Mathematical and Simulation Models in the AnyLogic program
Contents
1. Figure 2 4 The agent based model From the graph obtained in the agent based model one can see the stochastic nature of the model but it is still very similar to results obtained in the system dynamics model Now a question naturally arises Why is it necessary to use agent models and what are its advantages In the system dynamics model the duration of the disease was shown by the exponential distribution It 1s not a real situation for some diseases In fact it would be better if the duration of disease 1s shown using a triangular distribution The transition between the state infected and the state recovered should be modified so that the rate of transition should be replaced by the Fire After timeout and in the Timeout field the following should be entered Triangular trajanjeBolesti 2 trajanjeBolesti trajanjeBolesti 1 5 Mathematical and Simulation Models in the AnyLogic Program 13 Now the simulation shows the following graph 1 000 0 900 0 300 0 700 0 600 0 500 0 0 0 10 0 20 0 30 0 40 0 50 0 60 0 70 0 30 0 90 0 Figure 2 5 Change in a system dynamics model This change is possible in a system dynamics model Changing the equation one can try to reproduce this behavior but it significantly complicates the equation Even if we achieve to get this behavior in a model of system dynamics it is just an attempt to reproduce known results while the agent model is a way to model the actual characteristics of some2. broj jedniki u populaciji isowo 7777 Yeme Figure 5 1 Graph of the Lotka Volterra model He observed two populations One was the predator and the other the prey population In the observed envionment he put several oranges prey s food and covered them ba wax which enabled Huffaker to control the amount of the eaten food In this environment he put also few rubber balls On the graph 5 2 below we can see the result of Huffaker s experiment obtained for one choice of oranges and balls 2500 2000 1500 J ai is at oe 500 i a o broj jedinki u populaciji rtve predator a eoepad ihioendod n iyuipal oa 745 7718 S o 37 26 9 15 1076 10 24 11714 12 75 12 27 1 16 2 6 2 28 1955 1956 Figure 5 2 Graph of the Lotka Volterra model obtained by following two populations in laboratory conditions The graph shows that the populations are exhibiting a cyclic behavior and that the maximum value for predators behind the maximum value for the preys In order to conclude that this experiment really reflects the assumptions of the Lotka Volterra model we must consider another fact Actually in this experiment there is a rather complex environment in which there are also other populations which is contrary to the assumption made in the model Thus also the assumption that the predators meet the preys only randomly is questionable 24 A Takaci Another experiment has been based on monitoring the number of
3. A Takaci put together a model with individual elements By adjusting the parameters of each element that are used one gets the desired behavior of the model For now on we only use four elements to make a simple queuing model queue del source oe delay The first element source is used for the generation of the entities client manufacturing parts etc arriving into the system The parameters for the source are simple and most often they are used for entering the random distributions according to which the entities were generated x Properties General Replication Description Name Source Type com xj anylogic lib enterprise_library Source Parameters Name Value on Exit new Entity Entity class generation Type distribution first Anival Time 0 interanival Time exponential 0 67 entities PerAnival 1 amvals Max infinity can WaitAtOutput true In our example the arrival of entities is accomplished by an exponential distribution with parameter 0 67 and by each generation of new entities in the system exactly one entity is inserted because in the field entitiesPerArriva when one enters arrivalMax defines the maximum number of generated entities and by entering infinity we get an infinite number of generated entities More precisely the entities will be inserted into the system throughout the duration of the simulation model The next element in the model is the queue which enables t
4. Recovered 228 0 Vactinated 58 Figure 2 8 The simulation of the model 3 Mathematical pendulum From a mathematical point of view the pendulum is an oscillating system consisting of a nonstretchable string of some length but of negligible mass and a negligibly small suspended ballot with larger mass than that of the string The problem is to observe the movement of the pendulum under the influence of gravity For simplicity for now we assume that the pendulum s length is L 1 and that the pendulum s mass is m 1 The first assumption is that gravity has a constant impact Mathematical and Simulation Models in the AnyLogic Program 17 force to the pendulum force g acting vertically downward In order to analyse the movement of the pendulum in a positive or negative direction we have to measure the angle a compared to the pendulum at rest and the pendulum angular velocity The equation of the motion of the pendulum is 2 Ot a gsina 3 1 Using the definition of the angular velocity we can put it in the previous term and obtain a O 3 2 gsina By analyzing the behavior of the pendulum we can conclude that the model of an ideal mathematical pendulum appears in four states pendulum is at rest vertically down pendulum is moving between two points in which they now rest with the same deflection on both sides of the balanced position pendulum rotates continuously in the same direction
5. he immediately resets the time that indicates how many lynxes may be without food before they die The time that the lynxes can spend without enough food is just one day otherwise he does not survive The model defines algorithmic functions to determine whether the area around the hares is overcrowded while this function determines new areas for lynxes in a random way The result obtained in the simulation gives a graph that looks much like the graphs obtained by observing a population in nature The highest values for lynxes are a little behind the greatest value for hares Unlike to the previous models it can happen that the total population tends to extinction because of the parameter values The simulation was added a 2D image of the space where the lynxes and hares are staying The model is based on an X Tek company s model 6 A universal simple queuing model The model to be described can be applied to various types of queuing models Often we meet the situation when we have to wait in a line to get a service say visit a doctor cash a check or wait in a line in front of an ATM All these models have in common that the client enters the system gets in the line waits for the service receives the service he needed and then leaves the system This is a rather simple model but the main problem in making this model is finding the appropriate random Using the Enterprise Libary an important part of the AnyLogic programme one can
6. and never comes in the steady state pendulum is vertically upright at rest unstable This model of the pendulum is fairly easy to model in the AnyLogic Namely it is enough to enter the above equation and observe the behavior of the pendulum In the model we use variables to represent the x and y coordinates of the pendulum in animation namely the angle and the angular velocity To begin with we introduce two parameters the initial angle and the length of the pendulum and the length Ox Oy alfa jomega By introducing another parameter in AnyLogic one easily adds the resistance at the middle of the pendulum The parameter r represents the resistance of the environment Just change the equation for the radial velocity gsna ra A Takaci Instead of a detailed description of how the model is made in AnyLogic we rather use the report generated for each AnyLogic model The report is a good way to get someone to show detailed models of structures Active Object Main Parameters Name alfaO Type real Default value 3 14 Name l Type real Default value 100 Name g Type real Default value 9 81 Icon Structure Ox Oy Picture Malta omega Variable Omega Variable type real Equation type Integral or Stock Equation d omega dt g sin alfa 1 Initial value O Variable alfa Variable type real Equation type Integral or Stock Equation d alfa dt omega Mathemati
7. below the largest prey population This model has the following assumptions population of preys will grow exponentially if there are no predators population of hunters will starve disappear in the absence of the preys hunter can eat as many preys two sides are moving in a homogeneous confined space in a random way Let us introduce the following variables 22 A Takaci P predator population WN prey population ft time r rate of growth of the prey population a rate of attacks on the prey population q rate of mortality of the predator population c efficiency of conversion of food into descendants In some models another assumption is introduced namely the rate of mortality of the prey population In general it is assumed in the model that the prey does not become extinct naturally but rather gets eaten by the predators We add that in some models the product of the parameters c and a is called the productivity of predators Let us consider what happens to the predator population when the prey population is not present in the observed eco system Without predators food the number of predators declines exponentially since then it satisfies the following simple ODE dP a qgP 5 1 Note that gP is the product of the mortality rate of the predators and the population of predators The negative sign in the upper equation indicates the decline of the number of predators We
8. get a more complicated and perhaps a more realistic model in the form od ODE if we add the product caPN to the right hand side of the upper equation 5 1 see equation 5 2 below Note that the mentioned product describes the rate of attacks on the preys multiplied by the number of predators and the number of preys aE caPN qP 5 2 dt Next the population of preys 1s expected to grow exponentially in the absence of predators A rN 5 3 dt The presence of predators prevents the exponential increase of the preys The product aPN describes the mortality of the preys implying the following ODE a rN aPN 5 4 dt The obtained equations 5 1 5 4 describe the predator prey model which predicts cyclic dependencies As the number of predators P grows so does the product aPN also which additionally enhances the growth of the number of predators At the same time the aPN product reduces the number of preys N which indirectly reduces the number P of predators Now as the value of aPN in 5 4 decreases the prey Mathematical and Simulation Models in the AnyLogic Program 23 population is recovering and N starts to grow But then the parameter q increases and the cycle will start again The ideal curve is shown on the graph below The model was tested also in experimental conditions by following the number of both populations In particular one experiment done by Huffaker in 1958 can be accepted as valid ze evi
9. initial value of 0 05 and a parameter of type double stopaKontakata with an initial value 5 This defines the class Person The diagram of the class structure Person has to be inserted into the Main class by dragging the mouse In the diagram of the structure Main there will appear an orange rectangle that represents the agent defined over a class of persons More is needed into the properties of this class can change the name for a class of people in the Replication tab 1000 This means that in our model we shall make 1000 agents Moreover on the diagram structure there should be the variables broj NeZarazenih brojZarazenih and brojOporavljenih of the type Integer Mathematical and Simulation Models in the AnyLogic Program 11 ljudi Zz ObrojNeZarazenih brojZarazenih chartTime Obro a BOT brojOporavlienih The model is now almost finished but we still have to do something to start the work Recall that on the diagrams for the agents each agent will be placed on creating the uninfected state because the state shows the initial indicator Since there is no infected person nothing will happen Thus there is a need for one agent to manually enter a code that will start the epidemic It is necessary to open the Code field in the Main Class section of the Startup Code and then enter judi item 0 statechart fireEvent kontakt which ordered to the zeroth agent to send the signal event contact Animation can be kept simple for the time b
10. into the falling phase Name odskakanje Fire If change event occurs ha Change event a y lt 0 k vy lt 0 Guard Math abs vy gt 1 Action vy k v Next the state diagram enters its final state which means that the ball does not have enough energy to continue its bouncing and the system goes to a stand Mathematical and Simulation Models in the AnyLogic Program 21 Name transition Fire If change event occurs T Change event This example shows how a seemingly simple model can have a hybrid behavior thus it is essential to have a tool that allows you to easily create hybrid models 5 The Lotka Volterra model The Lotka Volterra model is composed of a pair of differential equations describing the relationship between the predators and preys in the simplest case The model was developed around 1925 independently by two authors a biologist Alfred Lotka and the famous Italian mathematician Vito Volterra Volterra has developed a model to explain his son in law the behavior of two fish populations in the Adriatic Sea where one species the preys is the food for the other one the predators but often in the literature called sharks The previous model is a natural extension of the so called logistic model introduced around 1845 by the Belgian scientist Verhulst In the model the population of predators and preys oscillates so that the largest population of predators is just
11. lt b and lt Q for each t gt 0 implying that in this case Mathematical and Simulation Models in the AnyLogic Program 5 the epidemic disappears There will be no epidemic unless a critical value b a is reached by the initial population of uninfected This value is small if b lt lt a which means that the immunity stops the spread of the disease The number of recovered R R t from the third equation monotonically increases but since SX the limit R o lim R t exists Next the limit 00 lim I t t 00 too also exists and the expression R oo N shows the strength of the epidemic that effected the population It is now necessary to determine these limits From the first and the third equation in 2 1 it follows Sis dR Hence S S O exp aR b From R lt N it follows that Z 5 exp aN b ang 5 gt 9_ This means that there will always remain a number of uninfected In fact several persons in the population will not get sick until the end of the epidemic dI i Ze A Z as dt From the equality 0 0 it follows that I N S Ei a S O Clearly S t S cc gt O and J t 0 t o As a consequence of this analysis we have S t O exp aRt b and the equality S c0 S O exp a N S b 2 2 Note that equation 2 8 determines S oo The equation is satisfied only for one positive value of S oo less than b a Once we find oo the limit R oo can
12. models where the behavior is defined for the agent the entity It is also important to note that this model does not include the use of resources which also represents a step towards a more realistic model for example if you model a doctor s surgery with a doctor a nurse and say an ECG machine These are the resources that the patient will need to use in the course of receiving the service Sharing of resources is actually the biggest problem to be addressed by the mathematical model Let us add that a natural question is whether the doctor s office will pay off purchasing an ECG device in order to reduce the waiting time for patients and enable more medical examinations The correct answer on this important question which means more revenue and more satisfied patients can be easily checked by the queuing model REFERENCES 1 2 3 4 5 6 7 8 9 10 J Caldwell and J Douglas K S Ng Mathematical Modeling Kluwer Academic Publishers 2004 H Caswell Matrix Population Models construction analysis and interpretation Sinauer Associates Inc Publishers 2001 V eri Simulacijsko modeliranje kolska knjiga Zagreb 1993 N D Fawkes J J Mahony An Introduction to Mathematical Modelling John Willey and Sons Second Edition 1996 R Haberman Mathematical Models Fourth Edition Prentice Hall 1977 S Ross Simulation Third Edition Academic Press 2002 M Haddin Modelling an
13. the softening of the epidemic and the duration of the disease In other words the epidemic disease of shorter duration will end sooner An example of this is the Ebola virus The reason why this extremely infectious and fatal disease is not spread around the world that it that quickly kills its victims the epidemic has very short duration and it is unlikely that the disease will spread to a larger territory By testing the parameters one can determine the parameter values appropriate for the required behavior of the model For now on we shall use the following values for the parameters c 5 p 0 05 n 1000 d 15 Entering the equations in AnyLogic one can construct the following simulation model a aSI dt as aSI bl dt dR _ py dt ee i n 2 4 pas d By entering the initial values 999 1 and R 0 and running a simulation we obtain the following graph where the number of uninfected is shown in blue the number of infected in red and the number of recovered in yellow A Takaci i 0 0 10 0 20 0 30 0 40 0 50 0 60 0 70 0 80 0 90 0 Figure 2 1 SIR model with vaccination The vaccination should greatly reduce the number of infected in the system In the model we include vaccinations by changing the system of equations compare 2 4 above and 2 5 below aSI a dt a aSI bI dt see dt cp 2 5 Now by running the simulati
14. AnyLogic model Mathematical and Simulation Models in the AnyLogic Program 29 Each agent is assigned the variable Location which contains information on the current position of the agent in space The initial position is assigned of each agent in a random way The value of the variable location site 1s updated when the agent moves in space and that influences its behavior The timers in this model are defined both for the birth and death of the lynxes and the hares The behavior of the timer is cyclic and creates a new agent by the exponential distribution Births of the hares depend also on their local density For the hares the diagram is simple and it describes the behavior of the hares The states in the diagram are alive and dead They are defined by two transition one for the signal message of the lynx I ate you The second transition is the definition of time and a life expectancy of a hare The state diagram for the lynxes is more complicated Namely the lynxes prefer a certain time period for hunt Whether a lynx finds a hare depends on the density of hares and the number of lynxes in the area If a lynx finds and eats a hare then it sends a signal message to the hare I ate eat you leaves the state of hunger and immediately reenters to the hungry state If the lynxes had no luck in hunting the hares and the lynxes do not catch a hare then they move in space but remain in a state of hunger When a lynx catches a hare
15. Teaching Mathematics and Statistics in sciences IPA HU SRB 0901 221 088 Interesting Mathematical Problems in sciences and Everyday Life 2011 Mathematical and Simulation Models in the AnyLogic program Arpad Takaci Dusan Mijatovi Department of Mathematics and Informatics Faculty of Sciences University of Novi Sad takaci dmi uns ac rs 1 Lorenz s weather model Lorenz s mathematical model of the weather made in 1963 describes the motion of the air through the atmosphere More precisely Lorenz observed the hot air rising up into the atmosphere then cooling down and eventually falling to the ground While observing the weather conditions Lorenz noticed that the weather does not always behave in a foreseen manner In his attempt of introducing a mathematical model by following the patterns of the weather conditions he discovered that the obtained model is of chaotic behavior Actually the model he introduced was a nonlinear system of differential equations with three unknown variables At the beginning Lorenz was studying a system of 12 equations and soon he learned that small changes of the variables initial conditions make large changes in the behaviour of the model This sensitivity of the initial conditions of the model nowadays is called the Butterfly effect One of the conclusions from this analysis is that all weather forecasts for a period longer than a week most often prove themselves wrong Fro
16. ain brojVakcinisanih The transition from the state neZarazen into the state vakcinisan waits the signal event vakcinisan In this way an agent can move into the state vakcinisan only from the state neZarazen Before starting the animation it is necessary to put chartTime into the Properties and show the variable brojVakcinisanih The obtained graph clearly shows the efficiency of the vaccination Namely the number of infected has much decreased compared with their number obtained in the previous simulation Mathematical and Simulation Models in the AnyLogic Program 15 0 0 10 0 20 0 30 0 40 0 50 0 60 0 70 0 80 0 30 0 Figure 2 6 The graphs of the S and R functions Perhaps the most important influence on the spread of the epidemic is the spatial environment in which the epidemic spreads Introducing the impact of the environment in the system dynamics model implies the work with partial differential equations and presumabely such models can take us too far However the introduction of an environmental impact in the agent model is not difficult Thanks to the agent modeling library one can easily create an agent with a quite a few different properties To the diagram of the structure one has to include the object agentBase from the library object AgentBase After that we have to define the The dimensions of the agent s environment will be 300 x 300 The default network of the agents will be all in range and contact r
17. ange should be set to 30 Next we have to change the code for the internal transition in the state infected as follows if randomTrue verovatnocaZaraze agentBase sendToRandomContact kontakt and then add into the property onReceive of the object agentBase the following expression statechart fireEvent message The result of this simulation shows an even greater reduction of the number of infected because now the environment affects the agents by the rate of contacts Then the epidemic is spreading slowlier through the population and thus the vaccination process becomes more efficient 16 A Takaci 1 000 0 300 0 800 0 700 0 4 600 0 500 0 300 0 4 200 0 100 0 0 0 10 0 20 0 30 0 40 0 50 0 60 0 70 0 30 0 90 0 Figure 2 7 In order to get a better presentation of the environment influence on the spreading of the epidemics one can add into the model the animation of each agent Since the agents are multiplied objects it is enough to make the animation just for one agent and then in the final animation all of 1000 agents will appear in the model Next for better visualization of the epidemic spreading one should add in the field Fill color an appropriate color gt 0ng 1 2 xi x2 x5 ary SIR Agent based model na APES aie 2 gt a vy 4 f ef a e i eye F oP KOA e Plt al E Oe le 20 0 30 0 40 0 50 0 60 0 70 0 80 0 90 0 100 0 110 0 wee Be mm Infected 2o
18. animals in nature The problem in monitoring the population in nature and modeling of the population is that most predators rely on multiple preys Fortunately there are several species that are strictly specialized in the course of evolution and the food they eat is of just one kind In Canadian forests there is an ideal example of this relationship between the population of lynx and snow rabbits hares The Hudson Bay Company has been closely monitoring the number of lynx and rabbits between 1800 and 1900 It was for a long time noticed that these two populations are very good for observation because of the frequent attacks of the lynx on the hares in view of their way of hunting The collected data show cyclic oscillations in both populations at approximately every 12 years The graph 5 3 presents the data obtained 160 Lecevi 140 Risovi 120 80 A i 60 i 4of5 FEA I 20 U hiljadama 1845 1855 1865 1875 1885 1895 1905 1915 1925 1935 5 3 The graph obtained by observing the populations of lynx and hares in the nature A lack of Lotka Volterra model is relying on unrealistic assumptions The prey population has a limited amount of food in nature and this affects the number of predators while on the other hand they can eat unlimited amounts of food From the mathematical point of view the cyclic behavior we got from a system of differential equations in the Lotka Volterra model tends to repeat itself endl
19. be found from R N S o and the spread of the epidemics equals to R co N 2 4 A Vaccination Model So far we have worked with the assumption that only few people have natural immunity and thus their number can be ignored We continue to assume this assumption but we will include consideration of the case when the epidemic is spreading so fast that it is necessary to stop the spread the epidemic by vaccination 2 5 A Takaci To simplify the model we assume that vaccination removes a person from S or R momentarily and that there is a way to avoid vaccination of those who are infected group J Now we have to introduce a new group V consisting of the vaccinated people Thus we have the following system a aSI a t di 2 3 si a t dt The first equation in the system 2 3 changes the first equation of the initial SIR model see equation 2 1 Function a t is the rate of vaccination It is not easy to choose the function representing the rate of vaccination Namely any vaccination program includes the cost of personnel commitment to conduct the vaccination their efficient use of equipment and other resources Moreover we must take into account the damage suffered by the whole society from the epidemic It is desirable that the vaccination program is limited by the total number of infected at some time and to keep the number of infected below a desired level To ensure that no more than JN indiv
20. cal and Simulation Models in the AnyLogic Program 19 Initial value alfaO Variable y Variable type real Equation type Formula Equation y cos alfa Variable X Variable type real Equation type Formula Equation x sin alfa Animation Name animation Picture Oval Ovall X x Y y Line Line2 End point X x End point Y y 4 The Bouncing Ball This model is one of the simplest hybrid models and is thus often used as an example of hybrid modeling The ball is released from a certain height to a solid base hitting the ground and moves up and then again falls down After each shot of the ball on the ground it loses a certain portion of energy so that eventually ceases jump The falling of the ball down is simply described by a system of diferential equations 4 1 given below 20 A Takaci dy _ dt 4 1 avy dt where V is the speed of the ball on the y axis and g is the gravity The moment when the ball hits the ground is the part when this model becomes hybrid In AnyLogic this is simply modeled using a state diagram odskakanje In the phase of falling the ball behaves according to the above equations The transition bouncing has to wait for the condition y lt 1 and V lt 0 provieded that v gt 1 If these conditions are met then we have V kV 4 2 where k in 4 2 1s the coefficient representing the loss of energy of the ball After that the ball goes back
21. contact We will assume that the rate of contact is as follows if randomTrue verovatnocaZaraze 10 A Takaci Osoba neko main judi random neko statechart fireEvent kontakt The transition from infected to recovered state also happens with a rate For the transition rates we take the value trajanjeBolesti the duration of the disease We also need to enter the input and output actions for each state The variables brojNeZarazenih number of suspected brojZarazenih number of infected and brojOporavljenih number of recovered monitor the number of agents who are in these states so that the input action in the state would be required to increase by one and the output action to reduce the number of agents in this state The picture shows how it was done for the state of infection In the same way one can analyze the status of not infected and recovered x Properties General Description Name 2arazen Deferred events Equations Entry action main brojZarazeniht Exit action nain brojZarazenih Returning to the diagram structure of class Person there is a need to define the main variable of type Main and has an initial value Main getOwner The class of persons should include the three parameters that are features of the agent It is necessary to define the parameter trajanjeBolesti that type double and has an initial value of 15 verovatnocaZaraze that type double and has an
22. d Quantitative Methods in Fisheries Chapman amp Hall CRC 2001 J A Hamilton Jr D A Nash U W Pooch Distributed Simulation CRC Press 1997 Mark M Meerschaert Mathematical Modeling Second Edition Academic Press 1998 D Mooney R Swift A Course in Mathematical Modeling Mathematoical Association of America 1999 Mathematical and Simulation Models in the AnyLogic Program 33 11 A Takac i with L Juhas and D Mijatovi Skripta za matemati ko modeliranje in Serbian WUS Belgrade and Faculty of Sciences Novi Sad 2006 12 B P Zeigler H Praehofer T G Kim Theory of Modelling and Simulation Second Edition Academic Press 2000 13 AnyLogic User Manual XJ Technologies 2005
23. d b 8 3 then for 0 lt r lt 1 the system is stable while for r gt 1 it is unstable If 1 lt r lt 24 74 then the equilibrium points denoted by c and c2 given by c Wbr Jbr r D c 4b r 1 yb r 1 r 1 are stable while if r gt 24 74 then the points c and c2 become unstable which means that the behavior of the whole system becomes unstable The model structure in the AnyLogic programme is very simple since the mathematical expressions can be easily put into the DEs In our case the structure in AnyLogic gets the following form ai eo chartTime ChartxY _Z ChartYZ_X Mathematical and Simulation Models in the AnyLogic Program 3 Lorencov model wu 5 amp B X g uS 5 Figure 1 1 Results of the running the Lorenz model in the AnyLogic programme 2 The Kermack McKendrick model SIR model 2 1 Introduction The epidemic model introduced by Kermack and McKendrick in the early years of the 20th century is a practical and reliable system of equations for monitoring the spread of infectious diseases This model was used for examining the spread of dangerous diseases such as plague Although simple the model is both complete and efficient The model is based on three variables representing a population of people divided into three groups uninfected suspected infected recovered and immune dead recovered immune and removed It is important to note that the model was basically designed t
24. diseases If we recall the system dynamics model previously described a model is contained in it and vaccinations How would it put the process of vaccination in the agentbased model In view of the possibility of combining paradigms in modeling the AnyLogic we may model the system dynamics through vaccination On the Main class diagram of the structure will bring the necessary variables for vaccination One variable should have a name with an initial value of razvojVakcine 10 while the other variable is given by the differential equation dVakcina dt razvojVakane Moreover the diagram would require a static timer which should assign the name immunization The timer will be cyclic with a timeout and actions to be performed is given by the following code while vakcina gt 1 Ijudi random statechart fireEvent vakcinacija vakcina 14 A Takaci This code says that every time the timer comes to its end perform the vaccinations as long as there is vaccine in stock Vaccine recipients will be selected in a random way but only uninfected agents can go into a state of vaccinations avakcinacija vakcina a X razvijanje akcine chartTime A liudi QO brojNeZarazenih OpbrojZarazenih ObrojOporavlienih Opbroj akcinisanih neZarazen yakecinisan zarazen So oporavijen In the state diagram for the agent we have to add another state vakcinisan The starting action for that state will be m
25. e delay element is a service delivery or for example the time the client spent waiting for his cashier to take or deposit some cash Esl Properties General Replication Description Name delay Type com ag anylogic lib enterprise_library Delay Parameters Name Value on Enter on Exit delay Time tniangular 0 8 1 1 3 capacity 1 stats Enabled false animation Shape animation Type AUTO animation Forward true schedule without_schedule In the delay time one has to place a random distribution which corresponds to the duration of the service in our case it is a triangular distribution with parameters 0 8 1 1 3 which means that the customer is retained for example on the bank desk at least 32 A Takaci 0 8 time units a maximum of 1 3 with an average time of a time unit Field capacity determines how many clients can be serviced simultaneously The element sink has nothing to adjust for the behavior of the model since it simply destroys the generated entities This model is a starting point for learning and understanding the queuing models By extending this model with other elements of the Enterprise Library we can get very complex models such as models of airports emergency services manufacturing etc It 1s noteworthy that in these models the behavior of the models associated with the elements in the model and not the entities situated in the model as is the case with agent
26. eing Next we shall draw a rectangle animation to be used for graphical representation of the infected uninfected and recovered Graphical display is enabled by using the Business Graphics Library where it 1s necessary to insert the object ChartTime into the class structure diagram Main The properties of chartTime are set on the next picture Parameters Name Value Placeholder animation rectangle BackgroundColor USE Placeholder FILL COLOR BorderColor USE Placeholder LINE COLOR UpdateM ode AUTO UPDATE Data EVERY TimeStep TimeStep 1 Update amp tT imel true TimeWindow 100 InterpolationT ype LINEAR ScaleT ype AUTO SAME FOR ALL DATA LineWidth 1 ShowGrid false FillUnderneath false ShowLegend false Datal brojNeZarazenih ColorO Datal Colorl Type real dynamic Data Yalue brojNeZarazenih Default value not specified Parameter Data0 Color Now running the simulations we obtain the following graph 12 A Takaci 1 000 0 300 0 7 00 0 4 700 0 7 600 0 4 500 0 4 400 0 7 300 0 4 EEN ee 20 0 30 0 40 0 50 0 60 0 70 0 80 0 90 0 200 0 4 Figure 2 3 System dynamics model Next comparing the graphics obtained in the system dynamics model Figure 2 3 with the same parameters for the Agent based model Figure 2 4 one can observe the differences
27. essly and thus a similar behavior will be obtained for each set of values for the four parameter model Let us conclude that the Lotka Volterra model is not enough to present the behavior of most populations in the nature In fact additional information specific for the analysed system has to be inserted in the system The presentation of models in AnyLogic we start with the simplest model Since the AnyLogic has a good method of solving differential equations the easiest way is to define two varuables that will represent the populations of hares and lynx These two variables appear in a system of differential equations for the Lotka Volterra model given below in this chapter The variables are placed in the class of an active object in the model At the class level the following parameters are defined natalitetrsova Mathematical and Simulation Models in the AnyLogic Program 25 stoparastazeceva stopanapada efikasnostkonverzije with initial values The animation shows the number of hares and lynx both graphically and in numbers Properties nx General Image Description Class name Model Base class Parameters stoparastazeceya real morbalitetrisova real stopanapada real efikasnostkonyerzije real Implements interfaces startup code Equations d risovi dt risevi mortalitetrisova stopanapada efikasnostkonverzije zecevi djzecevi dt zecevi stoparastazeceva stopanapada risovi Additional class c
28. he generated entities to stand in a line This element is essential for defining the maximum capacity line which in our case is 1 5 Mathematical and Simulation Models in the AnyLogic Program 31 Properties a x General Replication Description Mame queue Type com anylogic lib enterprise_library Queue Parameters Name Value on Enter on Exit Preempted on Exit Timeout ont Exit on Exit queue Type FIFO capacity 15 entities To Animate all preemption false timeout False stats Enabled false animation Shape animation Type AUTO animation Forward tue One can also select the type of the queue In our case the queue is of FIFO type first in first out The fields preemption and timeout are also very important since they allow us to model a simple typical customer behavior in such systems It might happen that some clients will not want to enter and or wait in a long queue but will rather leave the system If the field preemption enrolls the true value this option will be activated in the model What will happen next with this entity can be defined so that the output from the queue element corresponding to the option preemption connects a new element from the Enterprise Library e g with the sink entity By the same principle it can also be used to represent timeout options of those customers who stood in the line but their waiting times turned out to be too long and thus they wanted to give up Th
29. iduals from a population get the disease during the time interval 0 lt lt T it is necessary that RT IT lt N To keep the number of infected under a certain value N in the same period of time we have to assume max I t lt N3 O lt t lt T The control of the epidemic in this way implies that a t should be chosen small enough in order to meet the limits given in the previous two inequalities and thereby keep costs at a minimum Note that we got a problem of dynamic programming Model parameters The first step is to determine the formulas for the infection factor a and the disease withdrawal factor b For simplicity we use only the following four parameters namely d c p and n which have a significant impact on the system and on the behavior of the epidemic d duration of illness c rate of contacts p probability of infection n total population We will use the following formula for the parameter a cp n which shows that the spread of the epidemic increases whenever the rate of contacts occuring between Mathematical and Simulation Models in the AnyLogic Program 7 people from the observed population increases and the likelihood that the disease is transferred to direct contacts between the individuals Note that the number 1 n multiplied by the number of uninfected S gives the infected part of the population The parameter b satifies the equality b d which shows that there is an inverse proportion between
30. jiva cikli ni tajmer O lokacija A rodenje Vreme Signalni dogadaj Ekponencijaino Natalitet ze eva OCekivani pojeo sam te Kreira zeca ako lokalna gusitna ivotno vek dozvoljava Uginuo obri i agenta Ris Promenljiva 2 j Nije ulovio zeca lokacija Promeni lokaciju Vreme potrebno za lov Cikli ni tajmer A ro enje Eksponencijalno Natalitet risova Kreira novog risa na istoj lokaciji Prona ao zeca Signal pojeo sam te Resetuje ovu tranziciju Vreme Animacija o ekivani ivotni vek Vreme 4 koje ris mo e da bude bez hrane Uginuo obrisi agenta Rezultati simulacije ze evi iB risovi 5 7 Predator prey model created by agents The model based agents in the AnyLogic program represents rather well the real system Now we add the following hypotheses given below Both the hares and lynxes have a certain life expectancy Their deaths are caused by age starvation and lynx attacks Hares and lynxes are aware of the two dimensional space in which they are located The density of the hare population is limited and they reproduce themselves only if there is enough space around them Lynxes hunt only in the area around them and they do that with certain frequency If the lynx does not catch a hare in the course of hunting then it changes its position in space Ifin a certain time period a lynx fails to eat a hare then it dies Next we give some elements of the
31. m the technical point of view the Lorenz s oscillator is nonlinear treedimensional and deterministic Let us remark that still today this system is used for the demontsration of a system with chaotic behavior Next we give Lorenz s system of ODEs with the unknown functions x y 1 z depending on time t A Takaci ZG ax dt t dy rx y Xxz 1 1 a y 1 1 dz xyt bz dt The physical explanation of the unknowns is as follows The function x represents the proportional speed of the air motion as a result of the convection The function y represents the value of the temperature difference between the hot air moving up and the cold air falling down The function z is the value of the vertical temperature difference in the system from down to up The parameters a b and r in 1 1 represent the essential part of the system in view of its big influence on the system The parameter a corresponds to Prandtl s number which was obtained from the basic property of the observed air its usual value is 10 The parameter b represents the observed area in the model Lorenz took for b the value 8 3 1 e 2 666 The parameter r is the Raylog s number which defines the moment when the air convection starts in the system This number whose usual value is 28 is essential in the process of change in the system from stable into chaotic Depending on the value of the parameter r the model has the following behaviors If a 10 an
32. o monitor a limited population where the number of persons in the course of the epidemic changes only slightly In general the model reffers to diseases that rarely have a fatal outcome but with minor modifications it may be included in a model which allows changes in the size of the analyzed population Kermack McKendrick model allows the inclusion of many features and characteristics of epidemic diseases by using appropriate parameters Actually it is these parameters that determine the behavior of the model provided that the values for the parameters have been carefully selected and thoroughly tested 2 2 The system of equations used to describe the epidemic The system uses parameters that contain a variety of influences that essentially affect the progress or prevent an epidemic We denote with a the group of parameters that influence positively the epidemic and with b the group of parameters that reduce the epidemic Further on with S we denote the uninfected group with the first group of 4 A Taka i infected individuals and with R the recovered ones The original system of equations of the Kermack McKendrick model see 10 is given in equation 2 1 n aSl dt d _ ast bl 2 1 dt IR br dt 2 3 Analysis of the system 2 1 From the first two equations one can conclude that the number of uninfected individuals that will become infected is proportional to the number of contacts between people from the gro
33. ode Broj zeceva 1 29275397581597 Broj risova 803 8691163858218 100 0 Figure 5 5 A linx and its prey interaction in number and the corresponding graph The following mathematical model represents a shift in terms to some finer structure of the model and better use of opportunities of the AnyLogic program in which we have a classic example of system dynamics Differential equations appearing in 5 5 are 26 A Takaci given in a somewhat different way but they are still in line with the Volterra Lotka model d Zecevi dt d Risevi dt RodjeniZec evi UginuliZecevi RodjeniRis evi UginuliRis evi RodjeniZec evi Zecevi NatalitetZ eceva UginuliZecevi GustinaZeceva Risevi RodjeniRis evi Risevi NatalitetR iseva Zecevi gustinazeceva PovrsinaOblasti 5 5 Lynx mortality was given as a table search on the basis of these data was made using the spline interpolation function Lookup Table Data Scatter x Argument Function 0 005 20 10 O 10 20 30 40 100 110 50 60 70 80 90 120 Interpolation Approximation Constant interpolation C Linear interpolation Spline interpolation Approximation Order Discrete values ie If argument is out of range Raise error Use nearest valid argument C Extrapolate Extrapolate by repeating Return custom value cme The structure is inse
34. on we obtain the following graph Mathematical and Simulation Models in the AnyLogic Program 9 Figure 2 2 A smaller number of infected due to vaccination Since the parameters in both instances were the same one can compare the simulation results Of greatest interest is the number of infected In the first example the number of infected has come to almost 400 In the model with vaccination only a small number of infected moved over 100 2 6 The agent SIR model in AnyLogic As mentioned in the previous text each model created in system dynamics can be translated into an agent based model To that end first one has to create a new model and a new class of active objects called Person In this class the state diagram should be inserted Next a state diagram describing the behavior of an agent has to be made Obviously the diagram must have three states uninfected infected and recovered The next figure shows the state diagram as created in AnyLogic z neZarazen l zarazen A When a diagram is drawn it should set the conditions for transitions between states transition between uninfected and infected states will make over the signal events If an uninfected agent gets the signal contact which means that there has been contact with an infected agent then he switches to the state infected Internal transition from the state infected with the ability of an infected agent sends a signal
35. rted and two timers that are used for insertion of lynx and hares into the system during the execution model For each timer is set to operate in manual mode In executing a timer UbaciZeceve execute the Java command Rabbits 1000 Timer UbaciRiseve executed Lynx 50 In the active object class there are three real parameters namely povrsina oblasti area of the region NatalitetZeceva natality of the hares and NatalitetRiseva natality of the lynx with default values given below Mathematical and Simulation Models in the AnyLogic Program 27 ibacizeceve Wi baciniseve a jenizecevi Z kodjeniRisevi gustinazeceva m a 7 d reme u go d ina m a 1 1 i Rovrsina oblast Nataltat seva Sd Natalitet zeceva 5333 33 4666 67 gt 4000 0 3333 33 2666 67 rA 2000 0 1333 33 gt Ubaci 1000 zeceva Ubaci 50 riseva 666 67 jy 0 0 Figure 5 6 Another linx and prey interaction in numbers The animation is in this model more complex Moreover in order to display the number of individuals for the hare and lynx numbers and by using diagrams interactive forms have been inserted Two command buttons are connected to timers in the diagram structures that increase the number of individuals Three sliders are linked to parameters povrsinaoblasti NatalitetZeceva and NatlitetRisova Using the slider in the conduct we can change the values for these parameters 28 A Takaci Zec Promeni
36. up S and group both under the assumption that the number of contacts depends only on the number of persons in each group 1 e there is a uniform mixing in the population The third equation is based on the assumption that the number of recovered persons is proportional to the number of infected ones which represents an average time that people spend in a state of infection until you cross the state when they can neither transmit the infection nor can they become infected The assumption that there is a constant number of persons in the population implies the following simple equation S t 1 t R t N t where N N t is the total number of the population Note that the rate of increase of infected individuals is given by R a b Firstly we start with the equation S O 1 0 N 0 Of course there must be at least a few infected people 1 e O0 gt 0 We are interested only in those solutions where S Z and R are non negative which means that it holds dS FA lt 0 when there are people who are infected and those that were infected Since S t decreases as time goes by S t lt S lt SC for t gt t gt 0 In other words the value of S is constantly decreasing and must be nonnegative meaning that when t S must have a threshold which may be Q 1 e S lim S t dI S From the second equation we have mA lt 0 privided that aS lt b Since S decreases as time passes it follows that aS O
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