The disease is caused by infection of measles virus paramyxovirus cluster. It is a deadly disease. Vaccination is the most effective strategy to prevent the disease ...
Mathematical models (Linear) about controlling Measles with Vaccination
Presented by : Nadia Cikyta Maliangkay (1214100077)
Author : Maesaroh Ulfa (Mathematics Department of UIN Sunan Kalijaga Yogyakarta)
Lecturer: Prof. Dr. Basuki Widodo, M.Sc
MATHEMATICS DEPARTEMENT SEPULUH NOPEMBER INSTITUTE OF TECHNOLOGY 2016
Abstract Measles (also known as Rubeola, measles 9 day) is a highly contagious virus infection, characterized by fever, cough, conjunctiva (inflammation of the tissue lining of the eye) and skin rash. The disease is caused by infection of measles virus paramyxovirus cluster. It is a deadly disease. Vaccination is the most effective strategy to prevent the disease. It is generally given to children. This research aims to establish a model of the effect of measles vaccination, forming the point of equilibrium and analyze the stability, create a simulation model and interpret them, and to know the design to optimize the vaccination coverage required, so it can reduce the spread of this disease. This research was conducted by the method of literature study. It is expected to provide an overview of the mathematical model used to control measles vaccination with division of classes SEIR. The steps taken is identifying the problem, formulating assumptions to simplifying the model, making the transfer diagram, defining parameters, determining the equilibrium points and analyzing the stability, simulating the model, and forming the design to optimize the vaccination. Then from this research can be obtained free balance point of endemic and diseases and their stability. Based on the results obtained, the simulation is done by taking the data in Yogyakarta, and obtained vaccination coverage with two doses that can increase the herd immunity with lower vaccination coverage Keyword: Measles, Vaccination, SIR, SEIR, Linear Systems of Differential Equations
Introduction Measles is a highly contagious infection caused by the measles virus. Initial signs and symptoms typically include fever, often greater than 40 °C (104.0 °F), cough, runny nose, and inflamed eyes. Two or three days after the start of symptoms, small white spots may form inside the mouth, known as Koplik's spots. A red, flat rash which usually starts on the face and then spreads to the rest of the body typically begins three to five days after the start of symptoms. Symptoms usually develop 10–12 days after exposure to an infected person and last 7–10 days . Complications occur in about 30% and may includediarrhea, blindness, inflammation of the brain, and pneumonia among others. Rubella (German measles) and roseola are different diseases. The measles vaccine is effective at preventing the disease. Vaccination has resulted in a 75% decrease in deaths from measles between 2000 and 2013 with about 85% of children globally being currently vaccinated. No specific treatment is available. Supportive care may improve outcomes. This may include giving oral rehydration solution (slightly sweet and salty fluids), healthy food, and medications to control the fever. Antibiotics may be used if a secondary bacterial infection such as pneumonia occurs. Vitamin A supplementation is also recommended in the developing world. Measles affects about 20 million people a year, primarily in the developing areas of Africa and Asia. It causes the most vaccine-preventable deaths of any disease. It resulted in about 96,000 deaths in 2013, down from 545,000 deaths in 1990. In 1980, the disease was estimated to have caused 2.6 million deaths per year. [ Most of those who are infected and who die are less than five years old. The risk of death among those infected is usually 0.2%, but may be up to 10% in those who have malnutrition. It is not believed to affect other animals. Before immunization in the United States, between three and four million cases occurred each year.[ As a result of widespread vaccination, the disease was eliminated from the Americasby 2016
Problem Statement 1. How Mathematical Models (Linear) on controlling Measles with Vaccination? 2. How to analysis the stability of the equilibrium point of mathematical models (Linear) on controlling Measles with Vaccination? 3. How to simulate the stability of the point of equilibrium models (Linear) on controlling Measles with Vaccination?
Mathematical Modelling In this model, the total population (N) divided into four classes: Susceptible class (S(t)) stated that the number of individuals who are vulnerable to measles, class Exposed (E(t)) stated that the number of individuals detected measles but not yet infected class
Infectious (I(t)) stated that the number of individuals infected with (has become measles active) and can transmit measles, and recovered class (R (t)) stated that the number of individuals who had been healed or cured through the vaccination that makes the permanent immunity. Large population revealed by N = S(t) + E(t) + I(t) + R(t) with S(t), E(t), I(t), dan R(t) states many individuals vulnerable S, individual infected but not yet contracting E, individuals infected with I and individuals who have been healed R. For the next S(t), E (t), I (t), and R(t) will be written S, E, I, and R. Assumptions used to formulate the model of measles with the influence of vaccination is as follows: 1. There is a birth and death in the population of 2. There is no migration 3. Each individual is born will be vulnerable 4. Each individual that contracting will become infected 5. The incubation measles (short)10-14 days 6. Dangerous disease, if infected can cause the death of 7. Individuals who are vulnerable if vaccinated will be immune to the disease. 8. A person who has been healed will be immune to measles and not be vulnerable back 9. . The constant population (closed), which means N S(t) E(t) I(t) R(t) The amount of the population in time t equals the number of individuals vulnerable, contracting, infection, and healed. From the assumptions on the obtained model flow diagram mathematics measles with vaccination.
Figure 1. Transfer Diagram mathematical models with measles vaccination
The parameters used are : b = birth rate = mortality rate = contact rate = infectious rate = recovery rate = differential mortality due to measles p = proportion of those successively vaccinated at birth with b , , , , , , > 0 dan 0 < p < 1. From the figure 1 and the explanation of the above obtained from measles mathematical models with the vaccination as follows :
Equilibrium Point Analysis and Stability of Model Equation system (1) above will reach equilibrium point when 𝑑𝑅 𝑑𝑡
= 0 , So the equation becomes:
𝑑𝑆 𝑑𝑡
=0,
𝑑𝐸 𝑑𝑡
=0,
𝑑𝐼 𝑑𝑡
= 0 , dan
Based on the common (2) obtained two types of equilibrium point, namely: 1. The disease free equilibrium point that is a condition where does not occur the spread of infectious diseases in the population. 2. Endemic equilibrium point, which is a condition in which the spread of infectious diseases in the population. Both of them will be discussed in the following lemma lemma. Lemma 1 If I = 0 then there is no individual infected with the virus and transmit measles to other individuals. This system has a balance point disease-free
Lemma 2
If I≠0 there are individuals who are infected and transmit measles to other individuals. This system has a balance point endemic E1=(S*,E*,I*,R*) with
Stability Analysis of Equilibrium Point The stability of the equilibrium point is used to know the behavior of the system by defining 𝑅𝑝 = (1 − 𝑝)
𝑏𝛽𝜎 𝜇 (𝜎+𝜇)(𝛾+𝜇+𝛿)
for the value of Rp < 1 where can be
found sebagat equilibrium point is free of disease and there is no genesis endemic. On the stability of the disease free equilibrium point, if in the population found no infected so stand does not occur epidemic, because the system will go back into the system fluids. If Rp > 1, disease free equilibrium point there but start unstable. If there are infected into free state disease, it will become an epidemic and the system will go to the state of the endemic in asimtotik, and stable to Rp >1 . The Parameters Rp = 1 can be interpreted as a minimum vaccination level. This number is the
minimum number of vaccination is needed to prevent the epidemic
Numerical simulation The simulation data in this research obtained from the health profile of the Special Region of Yogyakarta in 2011 to carry the data from the year 2010. population in Daerah Istimewa Yogyakarta numbered 3.457.491 people (individuals), the number of the birth of 43.242 people, the number of death 43.242 people, number of yangterkena measles 292 people, and the number of deaths due to measles 0. The average latent period 12 days, the average duration of infection 9 days, real reproduction figure R* is 15 (each infectious the average will infect 15 susceptibles) Table 3.1. The initial value Data Class SEIR with certain Assumptions
The early Data class SEIR in obtain from the regional data in Yogyakarta in 2010. Class vulnerable populations obtained from the number of birth, infection class population obtained from the number of patients with measles, total population obtained from the population in Yogyakarta in 2010. To further expose class population obtained by taking the number that approaches the number of class infection, then class healed obtained with reduce the number of overall population with the population of the vulnerable class, expose and infection. The parameters of the model (numbers of transition per unit of time (days)) calculated using these formulas the estimation of model parameters and obtained: 1. σ = ½ = 0,0833 per day 2. γ= 1/9 = 0,1111 per day 43,242
3. b = 3457491𝑥365 = 3,4265x10-5 4. δ = 0 5. β = 15/9 = 1,6667 per day
Conclusion and Recommendations From the discussion that has been done, can be concluded as follows. 1. Mathematical models for measles control using vaccination can be expressed
As
𝑏(1−𝑝)𝑁
2. The Model has two equilibrium point namely 𝐸0 = (
𝜇
, 0,0,
𝑏𝑝𝑁 𝜇
) and
E1=(S*,E*,I*,R*) with
3. Equilibrium point E0 stable local asimtotik to Rp < 1. Equilibrium point E1 stable asimtotik local to Rp >1 . 4. The level of vaccination is needed to prevent the spread of disease can be expressed as pc = 1- (1/Ro) 5. The scope of the optimal vaccine required so that can reduce the spread of the disease is 0.77 to perform two times the vaccination
References Journal of mathematical models, presented byMaesaroh Ulfa (Mathematics Departement of UIN Sunan Kalijaga Yogyakarta)
