THE INSURANCE PLAN DEPENDS ON THE COMPARTMENT MODEL TO THE RISK FACTORS FOR THE COST OF HEALTH CARE FOR INFECTIOUS DISEASES

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INTRODUCTION
Various infectious disease outbreaks have critically impacted the health sector and financial losses to society. For decades, scientists have worked hard to understand the transmission characteristics of the disease so as to design control strategies to prevent the further spread of infection. The field of science that studies these epidemic diseases and, in particular, the factors that influence the incidence, distribution, and control of infectious diseases in human populations is called epidemiology. The dynamics associated with the spread of infectious diseases, for example, predicting the number of cases in a given time period and the probability that the disease will attack can be understood through mathematical models [1]. Epidemiological mathematicians and statisticians have made many contributions to the mathematical modeling of infectious diseases. This is because mathematical modeling has proven useful in analyzing, predicting, evaluating pandemic risk, detecting, and implementing efficient control programs [2]. In some works of literature, epidemic models have been able to express disease dynamics in different population structures, such as the compartment model in which a population is divided into several compartments by providing relatively simple assumptions on the rate of flow between different classes of population members through a system of differential equations [3,37]. The compartment model was first proposed by Kermack and McKendrick in 1927, namely the susceptible compartment model for infection and recovery (SIR). After how many decades, the SIR model was popularized by Anderson et al. in 1979. Then the compartment model was expanded in [4], discussing a time-delayed non-autonomous SIRS epidemic model. The SIR model and the SIS and SEIR compartmental models can be found in the classic monographs of [5] and [6]. Pongsumpun [7] described the transmission of respiratory diseases with the SIRI model, in which the population consisted of three compartments, assuming that recovered patients could become re-infected.
Simon and Lefèvre [8] account for randomness in epidemiological dynamics with the formulation of the Markov chain through the expansion of the SIR deterministic compartment model.
The compartment model has its own advantages, among others, it does not require special knowledge for analysis and numerical methods for its calculations [9]. The infectious disease epidemiological compartment model has been widely used to evaluate public health policies, disease interventions, and is an element of public health decision-making to examine key risks 3 COST OF HEALTH CARE FOR INFECTIOUS DISEASES from epidemics [10,38]. Apart from that, the infectious disease compartment model has also started to become part of decision-making in actuarial science. The insurance model is a process to develop a form of contract between two parties, namely insuring and the insured, in which the insurer exchanges a fixed premium to pay the insured fixed or variable benefits on the basis of financial losses due to the occurrence of a specified event. People facing financial loss due to some particular event want to establish a market that can contribute premiums to insurance funds to obtain compensation in the future. Some of the questions faced by the actuarial profession relate to the possibility of an epidemic occurring. This question, in addition to the number of individuals who will be infected as well as how many will need treatment at any given time. So the actuarial profession is very reasonable to add economic considerations based on compartmental models to design insurance coverage in order to provide financial means to protect the public from the adverse economic impact of the epidemic. For example, for the SIR compartment model, different identifications can be made from an insurance point of view, with the three compartments playing significantly different roles in the insurance model. Members in the susceptible compartment face the risk of being infected by a market that can contribute premiums to insurance funds in exchange for coverage for direct medical costs incurred if infected. During an epidemic outbreak, infected policyholders will benefit from claim payments provided by the insurance fund. The idea of developing an insurance model is to determine how much we need to allocate today (today) to cover future financial losses.
In the infectious disease model, policyholders are committed to paying premiums continuously as long as they are still vulnerable and medical expenses are continuously reimbursed for each infected policyholder. During the treatment period, some literature on compartmental models in various actuarial applications, Feng and Garrido [11]  in accordance with its objectives, and can be parameterized by the available data. To insurance that the susceptible populations are exposed to infectious diseases, obtain the present value of the premiums paid by vulnerable groups and the present value of the annuity and benefit payments received.
With this in mind, we review the development of the application of the infectious disease compartment model to actuarial concepts in the financial setting of the costs of medical care for infectious disease patients. We systematically reviewed the literature review by tracing articles published from 2015 to 2021 using Dimensions, Science Direct and Google Scholar, with a focus on compartmental models of infectious disease dynamics. From the three databases, 45 articles were found in the initial search after removing duplicates. From 45 articles, and reviewed in full text, finally resulted in 14 studies discussing the compartment model. In this review we highlight various communicable disease compartment models and perform a quantitative analysis of 14 articles modeling insurance coverage e.g. annuities, benefits, and reserve levels to compensate for the impact of an infectious disease epidemic.

MATERIALS AND METHODS
The material used to conduct this systematic literature review consists of two different materials. The first material is conceptual material in the form of theories, mathematical models, and other supporting concepts. While the second material is publication media that presents the conceptual study, such as conference papers, books, and scientific journals.

Compartment Model
The compartment structure, which was first introduced by Kermack and McKendrick [15] and is the basis of the dynamics model of infectious diseases, was later developed by several biomathematicians. The formulation of dynamic models for the transmission of infectious disease epidemics is divided into different compartments. A picture of the movement of the relationship between one compartment and another is called a compartment model [16]. The process of individual flow from one compartment to another is referred to as a compartment system. Pangestu et al. [17] the most common epidemic model is the SIR model where the population consists of three compartments, namely Vulnerable-Infection-Recovered, assuming that people who are susceptible to infectious diseases, can be infected through exposure to infectious people, and after 5 COST OF HEALTH CARE FOR INFECTIOUS DISEASES a certain period of time recover. or die. Recovery is immunity to infection. This simplest version of the SIR model assumes homogeneous mixing and a fixed population size. Mahardika et al. [18] SEI1I2 HRD deterministic model, where the population size N consists of seven classes, namely susceptible (S), exposed but not infectious (E), infection class (I).

Insurance Plan
Insurance product planning goes through several important processes. There are several important coverages related to the planning process of an insurance product, namely determining the premium level, modeling premium reserves, and determining the number of claims protected based on these premiums. Planning insurance products, especially health insurance, is always related to the risk of disease that has the potential to make the insured sick. In addition, the health care financing factor of the sick insured also needs to be considered for determining the claims that will be covered in the insurance.

Infectious Diseases
Infectious diseases are diseases caused by bacteria, viruses, or parasites that can be transmitted through certain media. Infectious diseases are often also called infectious diseases because these diseases are suffered through viral, bacterial, or parasitic infections that are transmitted through various media such as air, syringes, blood transfusions, places to eat or drink, and so on. Infectious diseases are the result of a combination of various factors that influence each other. Noncommunicable diseases are diseases that are not caused by germs but are due to physiological or metabolic problems in human body tissues. There are several types of infectious diseases, such as covid-19, dengue fever, malaria, seasonal flu, ARI, SARS, etc.

Methods
We identify studies that report on the compartmental model applied in planning an insurance framework that delivers health care benefits. To identify articles, the initial search searched the Dimensions, Science direct, and Google Scholar databases on October 16, 2021, namely scientific articles related to epidemiological models and actuarial applications published from 2015 to 2021.
The review used a search strategy that would be tailored to each database by selecting the titles and abstracts consisting of the terms in Table 1 to mine the literature discussing the use of infectious disease models to determine insurance coverage. The keywords in Table 1 were used to perform a study search on several databases. The search strategy based on the database used is divided into 3 types, namely: Dimensions, Science Direct, and Google Scholar.

Literature Result from Dimensions
Search using Dimensions with keywords: Search number (9) contained in Table 1  Of the 47 articles that can be accessed, as many as 40.

Literature Result from Science Direct
Searching using Science Direct with the same keywords as Dimensions by accessing "title and abstract" and the search limits from 2015 to 2021-2021 obtained 20,385 results by clicking "type article", and 13,841 articles were obtained. Then continued the search by adding the keyword "AND Insurance" and obtained 11 articles.

Literature Result from Google Scholar
The search stages using Google Scholar are in the title and abstract with the keywords [9] in

PRISMA Selection
Articles that have been collected from searches in three databases are continued with article selection using the PRISMA flow. A detailed review is carried out on the article if the article meets the following criteria: 1. The article discusses the compartment model.
2. Articles that discuss the risk of health care costs.

Articles are written in English.
Based on the criteria for articles that are considered eligible for detailed review, only 14. The literature search process is presented in Figure 1.  Table 2.

RESULTS OF LITERATURE REVIEW
The search stages generated in Figure 1 found 14 articles eligible for systematic review. We are interested in identifying Types of compartmental models and types of communicable diseases.
Then, the insurance coverage was explored. Then, we identify many articles that use a particular compartment model. However, we do not critically evaluate each compartment model because the compartment model has many trade-offs and does not include a discussion of each article.

Literature Review Analysis based on Subtopics
The literature review analysis was carried out based on 3 subtopics of discussion, namely the type of compartment model, the type of infectious disease, and the insurance coverage being explored. Furthermore, the three subtopics were then analyzed further to see the distribution of the research. The results of the analysis can be used as an illustration of new opportunities for further research.

Compartmental Model Type
Literature analysis based on this subtopic was carried out by classifying articles using the  or as many as nine articles [19,21,22,23,24,25,28,31,32], while those discussing the SIS compartment model, SIVW, SEIR, SIRD, and SEIT only have one article each. Therefore, the most common compartment model is the SIR. Research other than the SIR model can be said to be still rare.

Compartment Model
In addition, when viewed based on the classification of compartment models, all models studied in the literature are homogeneous models. Therefore, there are new research opportunities in the exploration of non-homogeneous models. An example of the development of a nonhomogeneous model is S1S2EIR. An illustration of the development of homogeneous to nonhomogeneous models can be seen in Figure 3.  Based on Figure 4, using research case studies on infectious diseases, and Covid-19 became the topic, namely 4 articles [25,27,128,29]. In addition, studies that discuss cases of dengue fever, malaria, and seasonal flu each consist of 3 articles, 2 articles, and 1 article. Research other than cases of the disease can be said to be still rare. One of the cases that have not been discussed is ARI. ARI is a disease that attacks the lungs and is associated with shortness of breath. This ARI can be an interesting topic of discussion because it is known that many areas can be affected by this disease due to air pollution.

Exploration of Insurance Plan
Literature analysis based on insurance plan subtopics was conducted to classify articles based on the object of exploration of the insurance coverage under study. The insurance coverage used in these studies is the level of premiums, insurance products, premium reserves, medical expense claims, and insurance portfolios. Based on the results of the analysis, further classification of literature based on insurance coverage can be seen in Figure 5. Focuses on formulating epidemiological models for insuring susceptibility to infectious diseases and provides techniques for computing the present value of premiums paid by vulnerable groups, the present value of annuities and lump sum benefit payments received by infective deceased, as well as benefit reserves for insurance companies. A stochastic model containing standard Brownian motion was studied by Caraballo and Colucci [33].
Caraball and Keraani [34] explore the features of a stochastic SIR model with fractional Brownian motion. Lefèvre et al. [21] developed a recursive method to calculate cost components and premium rates according to the stochastic SIR insurance model and provided results on integral functions using the martingale argument.
Chernov et al. [27] proposed two different exponential models for infection rates before and after lockdown. The deterministic or stochastic SEIR compartment model with migration and vaccination fluxes is also a tool to measure the benefits gained through various reaction strategies and/or prevention, risk reduction of infection.
Lefèvre et al. [21] resolve financial risk with probability in the stochastic SIR insurance model.
Lefèvre and Picard [22] generalize the SIR model to a controlled epidemic model, in which infections will be isolated by health organizations to alleviate disease severity and study the representation of epidemic outcomes and path integrals in terms of pseudo polynomials. Lefèvre and Simon [22] consider cross-infection between two populations to be related. This general approach to studying the Laplace transform of integral functions was developed by Lefèvre and Picard [22]. Lefèvre dan Simon [21] mengusulkan proses Markov terstruktur blok umum untuk pemodelan epidemi. Perera [19] considers control strategies in a simple SIR model and premium variations with respect to model parameters. Nkeki and Ekhaguere [13]  The results showed that the most frequently occurring words were "model", followed by "insurance", "spread", and "risk". The visualization of the topic keyword mapping is shown in Figure 6. Furthermore, topic mapping analysis can be carried out based on keywords that are used as the main node. This topic mapping is done by selecting "insurance" as the main node of the mapping network. The keyword "insurance" was chosen to explain the relevance of insurance coverage to other topics. Visualization of mapping the topic "insurance" as the main node, can be seen in Figure 7. Based on Figure 7, we get a form of topic mapping based on the keyword "insurance" as the main node, connected to several other keywords. care cost factor can be used as an indicator to determine the size of the claim, as well as the premium reserve to cover the claim. Therefore, the topic of insurance research that uses the health care cost factor has not been carried out.

Visualization of "Disease Spread" Topic Mapping as Main Node
Furthermore, the topic mapping analysis is carried out based on other keywords that are used as the main node. This topic mapping is done by selecting "disease spread" as the main node of the mapping network. The keyword "disease spread" was chosen to explain the relevance of the topic of disease transmission to other topics. The visualization of mapping the "disease spread" topic as the main node can be seen in Figure 8.  Figure 8, we get a topic mapping form based on the keyword "disease spread" as the main node, connected to several other keywords. The keywords that are directly related are insurance, model, infection rate, risk, benefit, impact, seasonality, risk, parameter model, and spread. Based on this, it can be said that the opportunity for the topic of novelty related to disease transmission is to relate the topic to the risk factors of health care financing. In addition, it can also be linked to other insurance coverage such as premium reserves or claims based on the risk of transmission of the disease.

Visualization of "Insurance Coverage" Topic Mapping as Main Node
Furthermore, the topic mapping analysis is carried out based on other keywords that are used as the main node. This topic mapping is done by selecting "insurance coverage" as the main node of the mapping network. The keyword "insurance coverage" was chosen to explain the relevance of the topic of the scope of insurance covered to other topics. Visualization of mapping the topic "insurance coverage" as the main node, can be seen in Figure 9. Based on Figure 8, a topic mapping form is obtained based on the keyword "insurance coverage" as the main node, connected to several other keywords. The keywords that are directly related are infection rate, risk, spread, model, cost, case, impact, insurance, effect, removal, and insurance. Based on this, it can be said that the opportunity for renewable topics related to insurance coverage is to relate the topic to the risk of infectious diseases and the model of the spread of disease to create a model for the insurance product. The risk modeling factors for the spread of infectious diseases can be used as indicators for determining the size of claims, as well as premium reserves to cover these risk claims.

DISCUSSION
This study highlights the interaction between actuarial science and epidemiology and provides a literature review of the most recent results. The combination of modeling techniques from the two fields could revolutionize how we measure and model epidemic insurance. We hope to present a new source of inspiration for actuaries working in the dynamically evolving risk modeling and population assessment fields. We envision actuaries adding these epidemiological models and techniques to their growing risk modeling and analysis toolset.
From the results of this review, the compartmental model used in calculating the overall insurance coverage of the SIR model means that the population is homogeneous based on susceptibility, whereas, in reality, the level of susceptibility varies according to factors such as age or susceptibility to infection.

CONCLUSIONS
This study has reviewed the literature on compartmental models, infectious diseases, and their relation to insurance coverage. Furthermore, we have presented the results of a review of the compartment model that has been carried out by other researchers, namely SIS, SIVW, SEIR, SIRD, and SEIT. Based on the compartment model, no one has discussed a model with a nonhomogeneous form, so the development of a non-homogeneous compartment model is an opportunity for new research. Non-homogeneous compartmental models, such as S1S2EIR, linked to coverage and insurance modeling, have not yet been implemented. In addition, regarding the types of infectious diseases, we found several cases that were discussed, namely covid-19, dengue fever, malaria, and seasonal flu, so that other cases could become research novelties. We provide advice regarding the discussion of ARI disease as the object of disease risk, which is a factor in the formation of the insurance model. In addition, insurance coverage that has been carried out in the related literature mostly discusses the level of premiums, while premium reserves are still rare.
So that the combination of the discussion of premium reserves based on the non-homogeneous compartment model S1S2EIR which represents the spread of ARI disease, can be said as the latest research opportunity because no one has done it yet.

ACKNOWLEDGMENTS
We would like to thank to Universitas Padjadjaran.