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Local Stability Analysis of an SVIR Epidemic Model Harianto, Joko
CAUCHY Vol 5, No 1 (2017): CAUCHY
Publisher : Mathematics Department, Maulana Malik Ibrahim State Islamic University of Malang

In this paper, we present an SVIR epidemic model with deadly deseases. Initially the basic formulation of the model is presented. Two equilibrium point exists for the system; disease free and endemic equilibrium. The local stability of the disease free and endemic equilibrium exists when the basic reproduction number less or greater than unity, respectively. If the value of R0 less than one then the desease free equilibrium is locally stable, and if its exceeds, the endemic equilibrium is locally stable. The numerical results are presented for illustration.

SVIR Epidemic Model with Non Constant Population Harianto, Joko; Suparwati, Titik
CAUCHY Vol 5, No 3 (2018): CAUCHY
Publisher : Mathematics Department, Maulana Malik Ibrahim State Islamic University of Malang

In this article, we present an SVIR epidemic model with deadly deseases and non constant population. We only discuss the local stability analysis of the model. Initially the basic formulation of the model is presented. Two equilibrium point exists for the system; disease free and endemic equilibrium point. The local stability of the disease free and endemic equilibrium exists when the basic reproduction number less or greater than unity, respectively. If the value of R0 less than one then the desease free equilibrium point is locally asymptotically stable, and if its exceeds, the endemic equilibrium point is locally asymptotically stable. The numerical results are presented for illustration.

Local Dynamics of an SVIR Epidemic Model with Logistic Growth Harianto, Joko; Sari, Inda Puspita
CAUCHY Vol 6, No 3 (2020): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Maulana Malik Ibrahim State Islamic University of Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/ca.v6i3.9891

Discussion of local stability analysis of SVIR models in this article is included in the scope of applied mathematics. The purpose of this discussion was to provide results of local stability analysis that had not been discussed 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 less than one and m is positive, then there is one equilibrium point i.e. E0 is locally asymptotically stable. In the field of epidemiology, this means that the disease will disappear from the population. However, if the basic reproduction number more than one and b1 more than b, then there are two equilibrium points i.e. disease-free equilibrium point denoted by E0 and the endemic equilibrium point denoted by E1*. In this case the endemic equilibrium point E1* is locally asymptotically stable. In the field of epidemiology, this means that the disease will remain in the population. The numerical simulation supports these results.

ANALISIS KESTABILAN LOKAL TITIK EKUILIBRIUM MODEL EPIDEMI SEIV DENGAN PERTUMBUHAN LOGISTIK Harianto, Joko; Sari, Inda Puspita
Majalah Ilmiah Matematika dan Statistika Vol 22 No 1 (2022): Majalah Ilmiah Matematika dan Statistika
Publisher : Jurusan Matematika FMIPA Universitas Jember

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.19184/mims.v22i1.30174

The SEIV model uses population growth which is assumed to follow logistical growth. The model is studied then analyzed. The analysis shows that the non-endemic (disease-free) equilibrium point is locally asymptotically stable when the basic reproduction number less than one, while the endemic equilibrium point is locally asymptotically stable when the basic reproduction number greater than one. Then a numerical simulation was carried out using Maple software to support the results of the local stability analysis of the equilibrium point. Based on numerical simulations, it shows that a disease will disappear from the population when the basic reproduction number less than one and for a long time a disease will remain in the population (still an epidemic) when the basic reproduction number greater than one.Keywords: SEIV model, logistical growth, equilibrium point, basic reproduction numberMSC2020: 92C60

Analisis Kestabilan Lokal Titik Ekuilibrium Model Dinamik Kebiasaan Merokok Joko Harianto; Mira Aprilia Marcus; Jonner Nainggolan
KUBIK Vol 5, No 2 (2020): KUBIK: Jurnal Publikasi Ilmiah Matematika
Publisher : Jurusan Matematika, Fakultas Sains dan Teknologi, UIN Sunan Gunung Djati Bandung

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.15575/kubik.v5i2.9348

Dinamika kebiasaan merokok dalam artikel ini dianalisis dengan pendekatan model epidemiologi. Lingkungan perokok dibagi menjadi empat populasi, yaitu populasi (Potential) menyatakan populasi dari individu-individu yang tidak merokok, populasi (Light) menyatakan populasi dari perokok ringan, populasi (Smokers) menyatakan populasi dari perokok berat, populasi menyatakan populasi dari individu-individu yang berhenti merokok sementara dan populasi menyatakan populasi dari individu-individu yang berhenti merokok secara permanen. Model tersebut dimodifikasi kemudian dianalisis titik ekuilibriumnya. Langkah pertama, ditentukan titik ekuilibrium bebas rokok. Langkah kedua, ditentukan titik ekuilibrium kebiasaan merokok. Langkah ketiga, ditentukan the Smoking Generation Number (R0 ) dengan menggunakan next generation matrix yang melibatkan radius spektral. Langkah terakhir, kestabilan lokal setiap titik ekuilibrium pada modelnya dianalisis. Hasil analisis menunjukkan bahwa titik ekuilibrium bebas rokok stabil asimtotik lokal saat nilai the Smoking Generation Number kurang dari satu. Sebaliknya, jika nilai the Smoking Generation Number lebih dari satu dan b1(m+g) lebih dari b2(b1-m), maka titik ekuilibrium perokok ringan stabil asimtotik lokal. Sedangkan titik ekuilibrium perokok berat stabil asimtotik lokal jika nilai the Heavy Smoking Generation Number lebih dari satu. Kemudian dilakukan simulasi numerik menggunakan Software Maple untuk mengecek hasil analisis kestabilan lokal titik ekuilibrium tersebut.

Effect of Population Density on the Model of the Spread of Measles Joko Harianto; Katarina Lodia Tuturop; Venthy Angelika
Numerical: Jurnal Matematika dan Pendidikan Matematika Vol. 4 No. 2 (2020)
Publisher : Institut Agama Islam Ma'arif NU (IAIMNU) Metro Lampung

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.25217/numerical.v4i2.831

This study is expected to contribute to the health sector, specifically to describe the dynamics of the measles spread through the models that have been analyzed. One of the factors that became the focus of this study was reviewing the influence of population density on measles spread. The initial step formulated the model and then determined the primary reproduction number and analyzed the stability of the model equilibrium point. The results of the analysis of this model show that there are two conditions for the value of which is a requirement that the existence of two model equilibrium points as well as local stability is needed, namely and . When , there exists a unique equilibrium point, called the non-endemic equilibrium point denoted by . Conversely, when , there are two equilibrium points, namely and the endemic equilibrium point characterized by . The results of local stability analysis show that when , the equilibrium point is stable asymptotic locally. It means that if hold, then in a long time there will not be a spread of disease in the susceptible and vaccinated sub-population, or in other words, the outbreak of the disease will stop. Conversely, when equilibrium point is stable asymptotic locally. It means that if , then measles disease is still in the environment for an infinite time with the condition of the proportions of each sub-population approach to , , and .

Local Stability Dynamics of Equilibrium Points in Predator-Prey Models with Anti-Predator Behavior Joko Harianto; Titik Suparwati; Alfonsina Lisda Puspa Dewi
Jurnal ILMU DASAR Vol 22 No 2 (2021)
Publisher : Fakultas Matematika dan Ilmu Pengetahuan Alam Universitas Jember

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.19184/jid.v22i2.23991

This article describes the dynamics of local stability equilibrium point models of interaction between prey populations and their predators. The model involves response functions in the form of Holling type III and anti-predator behavior. The existence and stability of the equilibrium point of the model can be obtained by reviewing several cases. One of the factors that affect the existence and local stability of the model equilibrium point is the carrying capacity (k) parameter. If x3∗, y3∗ > 0 is a constant solution of the model and ∈ (0,x3∗), then there is a unique boundary equilibrium point Ek (k , 0). Whereas, if k ∈ (x4∗, y4∗], then Ek (k, 0) is unstable and E3 (x3∗, y3∗) is stable. Furthermore, if k ∈ ( x4∗, ∞), then Ek ( k, 0) remains stable and E4 (x4∗, y4∗) is unstable, but the stability of the equilibrium point E3 (x3∗, y3∗) is branching. The equilibrium point E3 (x3∗, y3∗) can be stable or unstable depending on all parameters involved in the model. Variations of k parameter values are given in numerical simulation to verify the results of the analysis. Numerical simulation indicates that if k = 0,92 then nontrivial equilibrium point Ek (0,92 ; 0) stable. If k = 0,93 then Ek (0,93 ; 0) unstable and E3∗(0,929; 0,00003) stable. If k = 23,94, then Ek (23,94 ; 0) and E3∗(0,929; 0,143) stable, but E4∗(23,93 ; 0,0005) unstable. If k = 38 then Ek(38,0) stable, but E3∗(0,929; 0,145) and E4∗(23,93 ; 0,739) unstable.Keywords: anti-predator behavior, carrying capacity, and holling type III.

Analisis Kestabilan Lokal Titik Ekuilibrium Model Dinamik Kebiasaan Merokok Joko Harianto; Mira Aprilia Marcus; Jonner Nainggolan
KUBIK Vol 5, No 2 (2020): KUBIK: Jurnal Publikasi Ilmiah Matematika
Publisher : Jurusan Matematika, Fakultas Sains dan Teknologi, UIN Sunan Gunung Djati Bandung

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.15575/kubik.v5i2.9348

Dinamika kebiasaan merokok dalam artikel ini dianalisis dengan pendekatan model epidemiologi. Lingkungan perokok dibagi menjadi empat populasi, yaitu populasi (Potential) menyatakan populasi dari individu-individu yang tidak merokok, populasi (Light) menyatakan populasi dari perokok ringan, populasi (Smokers) menyatakan populasi dari perokok berat, populasi menyatakan populasi dari individu-individu yang berhenti merokok sementara dan populasi menyatakan populasi dari individu-individu yang berhenti merokok secara permanen. Model tersebut dimodifikasi kemudian dianalisis titik ekuilibriumnya. Langkah pertama, ditentukan titik ekuilibrium bebas rokok. Langkah kedua, ditentukan titik ekuilibrium kebiasaan merokok. Langkah ketiga, ditentukan the Smoking Generation Number (R0 ) dengan menggunakan next generation matrix yang melibatkan radius spektral. Langkah terakhir, kestabilan lokal setiap titik ekuilibrium pada modelnya dianalisis. Hasil analisis menunjukkan bahwa titik ekuilibrium bebas rokok stabil asimtotik lokal saat nilai the Smoking Generation Number kurang dari satu. Sebaliknya, jika nilai the Smoking Generation Number lebih dari satu dan b1(m+g) lebih dari b2(b1-m), maka titik ekuilibrium perokok ringan stabil asimtotik lokal. Sedangkan titik ekuilibrium perokok berat stabil asimtotik lokal jika nilai the Heavy Smoking Generation Number lebih dari satu. Kemudian dilakukan simulasi numerik menggunakan Software Maple untuk mengecek hasil analisis kestabilan lokal titik ekuilibrium tersebut.

Analisis Sensitivitas Model Matematika Penyebaran Penyakit Tuberkulosis J Harianto; K L Tuturop
JURNAL ILMIAH MATEMATIKA DAN TERAPAN Vol. 19 No. 1 (2022)
Publisher : Program Studi Matematika, Universitas Tadulako

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.22487/2540766X.2022.v19.i1.15802

Model matematika dapat memberikan masukan bagi praktisi kesehatan masyarakat dalam mencegah penyebaran penyakit tuberkulosis. Namun, model matematika harus diparameterisasi dan divalidasi dengan hati-hati berdasarkan data epidemiologis dan entomologis. Penelitian ini bertujuan menganalisis indeks sensitivitas setiap parameter dalam model. Hal ini perlu dilakukan untuk menentukan kepentingan relatif dari parameter model terhadap penularan penyakit tuberkulosis. Model yang dibahas dalam penelitian ini berbasis SEIL (S untuk rentan, E untuk individu yang beresiko tinggi, I untuk infeksi dan L untuk individu beresiko rendah) yang melibatkan pertumbuhan logistik. Metode yang digunakan adalah literature review. Hasil penelitian ini menginformasikan bahwa parameter laju kontak dan daya tampung merupakan parameter yang paling dominan terhadap peningkatan penyebaran penyakit tuberkulosis. Di sisi lain, parameter laju pengobatan individu yang terinfeksi merupakan parameter yang paling dominan terhadap penurunan penyebaran penyakit tuberkulosis

Mathematical Modeling of Foot and Mouth Disease Spread on Livestock using Saturated Incidence Rate Imam Fahcruddin; Joko Harianto; Denny Fitrial
JTAM (Jurnal Teori dan Aplikasi Matematika) Vol 7, No 1 (2023): January
Publisher : Universitas Muhammadiyah Mataram

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.31764/jtam.v7i1.10264

Foot and Mouth Disease (FMD) is an acute infectious disease that attacks livestock, thus threatening the availability of food and the husbandry industry. This paper discusses the formulation of a mathematical model for the spread of FMD in livestock with a saturated incidence rate. The research method used is quantitative mathematical modeling with simulation, with stages including problem identification, determining assumptions, model formulation, analysis and model simulation. The discussion results obtained two equilibrium points, namely the non-endemic equilibrium point and the endemic equilibrium point, and then analyzed for stability. Numerical simulation is presented using Runge-Kutta approximation with MATLAB. Furthermore, after a sensitivity analysis, the parameters that greatly influenced the spread of FMD were direct or indirect contact (which led to the entry of the FMD virus) and the supporting capacity of livestock. Then the most influential parameter in reducing the spread of FMD is the application of culling on exposed animals and infected animals. The FMD modeling is a form of mathematical application to simulate the spread of disease on livestock.

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