CO-INFECTION MODEL FOR COVID-19 AND RUBELLA WITH VACCINATION TREATMENT: STABILITY AND THRESHOLD

This study aimed to explore a co-infection transmission model between Covid-19 and Rubella that involves administering vaccinations for both diseases. These two diseases not only have the same characteristics, but also has the same pattern in terms of the causes of disease, spread, clinical manifestations, and vaccines as prevention efforts. This model provided answers to the question of whether one of these diseases or both will disappear from the human population through several steps in the mathematical modelling analysis, which consists of: 1) critical point analysis, 2) stability analysis, 3) next generation matrix, and 4) threshold value analysis. This research resulted in four critical points, which is a critical point of disease-free, Rubella critical point, Covid-19 critical point, and the critical point for both diseases. Based on the next generation matrix and the disease-free critical point, two basic reproduction numbers were generated, namely RO1 for Rubella and RO2 for Covid-19. The first condition, R01 less than 1 and R02 less than 1, the disease-free critical point will be stable such that Rubella and Covid-19 will disappear from the human population. The second condition, R01 greater than 1 and R02 less than 1, the disease-free critical point become unstable, which means that Rubella-infected people will be found in the population. The third condition, R01 less than 1 and R02 greater than 1, Covid-19 will be found in the human population. 2 ARTIONO, WINTARTI, PRAWOTO, ASTUTI


INTRODUCTION
Since the end of 2019, there has been a sporadic spread of the corona virus not only in tropical countries but also in sub-tropical countries. Data obtained from the official WHO report until the end of March 2021 shows no less than 3. provinces in Indonesia. [1][2].
Since the first case was discovered in the city of Depok in early March 2020, the number of Covid-19 patients in Indonesia has been increasing rapidly. This has resulted in the government having to issue policies related to handling corona cases. The government issued circulars to study from home, work from home, and worship from home [3][4].
This case has also resulted in an increase in the function of the hospital, which is prioritized for handling Covid-19 patients. Likewise, medical personnel and medical equipment used to help treat Covid-19 patients. As a result, infectious diseases such as measles, rubella, and diphtheria are no longer the main concern of health workers. Unfortunately, at the official WHO report, Rubella is one of the diseases that occupies the highest position in Indonesia with more than 500 patients per year [5]. Unlike other infectious diseases, rubella will cause more serious problems when it attacks pregnant women. Babies born to mothers with rubella will die or experience CRS (Congenital Rubella Syndrome). This syndrome will cause the baby to have congenital defects such as impaired vision, impaired growth to impaired heart function [6]. The growth of babies with this condition is a concern for the government, therefore the Government of Indonesia has also issued a policy related to handling this disease, namely the provision of vaccines at school age. Based on an analysis of the Ministry of Health's data, it shows that during the Covid-19 pandemic there was a decline in control and surveillance performance of Diseases Preventable by Immunization (PD3I). Compared to 2019 data, the coverage from January to April 2020 shows a decline from 0.5% to 87% [7]. This research will construct a mathematical model of co-infection between Rubella disease and Covid-19. Through the mathematical model, the dynamics of the existence for these two diseases can be predicted through analytical techniques in mathematics. The advantages of mathematical models can show the possibilities that occur to the dynamics of the two diseases in a population such as when there are no two diseases, conditions when there is only rubella disease in a population, conditions when there is only Covid-19 disease, or conditions when two diseases are present. Rubella and Covid-19 can coexist in a population. These two diseases not only have the same characteristics but also have the same pattern in terms of causes, spread, clinical manifestations, and disease administration as a preventive measure.
Specifically, Covid-19 is well-known as a disease caused by the corona virus which was first discovered in the Huanan seafood wholesale market located in Wuhan City, the capital of Hubei Province, Central China. The virus that causes this disease is known as Severe Acute Respiratory Syndrome Coronavirus-2 (SARS-CoV-2). SARS-CoV-2 will attack the human respiratory system to cause severe lung infections [8][9]. According to Roethe (2020) it was reported that Covid-19 transmission can occur from individuals who have been infected with the virus or individuals who are not yet symptomatic or known as people without Symptoms [10].
Clinical manifestations of a person infected with Covid-19 will appear on the 5th to 7th day with several symptoms including fever (temperature > 38°C), coughing, sneezing, and shortness of breath. The spread of this disease occurs through droplets that come out when people infected with Covid-19 cough and sneeze [11][12][13].
Meanwhile, Rubella disease, also known as German Measles, is a disease that can be transmitted very quickly but is not deadly. A person with this disease will recover on its own, but this disease will require more attention when infecting a pregnant woman. The disease is caused by the Rubella virus from the Togaviridae virus family and the Rubivirus genus. It has several clinical manifestations that appear after a person is infected with the rubella virus, including fever, red rash on the skin, and enlarged lymph nodes behind the ear. Symptoms of a red rash on the skin only occur in 1-5% of patients [14][15][16]. From the results of mathematical analysis of the developed model for both disease, this study will answer the question of whether one of these diseases or both will disappear from the human population after being given the vaccine.

PRELIMINARIES
SIR mathematical modelling was used as a baseline in this research. At first, all the human population have been looked as healthy people but susceptible to disease. Furthermore, there is someone infected with Rubella and someone infected with Covid-19. These infected people will then infect other humans and form a sub-population of humans infected with Rubella and a subpopulation of humans infected with Covid-19. With the handling of Covid-19 patients who are not really good (not self-isolated or not receiving treatment at the hospital) and Rubella patients who are no longer the main concern of treatment in hospitals, this results in the occurrence of coinfection disease between Covid-19 and Rubella. Regard to the characteristics of the two diseases that allow patients of both diseases to recover, this results in the formation of seven human subpopulation compartments, namely the susceptible human sub-population ( ), the Rubella-infected human sub-population ( ), the Covid-19-infected human sub-population ( ), the infected human sub-population with Rubella and Covid-19 ( ), sub-population of humans who recovered from Covid-19 ( ), sub-population of humans who recovered from Rubella ( ) and sub-population of humans who recovered from both diseases ( ). By taking some assumptions that can provide limitations in making mathematical models, the pattern of the spread of the two diseases can be described in a compartment diagram as shown in Figure 1.
Moreover, some of the assumptions used to construct the mathematical model were mentioned as  The rate of change for each compartment at time is represented by the following system of nonlinear ordinary differential equations: With the initial value for each sub-population is shown as follows

a. Equilibrium Point and its existence
Based on the mathematical model that had been constructed, then we determined the solution of the system of nonlinear ordinary differential equations (1) -(7). Determination of this solution had been done by taking the left-hand side for each equation equal to zero so that four critical points were obtained as follows.

b. Stability Analysis
Moreover, we analyzed the stability system through the eigenvalues of Jacobian matrix from each equilibrium point. Based on the system of nonlinear ordinary differential equations (1) -(7) and the disease-free equilibrium point, we derived the Jacobian matrix as follows.
Regard to the eigenvalues of the Jacobian matrix, it could be concluded that the disease-free equilibrium point will be asymptotically stable if ( + 2 )( 1 + 2 + ) < 1 and Λ 1 (1− 1 ) ( + 1 )( 1 + 2 + ) < 1. It means that when the conditions have been met then both disease will be disappeared from the human population forever.

c. Two Basic Reproduction Number
Furthermore, basic reproduction numbers have been obtained through next generation matrix which is defined as = − −1 . It consists of matrix as a Jacobian matrix with non-linear elements from the sub-population that can transmit the disease ( , , and ) and as a Jacobian matrix with linear elements from the sub-population that can transmit the disease ( , , and ). Based on the non-linear system of ordinary equation (1) -(7), we derived these equations for .
While, these equation for are as follows.
So that it is obtained gone from the population. The condition 1 > 1 and 2 < 1 make the disease-free equilibrium point unstable, which means that people infected by Rubella will be found in the population. While, the condition 1 < 1 and 2 > 1 also make the disease-free equilibrium point unstable so then people infected by Covid-19 will also be found in the population.