CAUCHY –Jurnal Matematika Murni dan Aplikasi Volume 6(3) (2020), Pages 122-132 p-ISSN: 2086-0382; e-ISSN: 2477-3344 Submitted: July 08, 2020 Reviewed: October 02, 2020 Accepted: November 05, 2020 DOI: http://dx.doi.org/10.18860/ca.v6i3.9891 Local Dynamics of an SVIR Epidemic Model with Logistic Growth Joko Harianto1, Inda Puspita Sari2 1Department of Mathematics, Cenderawasih University, 2Faculty of Medicine, Cenderawasih University, Email: joharijpr88@gmail.com, indasarira@yahoo.co.id ABSTRACT Discussion of local stability analysis of SVIR models in this article included in the scope of applied mathematics. The purpose of this discussion was to provide results of local stability analysis that had not to discuss in some articles related to the SVIR model. The SVIR models discussed in this article involve logistics growth in the vaccinated compartment. The results obtained, i.e., if the basic reproduction number β„œ0 < 1 and π‘š > 0, then there is one equilibrium point i.e. 𝐸 0 is locally asymptotically stable. In the field of epidemiology, this means that the disease will disappear from the population. However, if β„œ0 > 1, 𝛽1 > 𝛽 and π‘š > 0, then there are two equilibrium points, i.e., disease-free equilibrium point denoted by 𝐸0 and the endemic equilibrium point denoted by 𝐸1 βˆ—. In this case, the endemic equilibrium point 𝐸1 βˆ— Locally asymptotically stable. In the field of epidemiology, this means that the disease will remain in the population. The numerical simulation supports these results. Keywords: Local Dynamics; SVIR Model; Logistic Growth INTRODUCTION The mathematical modeling of infectious disease spread is one of the areas of research for mathematicians and epidemiologists. The formulated model is often said to be a model epidemic. Epidemic models sometimes involve complexity and non-linearity. Therefore, not all epidemic models can found solutions with analytical methods. Epidemic models usually formulate with a deterministic or stochastic approach. Kermack and Mckendrick first introduced a deterministic model in the article [1], which was later generalized by Capasso and Serio in the paper [2]. In the article [2], The population is divided into three parts, i.e., susceptible, infected, and recovered. Each population contains some people whose health conditions correspond to the name of the population. The model is known as the SIR model. The SIR model represents the dynamic epidemic of a person infected to a susceptible person through direct contact with the closed population. The SIR model formulate as an initial value condition of the ordinary differential equation system, and it is analyzed mathematically. This SIR Model is then continuously developed by many researchers for various cases occurring today. Based on the SIR model developed by Capasso and Serio, the spread of infectious diseases can formulate by dividing the population into some compartment that depends on health conditions. Therefore, the vaccine given to a person can also be considered a specific situation to add to its compartment. People who have given vaccines are included in the V population [3]. The V population is then added to the SI model to study the dynamics of the spread of tuberculosis disease. The SIV model later generalizes into the SVIR model in the article [4], which discusses the possibility that http://dx.doi.org/10.18860/ca.v6i3.9891 mailto:joharijpr88@gmail.com mailto:indasarira@yahoo.co.id Local Dynamics of an SVIR Epidemic Model with Logistic Growth Joko Harianto 123 individuals can recover from the disease. Furthermore, the SVIR model develops to study the transmission dynamics of influenza disease discussed in articles [5] and [6]. The vaccination strategy discussed in some of the above articles is a continuously given vaccination. Some cases found that vaccines sometimes administer for a certain period or seasonal. Vaccination strategies for this state are usually called Pulse Vaccination Strategy (PVS). A comparison between the continuous vaccination model and the PVS discuss in the article [7]. Later, the model of the continuous vaccination strategy mentioned by Liu et al. was developed by Khan et al., Islam, Harianto, and Suparwati. In the article [8], the non-linear saturation rate adds to the SVIR model with a continuous vaccination strategy. The addition of population migration factors in the susceptible population of the SVIR model with a continuous vaccination strategy discuss in the article [9]. In the paper [10], the SVIR model with a continuous vaccination strategy was developed by adding a death factor due to disease infection. In other words, the infectious disease is a deadly disease. The discussion also provided numerical simulation to support the analysis results. The death rate of each population in the SVIR model is assumed to be identical. A year later, Harianto and Suparwati continued discussing the SVIR model by adding different death rate assumptions for each population in the SVIR model. Its numerical analysis and simulation discuss in the article [11]. In the paper [1] until [11], The population growth assumes to increase exponentially. The models in these articles are more realistic if their population growth uses logistic growth. In other words, the population assumes to be growing in logistics. The logistics growth factor on the SIR model discusses in the articles [12],[13],[14],[15], and [16]. The paper in [17] and [18] also discusses other compartment models that use logistical growth. Li et al. discuss the paper [12] as a SIR model with a saturation incidence rate. According to the article, these discussions will be modified SVIR models in [7] by involving the logistic growth in the population studied in [12]. Furthermore, it discussed the existence of the equilibrium point and local stability. The last part makes a simulated model that has analysis. METHODS This study uses the literature review method. The references used are in the form of several reputable international scientific books and journals. The theories used to support the results of this study are the equilibrium point stability theory of ordinary differential equation systems described in a book written by Perko [19] and the theory related to the construction of the basic reproduction number of an epidemiological model in article [20]. The flow of thought in this discussion, first adds assumptions to the model that has been studied, then forms the model. Next, determine the equilibrium point of the system and analyze its stability. Finally, a numerical simulation is given to check the correctness of the analysis results obtained and a sensitivity analysis of the basic reproduction number. RESULTS AND DISCUSSION Model The model discussed in this section is a modification of the SVIR model with a continuous vaccination strategy adopted from the article [7]. The modification of the model is the growth of its population using logistics growth and the death rate caused by disease involved. This model divides the human population into four epidemiological classes, i.e., susceptible (S), infected (I), vaccinated (V), and recovered (R). The dynamic spread of the disease for the case formulate as the following ordinary differential equation system. Local Dynamics of an SVIR Epidemic Model with Logistic Growth Joko Harianto 124 𝑑𝑆 𝑑𝑑 = π‘Ÿπ‘† (1 βˆ’ 𝑆 𝐾 ) βˆ’ 𝛽𝑆𝐼 βˆ’ (πœ‡ + 𝛼)𝑆 𝑑𝑉 𝑑𝑑 = 𝛼𝑆 βˆ’ 𝛽1𝑉𝐼 βˆ’ (πœ‡ + 𝛾1)𝑉 (1) 𝑑𝐼 𝑑𝑑 = 𝛽𝑆𝐼 + 𝛽1𝑉𝐼 βˆ’ (𝛾 + πœ‡ + πœ”)𝐼 𝑑𝑅 𝑑𝑑 = 𝛾1𝑉 + 𝛾𝐼 βˆ’ πœ‡π‘… with the initial conditions 𝑆(0) = 𝑆0 > 0, 𝑉(0) = 𝑉0 > 0, 𝐼(0) = 𝐼0 > 0 dan 𝑅(0) = 𝑅0 β‰₯ 0. π‘Ÿ denotes the intrinsic growth rate of susceptible population, 𝐾 is the carrying capacity of the area. 𝛽 is the contact rate between susceptibles and infected, πœ‡ is the natural death rate, 𝛼 is the rate at which an individual leaves the class of the susceptible by becoming the vaccination process. 𝛽1 is the contact rate between vaccinated and infected, 𝛾1 is the recovery rate from the vaccinated class, 𝛾 is the natural recovery rate from the infected class. πœ” is the rate of death caused by disease. Theorem 1 All solutions of the System (1) with initial conditions (𝑆0, 𝑉0, 𝐼0, 𝑅0) ∈ ℝ+ 4 are positive and bounded for all 𝑑 β‰₯ 0. Proof: The System (1) can be written 𝑑�̃� 𝑑𝑑 = 𝑓(οΏ½ΜƒοΏ½) with οΏ½ΜƒοΏ½ = (𝑆, 𝑉, 𝐼, 𝑅) ∈ ℝ4 and 𝑓(οΏ½ΜƒοΏ½) = (𝑓1(οΏ½ΜƒοΏ½), 𝑓2(οΏ½ΜƒοΏ½), 𝑓3(οΏ½ΜƒοΏ½), 𝑓4(οΏ½ΜƒοΏ½)) ∈ ℝ 4. Clearly 𝑓(οΏ½ΜƒοΏ½) differentiable continuous for all οΏ½ΜƒοΏ½ ∈ ℝ4 such that the System (1) has a unique solution. If we let 𝐺(𝑑) = 𝛽1𝐼(𝑑) + πœ‡ + 𝛾1, then the second equation of the System (1) can be written 𝑑𝑉(𝑑) 𝑑𝑑 + 𝐺(𝑑)𝑉(𝑑) = 𝛼𝑆(𝑑) (2) multiplying both sides of the Equation (2) by exp (∫ 𝐺(𝑒)𝑑𝑒 𝑑 0 ) gives 𝑑𝑉(𝑑) 𝑑𝑑 exp (∫ 𝐺(𝑒)𝑑𝑒 𝑑 0 ) + 𝐺(𝑑)𝑉(𝑑) exp (∫ 𝐺(𝑒)𝑑𝑒 𝑑 0 ) = 𝛼𝑆(𝑑) exp (∫ 𝐺(𝑒)𝑑𝑒 𝑑 0 ) By using multiplication rules on derivatives we have 𝑑 𝑑𝑑 [𝑉(𝑑) exp (∫ 𝐺(𝑒)𝑑𝑒 𝑑 0 )] = 𝛼𝑆(𝑑) exp (∫ 𝐺(𝑒)𝑑𝑒 𝑑 0 ) (3) By integrating two sides of the Equation (3) with 𝑑 gives 𝑉(𝑑) = 𝑉0 exp (βˆ’ ∫ 𝐺(𝑒)𝑑𝑒 𝑑 0 ) + [∫ 𝛼𝑆(𝜏) exp (∫ 𝐺(𝑒)𝑑𝑒 𝜏 0 ) π‘‘πœ 𝑑 0 ] exp (βˆ’ ∫ 𝐺(𝑒)𝑑𝑒 𝑑 0 ) > 0. Similarly, we can prove that 𝑆(𝑑), 𝐼(𝑑) dan 𝑅(𝑑) are nonnegative for all 𝑑 > 0. Consequently, from the first equation of the System (1), we have 𝑑𝑆 𝑑𝑑 = π‘Ÿπ‘† (1 βˆ’ 𝑆 𝐾 ) βˆ’ 𝛽𝑆𝐼 βˆ’ (πœ‡ + 𝛼)𝑆 < π‘Ÿπ‘† (1 βˆ’ 𝑆 𝐾 ) Hence, there exists 𝑀 = max{𝑆0, 𝐾} such that lim π‘‘β†’βˆž sup 𝑆(𝑑) < 𝑀 Let 𝑁(𝑑) = 𝑆(𝑑) + 𝑉(𝑑) + 𝐼(𝑑) + 𝑅(𝑑), total population change is 𝑑𝑁 𝑑𝑑 = 𝑑𝑆 𝑑𝑑 + 𝑑𝑉 𝑑𝑑 + 𝑑𝐼 𝑑𝑑 + 𝑑𝑅 𝑑𝑑 Local Dynamics of an SVIR Epidemic Model with Logistic Growth Joko Harianto 125 = π‘Ÿπ‘† (1 βˆ’ 𝑆 𝐾 ) βˆ’ πœ‡π‘ βˆ’ πœ”πΌ < π‘Ÿπ‘† βˆ’ πœ‡π‘ < π‘Ÿπ‘€ βˆ’ πœ‡π‘ Then we have 0 < 𝑁(𝑑) < π‘Ÿπ‘€ πœ‡ for 𝑑 β†’ ∞. Therefore, all solutions of the System (1) are positive and bounded all 𝑑 β‰₯ 0. Equilibria and Basic Reproduction Number We see that 𝑅 does not appear in three equations in the System (1). Hence, the fourth equation on the System (1) can be ignored, and the discussion is focused on three equations as follows: 𝑑𝑆 𝑑𝑑 = π‘Ÿπ‘† (1 βˆ’ 𝑆 𝐾 ) βˆ’ 𝛽𝑆𝐼 βˆ’ (πœ‡ + 𝛼)𝑆 𝑑𝑉 𝑑𝑑 = 𝛼𝑆 βˆ’ 𝛽1𝑉𝐼 βˆ’ (πœ‡ + 𝛾1)𝑉 (4) 𝑑𝐼 𝑑𝑑 = 𝛽𝑆𝐼 + 𝛽1𝑉𝐼 βˆ’ (𝛾 + πœ‡ + πœ”)𝐼 The equilibrium point of the System (4) obtained by resolving the following equations: π‘Ÿπ‘† (1 βˆ’ 𝑆 𝐾 ) βˆ’ 𝛽𝑆𝐼 βˆ’ (πœ‡ + 𝛼)𝑆 = 0 (5) 𝛼𝑆 βˆ’ 𝛽1𝑉𝐼 βˆ’ (πœ‡ + 𝛾1)𝑉 = 0 (6) 𝛽𝑆𝐼 + 𝛽1𝑉𝐼 βˆ’ (𝛾 + πœ‡ + πœ”)𝐼 = 0 (7) Clearly from the Equation (3), (4), and (5) that if π‘š = π‘Ÿ βˆ’ (πœ‡ + 𝛼) > 0, then System (4) has a unique disease-free equilibrium point. The disease-free equilibrium point of the System (4) denoted by 𝐸0 = (𝑆0, 𝑉0, 𝐼0) with 𝑆0 = πΎπ‘š π‘Ÿ , 𝑉0 = πΎπ›Όπ‘š π‘Ÿ(πœ‡+𝛾1) , 𝐼0 = 0. The basic reproduction number in epidemiology used to indicate the number of diseases and control spread. The basic reproduction number denoted by β„œ0. If β„œ0 < 1 then the spread of disease in the population predicted to disappear. However, if β„œ0 > 1 then the virus will remain in the population. The basic reproduction number of the System (4) in this discussion is determined based on the theory learned in the article [21]. Let 𝑋 = (𝑆, 𝑉), 𝑋0 = (𝑆0, 𝑉0), 𝑍 = (𝐼), β„Ž(𝑋, 𝑍) = 𝛽𝑆𝐼 + 𝛽1𝑉𝐼 βˆ’ 𝑛𝐼 with 𝑛 = 𝛾 + πœ‡ + πœ” > 0 and 𝑓(𝑋, 𝑍) = [ π‘Ÿπ‘† (1 βˆ’ 𝑆 𝐾 ) βˆ’ 𝛽𝑆𝐼 βˆ’ (πœ‡ + 𝛼)𝑆 𝛼𝑆 βˆ’ 𝛽1𝑉𝐼 βˆ’ (πœ‡ + 𝛾1)𝑉 ] Derivating β„Ž(𝑋0, 𝑍) by 𝑍 gives π·π‘§β„Ž(𝑋 0, 𝑍)|𝑧=0 = 𝛽𝑆 0 + 𝛽1𝑉 0 βˆ’ 𝑛 We have 𝐴 = π·π‘§β„Ž(𝑋 0, 𝑍)|𝑧=0 = 𝛽𝑆 0 + 𝛽1𝑉 0 βˆ’ 𝑛 because 𝐴 = 𝑀 βˆ’ 𝐷 then gives 𝑀 = 𝛽𝑆0 + 𝛽1𝑉 0 and 𝐷 = 𝑛. Hence, we have β„œ0 = 𝜌(𝑀𝐷 βˆ’1) = 𝛽𝑆0 + 𝛽1𝑉 0 𝑛 = π›½πΎπ‘š(πœ‡ + 𝛾1) + 𝛽1πΎπ›Όπ‘š π‘Ÿπ‘›(πœ‡ + 𝛾1) with π‘š = π‘Ÿ βˆ’ (πœ‡ + 𝛼) and 𝑛 = 𝛾 + πœ‡ + πœ”. The following discussed the existence of the endemic equilibrium point denoted by πΈβˆ— = (π‘†βˆ—, π‘‰βˆ—, πΌβˆ—). Assume that π‘†βˆ—, πΌβˆ—, π‘‰βˆ— β‰  0. From Equation (5), we have π‘Ÿ (1 βˆ’ 𝑆 𝐾 ) βˆ’ 𝛽𝐼 βˆ’ (πœ‡ + 𝛼) = 0 then Local Dynamics of an SVIR Epidemic Model with Logistic Growth Joko Harianto 126 π‘†βˆ— = 𝐾 π‘Ÿ (π‘š βˆ’ π›½πΌβˆ—) From Equation (6), we have π‘‰βˆ— = π›Όπ‘†βˆ— 𝛽1𝐼 βˆ— + πœ‡ + 𝛾1 = 𝐾𝛼(π‘š βˆ’ π›½πΌβˆ—) π‘Ÿ(𝛽1𝐼 βˆ— + πœ‡ + 𝛾1) From Equation (7), we have π›½π‘†βˆ— + 𝛽1𝑉 βˆ— βˆ’ 𝑛 = 0 (8) with 𝑛 = 𝛾 + πœ‡ + πœ”. If π‘†βˆ— = 𝐾 π‘Ÿ (π‘š βˆ’ π›½πΌβˆ—) and π‘‰βˆ— = 𝐾𝛼(π‘šβˆ’π›½πΌβˆ—) π‘Ÿ(𝛽1𝐼 βˆ—+πœ‡+𝛾1) substituted to Equation (8), then 𝐴1𝐼 βˆ—2 + 𝐴2𝐼 + 𝐴3 = 0 (9) with 𝐴1 = 𝐾 π‘Ÿ 𝛽2𝛽1 𝐴2 = 𝐾 π‘Ÿ 𝛽2(πœ‡ + 𝛾1) + 𝐾 π‘Ÿ 𝛽𝛽1𝛼 + 𝑛𝛽1 βˆ’ 𝐾 π‘Ÿ 𝛽𝛽1π‘š 𝐴3 = 𝑛(πœ‡ + 𝛾1) βˆ’ 𝐾 π‘Ÿ π›½π‘š(πœ‡ + 𝛾1) βˆ’ 𝐾 π‘Ÿ 𝛽1π‘šπ›Ό = 𝑛(πœ‡ + 𝛾1)(1 βˆ’ β„œ0) Hence, πΌβˆ— is the root of the Equation (9), i.e.: 𝐼1,2 βˆ— = βˆ’π΄2 Β± √𝐴2 2 βˆ’ 4𝐴1𝑛(πœ‡ + 𝛾1)(1 βˆ’ β„œ0) 2𝐴1 Clearly that if β„œ0 < 1 then π‘Ÿπ‘› > πΎπ›½π‘š, consequently 𝐴2 > 0. Thus, the existence of the endemic equilibrium point depends on β„œ0. The existence of disease-free and the endemic equilibrium points associated with basic reproductive numbers summarized in the following theorems. Theorem 2 We define β„œ0 = π›½πΎπ‘š(πœ‡ + 𝛾1) + 𝛽1πΎπ›Όπ‘š π‘Ÿπ‘›(πœ‡ + 𝛾1) 𝑔(πΌβˆ—) = 𝐴1𝐼 βˆ—2 + 𝐴2𝐼 + 𝑛(πœ‡ + 𝛾1)(1 βˆ’ β„œ0) 𝐴1 = 𝐾 π‘Ÿ 𝛽2𝛽1 𝐴2 = 𝐾 π‘Ÿ 𝛽2(πœ‡ + 𝛾1) + 𝐾 π‘Ÿ 𝛽𝛽1𝛼 + 𝑛𝛽1 βˆ’ 𝐾 π‘Ÿ 𝛽𝛽1π‘š with π‘š = π‘Ÿ βˆ’ (πœ‡ + 𝛼) and 𝑛 = 𝛾 + πœ‡ + πœ” > 0. i. If β„œ0 < 1 and π‘š > 0 then 𝐴2 > 0 such that there exists a unique equilibrium point of the System (4), i.e. the disease-free equilibrium point denoted by 𝐸0 = ( πΎπ‘š π‘Ÿ , πΎπ›Όπ‘š π‘Ÿ(πœ‡+𝛾1) , 0). ii. If β„œ0 > 1 and π‘š > 0, then System (4) has only two equilibrium point, i.e. the disease-free equilibrium point denoted by 𝐸0 = ( πΎπ‘š π‘Ÿ , πΎπ›Όπ‘š π‘Ÿ(πœ‡+𝛾1) , 0) and the endemic equilibrium point denoted by 𝐸1 βˆ— = (𝑆1 βˆ—, 𝑉1 βˆ—, 𝐼1 βˆ—) with 𝑆1 βˆ— = 𝐾 π‘Ÿ (π‘š βˆ’ π›½πΌβˆ—), 𝑉1 βˆ— = 𝐾𝛼(π‘šβˆ’π›½πΌβˆ—) π‘Ÿ(𝛽1𝐼 βˆ—+πœ‡+𝛾1) and 𝐼1 βˆ— is the positive root of 𝑔(πΌβˆ—). iii. if β„œ0 = 1, π‘š > 0 and 𝐴2 < 0, then System (4) has only two equilibrium point, i.e. the disease-free equilibrium point denoted 𝐸0 = ( πΎπ‘š π‘Ÿ , πΎπ›Όπ‘š π‘Ÿ(πœ‡+𝛾1) , 0) and the endemic Local Dynamics of an SVIR Epidemic Model with Logistic Growth Joko Harianto 127 equilibrium point denoted 𝐸2 βˆ— = (𝑆2 βˆ—, 𝑉2 βˆ—, 𝐼2 βˆ—) with 𝑆2 βˆ— = 𝐾 π‘Ÿ (π‘š βˆ’ π›½πΌβˆ—), 𝑉2 βˆ— = 𝐾𝛼(π‘šβˆ’π›½πΌβˆ—) π‘Ÿ(𝛽1𝐼 βˆ—+πœ‡+𝛾1) and 𝐼2 βˆ— = βˆ’ 𝐴2 𝐴1 . Local Stability Analysis This section discusses the local stability analysis of disease-free equilibrium and endemic points of the System (4). The local stability of the equilibrium point of the System (4) is determined using the Jacobian matrix (Linearization method) studied in [19]. The Jacobian matrix of Systems (4) written as follows: 𝐽 = [ βˆ’2π‘Ÿπ‘† 𝐾 βˆ’ 𝛽𝐼 + π‘š 0 βˆ’π›½π‘† 𝛼 βˆ’π›½1𝐼 βˆ’ (πœ‡ + 𝛾1) βˆ’π›½1𝑉 𝛽𝐼 𝛽1𝐼 𝛽𝑆 + 𝛽1𝑉 βˆ’ 𝑛 ] with π‘š = π‘Ÿ βˆ’ (πœ‡ + 𝛼) and 𝑛 = 𝛾 + πœ‡ + πœ” > 0. Local stability analysis of the disease-free equilibrium point of the System (4) presented in the following theorems. Theorem 3 If β„œ0 < 1 and π‘š > 0, then the disease-free equilibrium point of the System (4), i.e. 𝐸 0 = ( πΎπ‘š π‘Ÿ , πΎπ›Όπ‘š π‘Ÿ(πœ‡+𝛾1) , 0) is locally asymptotically stable. Proof. The Jacobian matrix of the System (4) in 𝐸0, i.e. 𝐽(𝐸0) = [ βˆ’π‘š 0 βˆ’π›½π‘†0 𝛼 βˆ’(πœ‡ + 𝛾1) βˆ’π›½1𝑉 0 0 0 𝑛(β„œ0 βˆ’ 1) ] Hence, the characteristics equation of 𝐽(𝐸0), i.e. (πœ† βˆ’ 𝑛(β„œ0 βˆ’ 1))(πœ† + π‘š)(πœ† + πœ‡ + 𝛾1) = 0 Then we have eigenvalue πœ†1 = 𝑛(β„œ0 βˆ’ 1), πœ†2 = βˆ’π‘š dan πœ†3 = βˆ’πœ‡ βˆ’ 𝛾1 < 0. Therefore, if β„œ0 < 1 and π‘š > 0, then all the eigenvalues of 𝐽(𝐸 0) are negative consequently according to [19], 𝐸0 is locally asymptotically stable. Local stability analysis of the endemic equilibrium point of the System (4) presented in the following theorems. Theorem 4 If β„œ0 > 1, 𝛽1 > 𝛽 and π‘š > 0, then the endemic equilibrium point of the System (4) denoted by 𝐸1 βˆ— = (𝑆1 βˆ—, 𝑉1 βˆ—, 𝐼1 βˆ—) is locally asymptotically stable with 𝑆1 βˆ— = 𝐾 π‘Ÿ (π‘š βˆ’ π›½πΌβˆ—), 𝑉1 βˆ— = 𝐾𝛼(π‘šβˆ’π›½πΌβˆ—) π‘Ÿ(𝛽1𝐼 βˆ—+πœ‡+𝛾1) and 𝐼1 βˆ— is the root of 𝑔(πΌβˆ—) = 𝐴1𝐼 βˆ—2 + 𝐴2𝐼 + 𝑛(πœ‡ + 𝛾1)(1 βˆ’ β„œ0), 𝐴1 = 𝐾 π‘Ÿ 𝛽2𝛽1 , 𝐴2 = 𝐾 π‘Ÿ 𝛽2(πœ‡ + 𝛾1) + 𝐾 π‘Ÿ 𝛽𝛽1𝛼 + 𝑛𝛽1 βˆ’ 𝐾 π‘Ÿ 𝛽𝛽1π‘š. Proof: The Jacobian matrix of the System (4) in 𝐸1 βˆ—, i.e. 𝐽(𝐸1 βˆ—) = [ βˆ’ 2π‘Ÿ 𝐾 𝑆1 βˆ— βˆ’ 𝛽𝐼1 βˆ— + π‘š 0 βˆ’π›½π‘†1 βˆ— 𝛼 βˆ’π›½1𝐼1 βˆ— βˆ’ (πœ‡ + 𝛾1) βˆ’π›½1𝑉1 βˆ— 𝛽𝐼1 βˆ— 𝛽1𝐼1 βˆ— 𝛽𝑆1 βˆ— + 𝛽1𝑉1 βˆ— βˆ’ 𝑛 ] because 𝑆1 βˆ— β‰  0 and according to Equation (5) we have π‘Ÿ βˆ’ π‘Ÿ 𝐾 𝑆1 βˆ— βˆ’ 𝛽𝐼1 βˆ— βˆ’ (πœ‡ + 𝛼) = 0 β‡’ π‘š βˆ’ 2π‘Ÿ 𝐾 𝑆1 βˆ— βˆ’ 𝛽𝐼1 βˆ— = βˆ’ π‘Ÿ 𝐾 𝑆1 βˆ— Local Dynamics of an SVIR Epidemic Model with Logistic Growth Joko Harianto 128 From Equation (6) we have 𝛽1𝐼1 βˆ— + πœ‡ + 𝛾1 = 𝛼𝑆1 βˆ— 𝑉1 βˆ— And clearly 𝛽𝑆1 βˆ— + 𝛽1𝑉1 βˆ— βˆ’ 𝑛 = 0. Hence, 𝐽(𝐸1 βˆ—) equivalent with 𝐽(𝐸1 βˆ—) = [ βˆ’ π‘Ÿ 𝐾 𝑆1 βˆ— 0 βˆ’π›½π‘†1 βˆ— 𝛼 βˆ’ 𝛼𝑆1 βˆ— 𝑉1 βˆ— βˆ’π›½1𝑉1 βˆ— 𝛽𝐼1 βˆ— 𝛽1𝐼1 βˆ— 0 ] Thus the characteristics equation of 𝐽(𝐸1 βˆ—), i.e. π‘Ž1πœ† 2 + π‘Ž2πœ† + π‘Ž3 = 0 with π‘Ž1 = 𝛼𝑆1 βˆ— 𝑉1 βˆ— + π‘Ÿ 𝐾 𝑆1 βˆ— π‘Ž2 = π‘Ÿπ›Όπ‘†1 βˆ—2 𝐾𝑉1 βˆ— + 𝛽 2𝑆1 βˆ—πΌ1 βˆ— + 𝛽1 2𝑉1 βˆ—πΌ1 βˆ— π‘Ž3 = 𝛽𝛽1𝛼𝑆1 βˆ—πΌ1 βˆ— + 𝛽2𝛼 𝑆1 βˆ—2 𝑉1 βˆ— 𝐼1 βˆ— + π‘Ÿ 𝐾 𝛽1 2𝑆1 βˆ—π‘‰1 βˆ—πΌ1 βˆ— π‘Ž1π‘Ž2 βˆ’ π‘Ž3 = π‘Ÿπ›Ό2𝑆1 βˆ—3 𝐾𝑉1 βˆ—2 + π‘Ÿ2𝛼𝑆1 βˆ—3 𝐾2𝑉1 βˆ— + π‘Ÿπ›½2 𝐾 𝑆1 βˆ—2𝐼1 βˆ— + 𝛽1𝛼𝑆1 βˆ—πΌ1 βˆ—(𝛽1 βˆ’ 𝛽) Clearly π‘Ž1 > 0, π‘Ž2 > 0, π‘Ž3 > 0 and if 𝛽1 > 𝛽 then π‘Ž1π‘Ž2 βˆ’ π‘Ž3 > 0. Therefore, based on the Routh-Hurtwiz criteria if 𝛽1 > 𝛽, then all the eigenvalues of 𝐽(𝐸1 βˆ—) are negative consequently the endemic equilibrium point of the System (4) denoted by 𝐸1 βˆ— Locally asymptotically stable. Numerical Simulations The following given numerical simulations to confirm the results obtained analytic related to local stability of disease-free equilibrium points and endemic to the System (1). Figure 1 is the local dynamics of the solution on the system (1) with the parameter value 𝐾 = 1, π‘Ÿ = 0,04 , πœ‡ = 0,01 , 𝛼 = 0,01 , 𝛽 = 0,002, 𝛽1 = 0,00045 , 𝛾 = 0,0001 , πœ” = 0,02 , 𝛾1 = 0,004 and initial conditions 𝑆0 = 0,8 , 𝑉0 = 0,1, 𝐼0 = 0,1 , 𝑅0 = 0. From the value of the parameter then we can calculate that β„œ0 = 0,04 < 1, π‘š = 0,02 > 0 and 𝐸 0 = (𝑆0, 𝑉0, 𝐼0, 𝑅0) = (0,5 ; 0,36; 0 ; 0,14). According to theorem 3, the disease-free equilibrium point denoted by 𝐸0 is locally asymptotically stable. These results correspond to the numerical simulation given. It means that the disease will disappear from the population. Figure 2 is the local dynamics of the solution on the system (1) with the parameter value 𝐾 = 1, π‘Ÿ = 0,04 , πœ‡ = 0,01 , 𝛼 = 0,01 , 𝛽 = 0,1, 𝛽1 = 0,00045 , 𝛾 = 0,0001 , πœ” = 0,02 , 𝛾1 = 0,004 and initial conditions 𝑆0 = 0,8 , 𝑉0 = 0,1, 𝐼0 = 0,1 , 𝑅0 = 0. Clearly that 𝛽1 > 𝛽 then we can calculate that β„œ0 = 1,66 > 1, π‘š = 0,02 > 0 and the endemic equilibrium point denoted by 𝐸1 βˆ— = (𝑆1 βˆ—, 𝑉1 βˆ—, 𝐼1 βˆ—, 𝑅1 βˆ—), i.e 𝐸1 βˆ— = (0,3 ; 0,21; 0,08 ; 0,09). According to theorem 3, the endemic equilibrium point denoted by 𝐸1 βˆ— Locally asymptotically stable. These results correspond to the numerical simulation given. It means that the disease will remain in the population. Local Dynamics of an SVIR Epidemic Model with Logistic Growth Joko Harianto 129 Figure 1. Dynamics of the solution on the System (1) for the case β„œ0 = 0,04 < 1 Figure 2. Dynamics of the solution on the System (1) for the case β„œ0 = 1,66 > 1 Local Dynamics of an SVIR Epidemic Model with Logistic Growth Joko Harianto 130 Sensitivity Analysis of the Basic Reproduction Number The sensitivity analysis performed was a local sensitivity ([22], [23], and [24]). In this section, the parameters analyzed are 𝛽, 𝛼, 𝛾1, 𝛾, 𝛽1. The sensitivity index for the parameter 𝛽 is 𝐢 𝛽 β„œ0 = πœ•β„œ0 πœ•π›½ Γ— 𝛽 β„œ0 = πΎπ›½π‘š(πœ‡ + 𝛾1) πΎπ›½π‘š(πœ‡ + 𝛾1) + 𝛽1πΎπ›Όπ‘š With the same steps, we can obtain a sensitivity index for the other parameters. The following Table 1 is given the five parameter sensitivity to the basic reproduction number. The parameter 𝛽, 𝛼, 𝛾1, 𝛾, 𝛽1 values used are 𝐾 = 1, π‘Ÿ = 0,04 , πœ‡ = 0,01 , 𝛼 = 0,01 , 𝛽 = 0,002, 𝛽1 = 0,00045 , 𝛾 = 0,0001 , πœ” = 0,02 , 𝛾1 = 0,004. Table 1. Sensitivity index of parameters to the basic reproduction number Parameter Sensitivity index 𝛽 +0,861 𝛽1 +0.005 𝛼 -0,138 𝛾 -0,001 𝛾1 -0.132 Sensitivity index of parameter 𝛽 and 𝛽1 has a positive value to β„œ0, it's mean that when the parameter 𝛽 and 𝛽1 increases, then β„œ0 value also increases and conversely. The parameter sensitivity index of parameter 𝛼, 𝛾, 𝛾1has a negative value to β„œ0, it's mean that when the parameter 𝛼, 𝛾, 𝛾1 increases, then β„œ0 value decreases, and conversely. The most influential parameters of the five parameters are transmission rate (𝛽) and vaccine rate (𝛼). CONCLUSIONS This article is discussed the local stability analysis of the equilibrium point of the SVIR model adapted from Liu et al. The SVIR model discussed in this article was modified, assuming the growth rate on susceptible classes (V) approached with the growth of logistics. First, we determine the disease-free equilibrium point. Then, we find the basic reproduction number by using a disease-free equilibrium point that obtains. Next, we determine the existence of the endemic equilibrium point associated with the basic reproduction number (β„œ0). Lastly, we determine the stability of the equilibrium point and illustrate the numerical simulation. Our results show that if β„œ0 < 1 and π‘š > 0, then there exists a unique disease-free equilibrium point denoted by 𝐸0 is locally asymptotically stable. The numerical simulation shown in Figure 1 supports these results. In the field of epidemiology, this means that the disease will disappear from the population. However, if β„œ0 > 1, 𝛽1 > 𝛽 and π‘š > 0, then the System (1) has two equilibrium point, i.e., the disease-free equilibrium point denoted by 𝐸0 and the endemic equilibrium point denoted by 𝐸1 βˆ—. In this case, the endemic equilibrium point denotes by 𝐸1 βˆ— Locally asymptotically stable. The numerical simulation shown in Figure 2 supports these results. 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