ANALYSIS OF DIPHTHERIA DISSEMINATION BY USING MULTI GROUPS OF DYNAMIC SYSTEM METHOD APPROACH 1 NUR ASIYAH, 2 BASUKI WIDODO, 3 SUHUD WAHYUDI 1,2,3 Laboratory of Analysis and Algebra Faculty of Mathematics and Sciences Institut Teknologi Sepuluh Nopember- Indonesia E-mail : nora@ matematika.its.ac.id Abstract- Diphtheria Dissemination is an infectious disease on a region which is become endemic or not, it can be analyzed by using a mathematical model approach in the form of differential equation systems. The disease dissemination in those populations can be clustered into some groups including group of under 15 years old and group of above 15 years old. Therefore, the diphtheria dissemination that occurred is cross-analyzed between the groups. This research consideres about analyzing multi-grup epidemic model with rate of nonlinear transmission for preventing the occurrence of endemic and evaluating the threshold, i.e. the basic reproduction values R 0, through stability of the free disease equilibrium point and the endemic equilibrium point in East Java areas. Data is taken from the year of 2013. The result of this research is expected to be able to give information about the characteristics of diphtheria patient through multi group of dynamic system model. The criteria of Routh Hurwitz is used to analyze the local stability and construct Liapunov function for the global stability by applying directed graph. Based on the theory of graph approach, the system can be described as a network, where each vertex is a homogeneus group and an edge(j, i)is existsif only if can be transmitted from group i to group j. Keywords- multi-grup analysis, stability analysis, basic reproduction I. INTRODUCTION Real time problem in this life can be accommodated by using certain assumptions to build a mathematical model which help us to easily find solutions of that problem analytically or numerically. Coupled system from nonlinear differential function on a network has been used to model many things such as: inspecting coupled system of nonlinear oscillator, understanding the spreading of contagious disease, and analyzing the stability and complexity of coupled system in a complex ecosystem model [1].Researches have been conducted to analysethe stability model of the spreading of contagious disease by neglecting the heterogeneity of a population [2], where this heterogeneity can be caused by many factors. A group of individuals can be divided into groups of homogenous according to their different relational pattern ontosomething such as group classification based on their age on the spreading of diphtheria. Due to possibility of different infection rate within each group, therefore a network concept on the spreading of contagious disease is needed. By using approach from graph theory, the system can be drawn as a network where each vertex shows a homogenous group and an edge (j, i) will be present if and only if the disease is infectious from groupito group j. Some vertexes will be connected by a vector edge indicating connection between vertexes in the system [3]. In this model, a vertex can present as an oscillator, a big ecology of community or a path, and can also present as homogenous groups for a common contagious disease, while the interaction between vertexes can be a physical connection between those oscillator, a spread between small groups in the path, or even a cross relation infection between homogenous groups within a heterogeneous group [1]. In this paper local and global stability of the epidemic model of two groups with nonlinear infection rate will be analysed. A Routh-Hurwitz criterion is utilized for the local stability analysis, while graph theory is implemented for the global stability. II. EPIDEMIC MODEL OF TWO GROUP AND THE SOLUTION AREA Compartment is a flow that describes the spread of disease from individuals. There are phases in a compartment, they are: S :Susceptible; healthy individual but is not immune to disease. E :Exposed; infected individual but hasn t shown a symptom (incubation area). I :Infective; Infected individual and also able to infect others. Epidemic model that consist of two groups are differential equation of epidemic model system with i, j = 1,2or can also be called as epidemic 2-Group. If given the equation of the infection rate f S, I = I S, then system of differential equations become : 55
Compartmentdiagramofthe model(1) -(6) are givenin Figure1: Figure 1. Compartment Diagram of Epidemic Two Group Model with Nonlinear Infection Rate Assuming that ϵ, ϵ > 0, and d > 0where d = min {d, d, d + γ }, d = d, d, d + γ, f S, I = I S > 0for S > 0, I > 0, and p, q is a positive constant. By adding equation 1-3 will give S + E + I = Λ d S β I S I + β I S (d + ϵ )E + ϵ E (d + γ )I = Λ d S d E (d + γ )I Λ d (S + E + I ) Therefore lim sup (S + E + I ), and the equations (4) (6) will give S + E + I = Λ d S β I S + β I S (d + ϵ )E + ϵ E (d + γ )I = Λ d S d E (d + γ )I Λ d (S + E + I ) and lim sup (S + E + I ) Thus the solutions area of the epidemic two group model can be written as There are four equilibrium points achieved. They are equilibrium point of disease free, endemic point, a point where first group are disease free while second group are endemic, and a point where first group are endemic while second group are disease free. 1.1 Disease Free Equilibrium Point When I, I = 0, it will have free disease equilibrium point P, where all the individuals are Susceptible or will have no any spread of disease in both of groups. By substituting I, I = 0to equation (10) and (13) will givee = 0 and E = 0. This result substituted to equation (8) and (11) resulting S = and S = Thus the disease free equilibrium point for this epidemic two group model can be written as P = (S, 0,0, S, 0,0)with S =, S =. 1.2 Endemic Equilibrium Point Endemic equilibrium point is a condition where the disease will always be inside the population in both of group. Endemic equilibrium point P = (S, E, I, S, E, I ) depend on the infected populations from contagious disease with I, I 0. It is given from taking = 0, = 0, = = 0.Thus the endemic 0, = 0, = 0, equilibrium point from the this two group model is P = (S, E, I, S, E, I )where equations below: S, E, I satisfy III. EQUILIBRIUM POINT Equilibrium point is a point where it s characteristic is invariant with time. Then the equilibrium points are taken from = 0, = 0, = 0 and the equation (1) ( 6) become: 56
IV. BASIC REPRODUCTION NUMBER AccordingDriesscheandWatmoughmethod [1], the basicreproduction numberis formulatedwith β ϵ C (S ) R = ρ (d + ϵ )(d + γ ) Assuming that if R = ρ(m ) with β M = M(S, S ϵ C (S ) ) = (d + ϵ )(d + γ ), becausef S, I = I S then C (S ) = p I S Therefore R β ϵ p I S β ϵ p I S (d = ρ + ϵ )(d + γ ) (d + ϵ )(d + γ ) β ϵ p I S β ϵ p I S (d + ϵ )(d + γ ) (d + ϵ )(d + γ ) To find R it should be found the Eigen value of the matrix M ie det M λi = 0 giving R = = 1 β ϵ p I S 2 (d + ϵ )(d + γ ) + β ϵ p I S (d + ϵ )(d + γ ) + β ϵ p I S (d + ϵ )(d + γ ) β ϵ p I S (d + ϵ )(d + γ ) + 4 β ϵ p I S β ϵ p I S (d + ϵ )(d + γ )(d + ϵ )(d + γ ) This R represents basic reproduction number for the two group epidemic model with nonlinear infection rate. V. LOCAL EQUILIBRIUM POINT STABILITY Localstabilityof criticalpointsis divided into3conditions. The firstcondition is localequilibriumpointstability of freedisease, second condition is local equilibriumpoint of bothgroupsare endemic and thirdconditionis local pointstability of the firstgroup is freedisease and the secondgroup is endemic [4]. 1.3 Local Equilibrium Point Stability of Free Disease Condition On the free disease equilibrium point P = (S, 0,0, S, 0,0)with S =, S =, it is known that β = β = ϵ = ϵ = γ = γ = 0, so then its Jacobian matrix: d 0 0 0 0 0 0 d 0 0 0 0 0 0 d J(P ) = 0 0 0 0 0 0 d 0 0 0 0 0 0 d 0 0 0 0 0 0 d The Eigen value can be taken from det J(P ) λi = 0, therefore resulting: λ = d, λ = d, λ = d, λ = d, λ = d, λ = d Due to Eigen value of (λ, λ, λ, λ, λ, λ )are negative on the real components, therefore the equilibrium point P =, 0,0,, 0,0is asymptotic stable [4]. 1.4 Local Equilibrium Point of Both Groups Are Endemic On the equilibrium point of P = (S, E, I, S, E, I ), both groups are on endemic condition. So that its Jacobian matrix is: with : Then it should be found det J λi = 0 Resulting characteristic equation with model of a λ + a λ + a λ + a λ + a λ + a λ + a = 0 with a =1 Then Routh-Hurwitz criterion stability is used to analyze the stability of the endemic equilibrium point. 57
VI. RESULT AND DISCUSSION To interpret the analysis result of this two group epidemic model, a simulation is made by using Matlab. And the analysis of simulation is done on endemic equilibrium point where R > 1.In this casewedo an analysisof datafrom diphtheriaineastern Javain 2013. Group division is performed as follows: - The first group is children under the age of 15 years - The second group is adults with age above 15 years Epidemicmodels is: ϵ : The rate of incubation in children ϵ : The rate of incubation in adult Result of the data processing diphtheria East Java in 2013 are presented in Table 1 Table 1. Data processing diphtheria East Java in 2013 Where : S : Population of childrenwho aresusceptible tothe disease (susceptible) S : Adult populationsusceptible tothe disease (susceptible) E : Population in children who contract the disease and can transmit the disease but has not shown any symptoms of the disease early (Exposed) E : Theadultpopulationinfectedandcantransmit the diseasebuthas not shown anysymptoms ofthe diseaseearly (Exposed) I : Populationin childrenwhohave symptoms of(infected, contagiousandundiagnosed) I : The adult population experience symptoms (infected, contagious and undiagnosed) d : Natural mortality rate of S d : Natural mortality rate ofs d : Natural mortality rate ofe d : Natural mortality rate of E d : Natural mortality rate of I d : Natural mortality rate of I Λ : The rate of recruitment of the children population Λ : The rate of recruitment of the adult population β :The chances of cross infection between group S and I β :The chances of cross infection between group S andi β :The chances of cross infection between group S andi β :The chances of cross infection between group S andi γ : The rate of healing of infected individuals in children γ : The rate of healing of infected individuals in adult By using Table 1, we obtain Ro = 6.37635. While the simulation of the model we obtain the results as shown in Figure 2 Figure 2. Analysis of Diptheria Dissemination by Using Multi Group of Dynamic System Method Approach in East Java 2013 In Figure 2, we obtain that when S 1 (the number of children vulnerable populations - children) decreases, then I 1 (the number of the infected population in children - children) becomes increase, and E 1 (the population of carriers of the disease in children - children) increases due to the individual vulnerable populations into the population are carriers of the disease and infected. This also applies to the adult population. CONCLUSION 1. The number of patients with diphtheria always Ro = 6.37635, which means that every month the patient can transmit the disease to 6 up to 7 people. 2. Based on the graph 2, it may be seen that the human population is infected and exposed to a sharp 58
increase in the first 1.5 months. It needs to be taken the reduction of the rate of transmission.. REFERENCE [1] Li, M. Y., Shuai, Z., 2010.Global-stability for Coupled Systems of Differential Equation on Network. J. Differential Equation 248 1-20. [2] Wiggins, S. 1990. Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York : Splinger-Verlag. [3] J. L. Gross and J. Yellen, 2006, Graph Theory and Its Application, Chapman Hall C R C, 2-nd edition [4] Finizio N.and Ladas G. 1988. Ordinary Differential Equations with Modern Applications. California : Wadsworth Publishing Company Belmont. 59