNT1-S2∗∗S2+(δ2+ϑ)S2∗∗1-S2S2∗∗+α21-S2-E2-I2-Q21-S2∗∗S2-δ2+ϑ+1IIP2E2-(q3T+δ2+ρ2+ϑ)I2+q3T-φ2S2∗∗)I2-(δ2+ρ2+ϑ+γ2-φ2S2∗∗)Q2

SinceS1∗∗=R1∗α1+μφ1E1∗+I1∗+δ1+φ12E2∗+I2∗andS2∗∗=C+R2∗α2φ2E2∗+I2∗-φ21E1∗+I1∗+δ2+ϑ

ω˙t=μNh1-S1∗∗S1+δ1R1∗α1+μφ1E1∗+I1∗+δ1+φ12E2∗+I2∗1-S1S1∗∗+α11-S1-E1-I1-Q11-S1∗∗S1-δ1E1-1IIP1E1-δ1+ρ1-φ1S1∗∗I1-γ1+δ1-φ1S1∗∗Q1+CNT1-S2∗∗S2+(δ2+ϑ)(C+R2∗α2φ2E2∗+I2∗+φ21E1∗+I1∗+δ2+ϑ)1-S2S2∗∗+α21-S2-E2-I2-Q21-S2∗∗S2-δ2+ϑ+1IIP2E2-(q3T+δ2+ρ2+ϑ)I2+q3T-φ2S2∗∗)I2-(δ2+ρ2+ϑ+γ2-φ2S2∗∗)Q2

Substituting the relations in Eqs. (18), we haveμNh=δ1(R1∗α1+μφ1E1∗+I1∗+δ1+φ12E2∗+I2∗),

CNT=(δ2+ϑ)(C+R2∗α2φ2E2∗+I2∗-φ21E1∗+I1∗+(δ2+ϑ)),

ω˙t=μNh1-S1∗∗S1+μNh1-S1S1∗∗δ1+α11-S1-E1-I1-Q11-S1∗∗S1+α11-S1-E1-I1-Q11-S1S1∗∗-δ1E1-1IIP1E1-γ1+δ1-φ1S1∗∗Q1+CNT1-S2∗∗S2+CNT1-S2S2∗∗+α21-S2-E2-I2-Q21-S2∗∗S2+α21-S2-E2-I2-Q21-S2S2∗∗-δ2+ϑ+1IIP2E2--(δ2+ρ2+ϑ+γ2-φ2S2∗∗)Q2,

ω˙t=-μNhS1∗∗-S12S1S1∗∗-α11-S1-E1-I1-Q1S1∗∗-S12S1S1∗∗-δ1E1-1IIP1E1-CNTS2∗∗-S22S2S2∗∗-α21-S2-E2-I2-Q2S2∗∗-S22S2S2∗∗-δ2+ϑ+1IIP2E2.

Substituting the relations in Eq. (18a), we have ω˙t=-μNhS1∗∗-S12S1S1∗∗-α11-S1-E1-I1-Q1S1∗∗-S12S1S1∗∗-δ1E1-1IIP1E1-CNTS2∗∗-S22S2S2∗∗-α21-S2-E2-I2-Q2S2∗∗-S22S2S2∗∗-δ2+ϑ+1IIP2E2,

ω˙t=-μNhS1∗∗-S12S1S1∗∗+α11-S1-E1-I1-Q1S1∗∗-S12S1S1∗∗+δ1E1+1IIP1E1+CNTS2∗∗-S22S2S2∗∗α21-S2-E2-I2-Q2S2∗∗-S22S2S2∗∗+δ2+ϑ+1IIP2E2≤0.

Hence, the condition (16) show that ω˙t≤0 of all terms. Then the equilibrium steady state B1=S1∗,E1∗,I1∗,Q1∗,R1∗,S2∗,E2∗,I2∗,Q2∗,R2∗ is the globally asymptotically stable in the Ω.

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 of30–36.

Definition 1:

The normalized forward sensitivity index of the variable (R0), depending on the parameter difference, is given as: Eζ∅=∂∅∂ζ×ζ∅. A new expression for R0 is introduced as:R0=α3C1+α2IIP2φ2γ1+δ1+q2T+γ1+δ1μ1+IIP1δ1+ρ1))φ1α1α2α3+g2q3TIIP2δ2+ϑδ11+IIP1δ1g1q2T+q2T+γ1+δ1δ1+ρ1.

Then the sensitivity indices of the basic reproduction number (R0), with respect to the system model depends on the nineteenth parameter are computed as below.χα10=∂R0∂α1α1R0=0,χα20=∂R0∂α2α2R0=0,χC0=∂R0∂CCR0=0,

χμ0=∂R0∂μμR0=IIP1q2T+γ1+δ1μ1+IIP1δ1+ρ1))δ2+ϑ+ρ2φ1φ21+IIP1q2T+γ1+δ1μ1+IIP1δ1+ρ1))(δ2+ϑ+ρ2)φ1φ2,

χδ10=(IIP1-g1q2T+q2T+γ1+δ1δ1+ρ1(δ1+1+IIP1δ1μ1IIP2+δ1+ϑ-g1q2T+q2T+γ1+δ1δ1+ρ11+IIP1∂R0∂δ1δ1R0=q2T+γ1+2δ1+ρ1(δ2+ϑ+ρ2)φ1φ2-1IIP1δ11+IIP1δ11IIP2+δ1+ϑq2T+γ1+2δ1+ρ1(1+IIP1q2T+γ1+δ1μ1+IIP1δ1+ρ1(1+IIP1q2T+γ1+δ1μ1+IIP1δ1+ρ1(δ2+ϑ+ρ2)φ1φ2-1IIP1δ11+IIP1δ1-g1q2T+q2T+γ1+δ1δ1+ρ11+IIP1q2T+γ1+ρ1μ1+IIP1δ1+ρ1δ2+ϑ+ρ2)φ1φ2-δ11IIP2+δ1+ϑ-g1q2T+q2T+γ1+δ1δ1+ρ11+IIP1q2T+γ1+δ1μ1+IIP1δ1+ρ1δ2+ϑ+ρ2)φ1φ2-1IIP11+IIP1δ11IIP2+δ1+ϑ-g1q2T+q2T+γ1+δ1δ1+ρ11+IIP1q2T+γ1+ρ1μ1+IIP1δ1+ρ1-g1q2T+q2T+γ1+δ1δ1+ρ11+IIP1q2T+γ1+ρ1μ1+IIP1δ1+ρ1δ2+ϑ+ρ2)φ1φ2/1+IIP1δ1)1IIP2+δ1+ϑ(-g1q2T+q2T+γ1+δ1δ1+ρ1)2(1+IIP1q2T+γ1+δ1μ1+IIP1δ1+ρ1δ2+ϑ+ρ2)φ1φ2

χδ20=∂R0∂δ2δ2R0=δ2-g2q3T+q3T+γ2+δ2δ2+ϑ+ρ2(-q3T+γ2+δ2)22δ2+ϑ+ρ2+IIP1q2T+γ1+δ1μ1+IIP1δ1+ρ1(γ2+ϑ+δ2)2φ1φ2+g2q3Tq3T+γ2-ϑ-IIP1q2T+γ1+δ1μ1+IIP1δ1+ρ1ϑ+δ2)2-q3T+γ2ρ2+ϑ+ρ2φ1φ2)/(q3T+γ2+δ2(ϑ+δ2(g2q3T-q3T+γ2+δ2(ϑ+γ2+δ2))2(1+IIP1q2T+γ1+δ1μ1+IIP1δ1+ρ1δ2+ϑ+ρ2)φ1φ2

χIIP10=∂R0∂IIP1IIP1R0=(IIP21IIP2+δ1+ϑ-1-2IIP1δ1+IIP12g2T+γ1+δ1μρ1δ2+υ+ρ2φ1φ2((1+IIP1δ1(1+IIP2(δ1+ϑ))(1+IIP1(q2T+γ1+δ1)μ(1+HP1(δ1+ρ1))((δ2+ϑ+ρ2)φ1φ2)),

χIIP20=∂R0∂HP2HP2R0=11+IIP2δ1+ϑ,

χρ10=∂R0∂ρ1ρ1R0=-(IIP2(q2T+γ1+δ1)(1IIP2+δ1+ϑ)ρ1(-g1q2T+(q2T+γ1+δ1)(δ1+ρ1))(1+IIP1(q2T+g1q2T+γ1+δ1)μ(δ2+ϑ+ρ2)φ1φ2)/((1+IIP2(δ1+ρ1))(g1q2T-(q2T+γ1+δ1)(δ1+ρ1))2(1+IIP1(q2T+γ1+δ1)μ(1+IIP1(δ1+ρ1))(δ2+ϑ+ρ2)φ1φ2)),

χρ20=∂R0∂ρ2ρ2R0=-(IIP2(1HP2+δ1+ϑ)ρ2(-g2q3T+(q3T+γ2+δ2)(δ2+υ+ρ2))(q3T+γ2+δ2+g2q3TIIP1(q2T+γ1+δ1)μ(1+IIP1(δ1+ρ1))φ1φ2))/((1+IIP2(δ1+ϑ))(g2q3T-(q3T+γ2+δ2)(δ2+ϑ+ρ2))2(1+IIP1(q2T+γ1+δ1)μ(1+IIP1(δ1+ρ1))(δ2+ϑ+ρ2)φ1φ2)),

χγ10=∂R0∂γ1γ1R0=γ1(q2T+δ1+ρ)(q2T+γ1+δ1)((γ1+δ1)+(q2T+γ1+δ1)(δ1+ρ)),

χφ10=∂R0∂φ1φ1R0IIP1((q2T+γ1+δ1)μ(1+IIP1(δ1+ρ1))φ1φ2))(1+IIP1(q2T+γ1+δ1)μ(1+IIP1(δ1+ρ1))(δ2+ϑ+ρ2)φ1φ2)),

χφ120=∂R0∂φ12φ12R0=0,

χφ210=∂R0∂φ21φ21R0=0,

χg10∂R0∂g1g1R0=-g1q2Tg1q2T-(q2T+γ1+δ1)(δ1+ρ1),

χφ20=∂R0∂φ2φ2R0=IIP1((q2T+γ1+δ1)μ(1+IIP1(δ1+ρ1))(δ2+ϑ+ρ2)φ1φ2(1+IIP1(q2T+γ1+δ1)μ(1+IIP1(δ1+ρ1))(δ2+ϑ+ρ2)φ1φ2,

χg20=∂R0∂g2g2R0=-g2q3Tg2q3T-(q3T+γ3+δ3)(δ2+ρ2+ϑ),

χγ10=∂R0∂γ1γ1R0=-(IIP2γ1(1HP2+δ1+ϑ)(-g1q2T+(q2T+γ1+δ1)(δ1+ρ1))(δ1+ρ1+g1q2TIIP1μ(1+IIP1(δ1+ρ1))(δ2+ϑ+ρ2)φ1φ2))/((1+IIP1(δ1+ρ1))(-g1q2T+(q2T+γ1+δ1)(δ1+ρ1))2(1+IIP1(q2T+γ1+δ1)μ(1+IIP1(δ1+ρ1))(δ2+ϑ+ρ2)φ1φ2)),

χγ20=∂R0∂γ2γ2R0=-g2q3Tγ2(q3T+γ2+δ2)(-g2q3T+(q3T+γ2+δ2)(δ2+ϑ+ρ2)),

∂R0∂q2Tq2TR0=(q2TIIP2(1HP2+δ1+ϑ)(-g1q2T+(q2T+γ1+δ1)(δ1+ρ1))(g1-δ1-ρ1+g1IIP1(γ1+δ1)μ(1+IIP1(δ1+ρ1))(δ2+ϑ+ρ2)φ1φ2))/((1+IIP1(δ1+ρ1))(-g1q2T+q2T+γ1+δ1δ1+ρ1)21+IIP1q2T+γ1+δ1μ1+IIP1δ1+ρ1δ2+ϑ+ρ2φ1φ2),

χq3T0∂R0∂q3Tq3TR0=g2q3T(γ2+δ2)(q3T+γ2+δ2)(-g2q3T+(q3T+γ2+δ2)(δ2+ϑ+ρ2)),

χϑ0=∂R0∂ϑϑR0=(ϑ(-g2q3T+(q3T+γ2+δ2)(δ2+ϑ+ρ2))(-(q3T+γ2+δ2)(2δ2+2IIP2δ1δ2+IIP2δ22+2ϑ+2IIP2δ1ϑ+4IIP2δ2ϑ+3IIP2υ2+ρ2+IIP2δ1ρ2+IIP2δ2ρ2+2IIP2ϑρ2+IIP1(q2T+γ1+δ1)μ(1+IIP2(δ1+δ2+2ϑ))(1+IIP1(γ1+δ1))(δ2+ϑ+ρ2)2φ1φ2)+g2q3T(1+IIP2(δ1+δ2+2ϑ)+IIP1(q2T+γ1+δ1)μ(1+IiP1(γ1+δ1))(ρ2+IiP2((δ2+ϑ)2+(δ1+δ2+2ϑ)ρ2))φ1φ2)))/(IiP2(1IIP2+δ2+ϑ)(g2q3T-(q3T+γ2+δ2)(δ2+ϑ+ρ2))2(1+IIP1(q2T+γ1+δ1)μ(1+IIP1(δ1+ρ1))(δ2+ϑ+ρ2)φ1φ2)).

We can estimate the sensitivity indices (S.I) of the basic reproduction number (R0), 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.Table 3 The sensitivity index (S.I).

Parameters	Sensitivity indices	Parameters	Sensitivity indices	
δ1	Negative	δ2	Negative	
α1	Positive	α2	Positive	
IIP1	Negative	IIP2	Positive	
g1	Positive	g2	Positive	
q2T	Negative	q3T	Positive	
γ1	Negative	γ2	Negative	
ρ1	Negative	ρ2	Negative	
μ	Positive	C	Positive	
φ1	Positive	φ12	Positive	
φ2	Positive	φ21	Positive	
ϑ	Negative			

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 R0 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.α1,α2,IIP2,g1,g2,q3T,μ,C,φ1,φ12,φ2 and φ21 have a positive effect on R0. The determine that the increase in the number of two exposed population E1,E2 and two infectious host population I1,I2 with the value IIP1,g1,g2,IIP2 may lead to an outbreak. On the other hand, the negative sign of the sensitivity indices (S.I) in the R0 i.e. δ1,δ2,IIP1,q2T,ρ1,γ1,ϑ,γ2 and ρ2 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 IIP1 (capita rate of progression of Thais population from the exposed state to the infectious state), IIP2 (capita rate of progression of foreign human from the exposed state to the infectious state), q2T ( the number of infected Thais that leave the quarantine period with the virus intact) and q3T (the number of infected Foreign that leave the quarantine period with the virus intact), g1 (the rate at which the exposed Thais are put into quarantine from the exposed and infected Thais) and g2 (the rate at which the exposed Foreign (tourists) are put into quarantine from the exposed and infected Foreign (tourists)), φ1( transmission rate of virus between population in Thais population), φ2 (transmission rate of virus between population in Foreign (tourists) population), μ (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.

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.

Author contributions

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).

Funding

Research and Development Institute and Faculty of Science and Technology, Phuket Rajabhat University.

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).

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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