Model Analysis of Diphtheria Disease Transmission with Vaccination, Quarantine, and Hand-Washing Behavior

ABSTRACT


2019-2021 period
. However, diphtheria is still a world problem that needs to be resolved.
Diphtheria is an acute respiratory infection caused by Corynebacterium diphtheria with most infections being asymptomatic or having a relatively slight clinical course. It takes two to five days for symptoms and signs to appear in sufferers. However, many sufferers are afflicted by breathing obstruction (World Health Organization, 2017;World Health Organization, 2023). Bacterial transmission can move from droplets or contact with an infected object of an infected person (Centers for Disease Control and Prevention, 2022). The highest risk factors are unvaccinated, visiting or living in an endemic country, living in densely populated and unhealthy areas, weak immunity due to disease or age (infants, children, and the elderly), or being in contact with the exposed or the infected individual. (Dr. Cameron Troup, 2023). Exposed individuals are all individuals who are in close contact for at least 15 minutes at a distance of fewer than two meters during the estimated duration of infection (Tang, et al., 2020). The time for the exposed individuals to transmit the infection to others is up to four weeks (Anderson, 2022).
To eradicate diphtheria disease, vaccination is a strategy. The most effective way to prevent diphtheria disease is a fully three-dose-diphtheria vaccination. (World Health Organization, 2017). Besides that, hand hygiene and quarantine interventions are essential strategies to reduce diphtheria spread. (McGuinness, et al., 2018;Truelove, et al., 2020). Fortunately, mathematical models have great roles in the field of health which can describe the dynamics of some of the factors or variables related to biological phenomena (Cassidy, et al., 2019;Alridha, Al-Jilawi, & Alsharify, 2022;Ratti, 2018). Hence, a mathematical model of the dynamics of the spread of diphtheria is needed as well to make a rational recommendation to public health, policy or decision makers, and program implementers.
Studies about the diphtheria model have been carried out (Chao, et al., 2021;Tosepu, Gunawan, Effendy, Ahmad, & Farzan, 2018;Valek & Tegeder, 2021). For the diphtheria disease transmission model, Sornbundit et al. (2017) offered a diphtheria model by considering asymptomatic infectious, full natural-acquired immunity, and partial natural-acquired immunity of carrier, individuals with full vaccine-induced immunity, and individual with partial vaccine-induced immunity. Izzati & Andriani (2021) discussed the SEIQR model by considering the rate of natural immunity of the exposed individuals in the spread of diphtheria. While other studies about the diphtheria disease model with the quarantine of exposed individuals analyzed the global stability and parameter estimation (Adewale, et al., 2017;Islam, Ahmed, Rahman, Karim, & Amin, 2022). Therefore, this study aims to construct and analyze the diphtheria spread model by considering transmission by contacting an exposed or infected individual, progression for vaccinated exposed individuals, quarantine for both exposed and infected people, and hand-washing behavior.

B. METHODS
The systematic steps used in this study are described as follows: The disease-free equilibrium point can be found by setting free disease in the population, which means, there are no exposed and infected numbers in the population. Then, determine the basic reproduction number of the model to explain the stability behavior (Diekman, Heesterbeek, & Metz, 1990). The basic reproduction number can be determined using the method of the next-generation matrix (Driessche & Watmough, 2002).

Determine Endemic Equilibrium Point
Considering the model system with the number of infected people as not null then solving the system of equations. 6. Stability Analysis First, determine Jacobian Matrix for the system. Then, determine Jacobian Matrix for disease-free equilibrium and endemic equilibrium points. After that, find the characteristic polynomials for both Jacobian matrices. Then, the stability is determined by identifying the eigenvalues of characteristic polynomials. The basic reproduction number is considered in the process.

Numerical Simulations
Numerical simulations were carried out on the model using maple18 software to show the dynamics of the spread of diphtheria with vaccination, quarantine, and handwashing behavior. The parameter values used in the simulation come from a journal article related to the diphtheria model Izzati & Andriani (2021) and from Indonesian population and health data (Countrymeter, 2023;Kementerian Kesehatan RI, 2017). Besides that, we make assumptions on several other parameters and variations in parameter values of vaccination, quarantine, and hand-washing behavior to show the effect of each variable on the system in simulation.

C. RESULT AND DISCUSSION 1. Mathematical Model
A new model of the spread of diphtheria was constructed which refers to the model SEIQR by considering transmission by contact with exposed individuals, vaccinated exposed individuals, quarantine for both exposed and infected individuals, and hand-washing behavior. Let the total population ( ) be divided into five compartments i.e. susceptible, exposed, infected, quarantine, and recovered. While the assumptions are explained as follows: a. The total human population is not constant. b. Individuals who are newly born/enter the population are susceptible to diphtheria. c. Susceptible individuals can become exposed through contact with exposed or infected persons. d. Vaccination of vulnerable individuals can make them recover. e. Vaccination of exposed individuals makes acceleration being infected. f. Quarantine is carried out for exposed and infected individuals. g. Infected individuals are being treated through the quarantine process. h. Hand-washing behavior reduces the risk of transmission from exposed and infected individuals. i. Diphtheria disease can cause death in the sufferer. j. Individuals who have recovered will not be re-infected because of their immune systems.
Based on the assumptions built, the diphtheria transmission model is given as follows: All the used parameters Λ, , , ℎ, , , 1 , , , 1 , 2 , are non-negative constants. Λ is the recruitment rate by natural birth and immigration, is the infections transmission rate (as the number of people a single case can infect in a day), is the reduction of infection transmission by the exposed individuals, ℎ is hand-washing behavior rate, is the natural mortality rate, is per capita rate at which susceptible individuals are vaccinated, 1 is per capita rate at which exposed individuals are vaccinated, is the rate of progression of exposed becoming infected, is per capita rate at which quarantine individuals are recovered, is diphtheria diseaseinduced death rate, 1 and 2 are the rates of quarantine for exposed and infected populations, respectively. Next, we have verified that system (1) is bounded, by following Lemma 1.

Epidemic Dynamic Analysis
The equilibrium point is reached when = = = = 0. In epidemic conditions, we derived the disease-free equilibrium point of system (2) by assumption = = 0. Based on model assumptions of (1), we get the number of quarantine people also null since exposed and infected people don't exist. Therefore, the disease-free equilibrium point is given by 0 ( 0 , 0 , 0 , 0 ) = ( Λ + , 0, 0,0).
To find the basic reproduction number, we determine the next-generation matrix of the system, as follows: After that, the disease-free equilibrium point, 0 , is substituted into and . Matrix The dominant eigenvalue of −1 is the basic reproduction number which is The biological explanation of the basic reproduction number is given as follows. The expression Λ/( + ) is the number of susceptible in the disease-free equilibrium. The expression ( + 1 ) is the progression of an exposed person to become infected. The expressions ( 1 + + 1 + ) and ( + + 2 ) are population reduction rates of exposed and infected people, respectively. The expressions (1 − ℎ) and (1 − ℎ), are the infectious rate of susceptible contact with exposed and infected people, respectively. The expressions (1 − ℎ)/( 1 + + 1 + ) and (1 − ℎ)( + 1 )/( + + 2 )( 1 + + 1 + ) are the number of new cases per unit of time caused by contact with an exposed and infected people, respectively. Theorem 1. The disease-free equilibrium of the system (2), 0 is locally asymptotically stable if 0 < 1, and unstable when 0 ≥ 1.
Secondly, based on equation (5), 0 > 1 causes 2 < 0, so that 3 4 < 0. The product of two eigenvalues can be negative if and only if the two eigenvalues differ in sign. Consequently, there is a positive eigenvalue. So, disease-free equilibrium 0 is unstable.
Further, if 0 = 1 then 1 > 0 and 2 = 0. This causes the eigenvalue equation (5) to be 2 + 1 = 0. Then, we obtained one eigenvalue that is zero and another one that is negative , for example, if 3 = 0 , then 4 will be negative. Hence, there is a non-negative eigenvalue so that disease-free equilibrium 0 is also unstable. In conclusion, disease-free equilibrium point 0 is locally asymptotically stable when 0 < 1, and unstable when 0 ≥ 1. ∎

Numerical Simulations
We do numerical simulations using Maple 18 to check the stability of the equilibrium points and to show the effect of vaccination, quarantine, and hand-washing behavior on diphtheria spread. In this simulations we vary the values of parameters , 1 1 , 2 , ℎ. The following shows the parameter values used in the simulation, as shown in Table 1. It is assumed that the initial condition of the variable (0) = 100; E(0)= 0.01; I(0)=0.235; Q(0)=0.001 (in thousand people). While, various parameters value of , 1 1 , 2 , ℎ and the basic reproduction numbers in simulations are shown in Table 2 and Figure 1.  Figure 1. (a) The effect of vaccination for susceptible to infected number, (b) The effect of vaccination for exposed to infected number, (c) The effect of quarantine for exposed to infected number, (d) The effect of quarantine for infected to infected number, and (e) The effect of hand-washing behavior to infected number The effect of vaccination for susceptible, vaccination for exposed, quarantine for exposed, quarantine for infected, also hand-washing behavior to the infected numbers can be observed in Figure 1 as well the value of parameters used in scenarios 1-3, 4-6, 7-9, 10-12, and 13-15 are shown in Table 2. We can find that, the greater the value of the lower the value of 0 and also the lower the value of the infected number. Similarly, the greater the value of 1 , 2 , and ℎ the lower the value of 0 and also the lower the value of the infected number. On the other hand, the greater the value of 1 the greater the value of 0 , and also the greater the value of the infected number.
When the basic reproduction number is less than one, the disease-free equilibrium is locally asymptomatically stable and the endemic equilibrium doesn't exist. When the basic reproduction number is greater than one, the endemic equilibrium is locally asymptomatically stable. It means that diphtheria disease can disappear when the parameter is suppressed until the basic reproduction number is less than one. The numerical simulations show that there is a significant effect of vaccination, quarantine, and hand-washing behavior for infected numbers, respectively. Vaccination, quarantine, and hand-washing behavior could significantly reduce the basic reproductive and the infected numbers. Whereas vaccination for exposed people could increase the basic reproductive and the infected numbers. The smaller the basic reproduction number, the faster the diphtheria disease will disappear from the population.

D. CONCLUSION AND SUGGESTIONS
The mathematical model of diphtheria transmission by considering vaccination, quarantine, and hand-washing behavior is already constructed. The analysis result proof of the epidemic and endemic stability. When the basic reproduction number is less than one, the disease-free equilibrium is locally asymptomatically stable and the endemic equilibrium doesn't exist. When the basic reproduction number is greater than one, the endemic equilibrium is locally asymptomatically stable. It means that diphtheria disease can disappear when the parameter is suppressed until the basic reproduction number is less than one. Besides that, the numerical simulations show that vaccination for susceptible, quarantine, and handwashing behavior could significantly reduce the basic reproductive and the infected numbers. Whereas vaccination for exposed people could increase the basic reproductive and the infected numbers. Hence, the smaller the basic reproduction number, the faster the diphtheria disease will disappear from the population. Thus, policies on vaccination, quarantine, and dissemination of hand-washing behavior, as well as monitoring of the exposed people are highly recommended to be carried out to reduce the spread of diphtheria in the population.