SVIRD EPIDEMIC MODEL WITH DISCRETE-TIME HYBRID MARKOV/SEMI-MARKOV ASSUMPTIONS

,


INTRODUCTION
The spread of infectious diseases can be modeled using deterministic and stochastic models according to Andersson and Britton [1]. The main purpose of communicable disease modeling is to analyze the spread, as well as provide solutions to the government about what actions can be taken to control the spread of the disease. As happened at the end of 2019, there was an outbreak of the coronavirus that spread throughout the world to cause a pandemic. The number of people infected has resulted in inadequate hospital capacity and excessive demand for healthcare workers. Due to the fact that the number of infected individuals is tied to the description and prediction of a group of susceptible individuals in a specific region, the spread of which is unexpected at the individual level, statistical patterns may be formed that lead to the usage of stochastic models.
One assumption of the Markov model is that the length of time a person spends in state i before moving to state j depends only on state i. Additionally, the sojourn time distribution exhibits a memoryless property, implying that it does not account for the length of probable occupancy in a specific condition. In the Markov model, the distribution of sojourn time is Geometrically distributed. However, in reality, such assumptions might result in implausible constraints. An alternative model that frames this concern is the semi-Markov model, which can be viewed as an extension of the Markov model. Discrete-time semi-Markov models have gained traction in recent years due to their usefulness in a number of different contexts, such as survival and reliability analysis by Barbu and Limnios [2], disability insurance by Stenberg, Manca, and Silvestrov [3], and credit risk by Vassiliou and Vasileiou [4] and Amico, Janssen, and Manca [5].
Most mathematical models for reliability assume that time is continuous. But in reality, the system has a discrete life span. Such as systems that work on demand, that work on a cycle basis, or that are only monitored at certain times (such as once a month or every day).
In other words, all of this life is intrinsically discrete. The same is true of the spread of COVID-19, where the number of infected people is reported on a daily basis which can be seen on Worldometer (https://www.worldometers.info/coronavirus/). Previously conducted research on COVID-19 includes the determination of reproductive numbers using transition intensity by Zuhairoh, Rosadi, and Effendie [6] and the prediction of COVID-19 distribution with the Richards curve model by Zuhairoh and Rosadi [7]. Bracquemond and Gaudoin [8] provides a good overview of the discrete probability distributions utilized in reliability theory. Additionally, insights from discrete-time semi-Markov models contribute to the application of time continuous semi-Markov models by Wu, Zheng, and Chen [9].
In recent years, discrete-time Markov models have become more widely used. As in the research Emmert and Allen [10] which investigated the spread of disease in a structured population using a discrete-time model. In comparison to semi-Markov processes, continuous-time and associated inference problems are of greater interest to many individuals. In this research, a discrete-time semi-Markov model will be employed. The research paper Limnios [11] provides an introduction to the discrete-time update procedure.
There are various advantages and disadvantages to using Markov and semi-Markov models, the main one being that Markov models are simpler and more visible. This simplifies the interpretation and understanding of conventional Markov models when used to simulate the spread of infectious diseases, for example. Simultaneously, the semi-Markov model allows quantification of duration under certain circumstances, due to a wider distribution of transit times. This provides a justification for developing a hybrid model that combines the two techniques as Verbeken and Guerry [12] has done in the case of manpower planning. The strengths and weaknesses of Markov and semi-Markov models for modeling the spread of infectious diseases are discussed in Section 1. Section 2 introduces a hybrid Markov/semi-Markov model, a combination of Markov and semi-Markov models with Geometric, negative Binomial, and discrete Weibull sojourn time distributions. One of the advantages of a hybrid Markov/semi-Markov model is that it captures a fixed effect duration, which is useful when trying to estimate as few parameters as possible. Consequently, the hybrid Markov/semi-Markov model facilitates the advancement of the semi-Markov model even when there is a lack of data. In section 2, we also describe a predictive model for the spread of infectious diseases. In Section 3, we apply the SVIRD epidemic model to COVID-19 data and then test whether the model meets the Markov assumptions. If not, the transition probability will be calculated using a statistical test assuming a hybrid Markov/semi-Markov. Furthermore, in Section 4, predicting COVID-19 cases in the short term with the prediction equation of the SVIRD epidemic model.
Finally, Section 5 contains the conclusions from the discussion presented previously.

HYBRID MARKOV/SEMI-MARKOV MODEL
A hybrid Markov/semi-Markov model is presented for each pair of (i, j) whether the transition from state i to state j can be considered a Markov or semi-Markov transition. The first step in building a hybrid Markov/semi-Markov model is to test the multi-state model to determine whether it meets the Markov assumptions. The Markov process introduced by a mathematician named Andrei A. Markov in 1906 is a stochastic process from the previous time that has no influence on the future time if the present time is known. Suppose X t is a stochastic process that has a discrete-state space S = {1, 2, . . . , m}. In general, according to Haberman and Pitacco [13] for any sequence of time points t 1 < t 2 < · · · < t n−1 < t n which corresponds to the state set i 1 , i 2 , . . . , i n−1 , i n , then the probability conditional must fulfill.
Pr(X t n = i n |X t 1 = i 1 , . . . , X t n−1 = i n−1 ) = Pr(X t n = i n |X t n−1 = i n−1 ) If i n−1 = i, i n = j and i n+1 = k then the first-order or 1-step transition probability can be written as follows.
A second-order or higher-order Markov chain is a Markov chain that depends on two or more of the previous values. According to Shamshad et. al. [14], the probability of a second-order transition or a 2-step transition can be written as follows.
Markov property tests need to be performed to verify whether the transition probability is Pr{X t+1 = k|X t = j, X t−1 = i} from the current state to the next does not depend on the previous state. Written in the notation p i jk which is the probability to state k with the previous condition that there has been a transition from state i to j (i ≤ j ≤ k). If the Markov property applies, then p i jk = p jk . Three tests can be used to test the properties of Markov, namely: (1) Test based on a contingency table, (2) Tests to verify whether a chain of a given order,

(3) Test to verify if the transition probability is constant over time.
A second test is used in this paper because it uses theoretical assumptions. According to Anderson and Goodman [15], the test to verify whether a chain is of second-order or not is defined as follows: H 0 : p 1 jk = p 2 jk = · · · = p m jk = p jk , j, k = 1, 2, . . . , m.
The chi-square test statistic for the null hypothesis (H 0 ) is Consider now the joint hypothesis that p i jk = p jk where i, j, k = 1, 2, . . . , m. By computing the sum, a test of this joint hypothesis can be obtained.
The standard limiting distribution has m(m − 1) 2 degrees of freedom.
Due to its simplicity, the homogeneous time Markov model is used in various domains and applications. In addition, the estimates made have relatively few parameters and are not very demanding of data availability. However, Markov chains cannot consider the duration of stay; it is also less flexible because there is a memoryless property which means sojourn time is Geometric distribution. This issue is solved by the semi-Markov model.
If it does not meet the Markov assumptions, then proceed with second step, which is to estimate the sojourn time distribution for each transition between states. The second step is known as the semi-Markov hypothesis test. Similar to the Markov process, a semi-Markov process can transition from one state to another. To be more precise, the length of time spent in each state before moving to the next is a random variable that is independent of the next state of the new process.
A stochastic process X t := J N(t) is called a semi-Markov process if it considers the sojourn time in state i before transitioning to state j, where this sojourn time is a random variable with a cumulative distribution function F i j (t).
The main difference for the Markov model is that the sojourn time distribution f can be any discrete probability distribution, combining the possible lengths of the sojourn. The Markov model with the P = (p i j : i, j ∈ S ) transition matrix can be called a semi-Markov model with a Geometric distribution of sojourn times where Then the semi-Markov hypothesis is tested at the level of the sojourn time distribution h i j .
The transition from state i to j satisfies the Markov assumption if the sojourn time is geometrically distributed. Under the geometric hypothesis, the equation should be seen as evidence to the contrary, that is, evidence in favor of a more general distribution of sojourn times. The statistical test used according to Stenberg, Manca, and Silvestrov [3] is as follows. The probability mass function of each distribution used is written in Nakagawa and Yoda [16]. Here is the relationship between the 3 distributions selected as the distribution of sojourn time of each transition that occurs between states.
(2) Negative Binomial distribution (X ∼ nbin(x; p, n)) If the value of n = 1 then it will be a Geometric distribution.
(3) Geometric distribution (X ∼ geo(x; p)) Proof. It is known that the SVIRD epidemic model consists of five states, namely susceptible, vaccinated, infected, recovered, and deceased. So it can be written as follows.
The infected state will increase if there is a transition from a susceptible or vaccinated state.
Meanwhile, it will decrease if there is a transition to a recovered or deceased state. Both statements can be written as Meanwhile, the value of S(t + 1) is obtained from If we know the number of individuals in each state at time t, then n i j (t) is a binomial random variable with parameters n i (t) and p i j . So that the following expected value is obtained.
Likewise, the number of people who leave the infected state due to recovery or death is denoted by n i,k+1 (t) with the following expected value.
Meanwhile, the number of newly infected people who come from vulnerable states is denoted n 0 j (t +1) with an expected value of S(t +1)s j . So the following prediction equation is obtained.
So that the SVIRD epidemic model is obtained The same proof holds for n I v (t + 1), until obtained

SVIRD EPIDEMIC MODEL
The SVIRD epidemic model has six states, Susceptible, Vaccinated, and Infected, divided into two states: infected without vaccination and infected after being vaccinated, Recovered, and Deceased. This model has random variables S(t),V (t), I s (t), I v (t), R(t), and D(t), whose meanings are explained in Table 1. An illustrative representation of the mathematical model can be seen in Figure 1. It can also be seen in the transitions that occur between states. Here, it is assumed that vaccination effectively prevents an individual from dying, meaning that death comes from individuals who are not vaccinated only. In addition, this model assumes that there is immunity in individuals infected with the disease, so there is no transition out of the recovered state. (1) Vaccination rateη of the problem in this study have been previously stated that we assume that people who have received at least 2 doses of vaccination will not die from COVID-19.
The first step is to test whether the SVIRD epidemic model meets Markov properties or not using the equation (4), with the following hypothesis.
where χ 2 tab = χ 2 (0.05;150) = 179.58. Because the value of χ 2 i jk > χ 2 tab then H 0 is rejected. So it is obtained that the SVIRD epidemic model does not meet the Markov characteristics.
The second step is testing the semi-Markov hypothesis at the h i sojourn time distribution using COVID-19 data, using the equation (10) and applying the S i j in the equation (9) for each SVIRD epidemic model transition. The results are summarized in Table 2.   March 2, 2022. The prediction results for the next 10 days can be seen in the Table 5. and  The sojourn time distribution used in this paper is three, namely Geometric, negative Binomial, and discrete Weibull distribution. So that in the application, it will be tested that each transition meets the sojourn time distribution, thus producing the appropriate transition probability. In the end, a prediction is used using the Theorem 1 for positive confirmed cases on support. Furthermore, thanks to all authors whose papers are cited in this paper as a valuable reference and resource for this paper.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.