Applying Inhomogeneous Probabilistic Cellular Automata Rules on Epidemic Model

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1 Alying Inomogeneous Probabilistic Cellular Automata Rules on Eidemic Model Wesam M. Elsayed 1 Deartment of Matematics, Faculty of Science,Mansoura University, Mansoura, 35516, Egyt. Mansoura, Egyt. Amed H. El-bassiouny 2 Deartment of Matematics, Faculty of Science,Mansoura University, Mansoura,35516, Egyt. Mansoura, Egyt. Elsayed F. Radwan 3 Deartment of Comuter Science, Faculty of Comuter and Infomation Sciences Mansoura Universty, Egyt, P.O. Box: Abstract Tis aer resents some of te results of our robabilistic cellular automaton (PCA) based eidemic model. It is sown tat PCA erforms better tan deterministic ones. We consider two ossible ways of interaction tat relies on a twoway slit rules eiter orizontal or vertical interaction wit 2 different robabilities causing more of te best ossible coices for te beavior of te disease. Our results are a generalization of tat Hawkins et al done. Keywords-Probabilistic Cellular Automata (PCA); Eidemic modeling; Otimization. I. INTRODUCTION Because of te sread of diseases, a tecnical innovative model sould be made to recover teir time regions. Many researces tried to solve tis roblem based on medical disease feature, wic suffer from unredictable ones. Wilst a single infected ost migt not be significant, a disease tat sreads troug a large oulation yields serious ealt and economic treats. In tis sense, matematical eidemiology is concerned wit modeling te sread of infectious disease in a oulation [see 2]. Te aim is generally to understand te time course of te disease wit te goal of controlling its sread. Traditionally, te majority of existing matematical models to simulate eidemics are based on ordinary differential equations. Tese models ave serious drawbacks in tat tey neglect te local caracteristics of te sreading rocess and tey do not include variable suscetibility of individuals. Secifically, tey fail to simulate in a roer way (1) te individual contact rocesses, (2) te effects of individual beavior, (3) te satial asects of te eidemic sreading, and (4) te effects of mixing atterns of te individuals. Matematical modeling in eidemiology was ioneered by Bernoulli in Neverteless, te work due to Kermack- McKendrick [7] can be considered as te starting oint for te design of modern matematical models. It consists of a SIR model (Suscetible-Infected-Recovered, SIR) wic is a set of Ordinary Differential Equations. Otimization metods ave been develoed for deterministic simulation models. However, tey ave not been done wit more comlex stocastic simulation models. For more details look in [2, 4, 5, 12]. Influenza is transmitted in a comlex way from erson to erson. In addition, given an introduction of influenza into a oulation, te robability of a major eidemic and te ossible size of an eidemic are igly variable. Tus, te matematical models for influenza eidemics sould ave a detailed contact structure and be stocastic. Besides, te eidemic rocess is non-linear since te incidence of new infections deends on te current number of bot infectives and suscetibles in te oulation at a articular time. All of tese factors make otimization based on traditional gradient metods, suc as te Newton Rason metod, difficult or even roibitive. [8] Develoed a stocastic aroximation metod wose convergence is guaranteed under mild conditions. Te metod, owever, requires knowledge of te analytic gradient of te considered objective function [11]. However, in terms of simulation otimization, te drawback of tese metods remains te unavailability of gradients. Wen trying to devise more realistic models we incororate satial arameters to better reflect te eterogeneous environment found in nature. An alternative to using deterministic differential equations is to use a two-dimensional cellular automaton (CA) to model location secific caracteristics of te suscetible oulation togeter wit stocastic arameters wic catures te robabilistic nature of disease transmission. Cellular Automata model sould be treated as a dynamical system tat involves a random variable. As a result, te suggested model sould be a stocastic dynamical model called robabilistic Cellular Automata model (PCA) wic is an extension of CA. Te state sace remains te same as well as te local and syncronous caracter of te dynamic. Te novelty is tat eac site udates its value randomly according to a robability distribution wic deends on te neigboring sites. Also, we will aly non-uniform cellular automata were eac cell satisfies te same or different rules according to wic cell states are udated in a syncronous and local manner. Usually, wen a CA-based model is considered to simulate an eidemic sreading, individuals are assumed to be distributed in te cellular sace suc tat eac cell stands for an individual of te oulation as in our case. In te case of our stocastic two-dimensional non uniform cellular automata model, we consider two ossible ways of interaction so, we ave two directions to get infected from oter cells eiter orizontal or vertical interaction as eac cell stands for an individual. Tis will reduce te ossibility to get infection from oters as in every direction tere are only two neigbors affecting te central cell in every direction as seen in following sections. Stocastic otimization metods will 39 P a g e

2 be used to model arameters for infectious diseases and oulation structures. Also, we aend every rule wit two factors namely certainty and coverage factors to sow te nonuniform way were eac cell affects anoter in it's neigborood and vice versa. Tis metod of two-way slit rules ave te way for more good coices of solutions as seen in sectioniv art B. In tis work we consider ow we can model te sread of an eidemic using a two-dimensional cellular automaton using rogramming features of te matrix algebra ackage MATLAB to develo an imlementation tat simulates a simle model. By adding our new robabilities of getting infection eiter from orizontal or vertical motion to te rogram mentioned in [3]. Also, te effect of oulation vaccination can be considered in tis model. A vaccination arameter V [0,1] must be considered in te objective function of te model. Suc arameter stands for te ortion of suscetible infected individuals at eac time ste wic are vaccinated. But in te rogram, te user ave te ability to vaccinate secific regions of te environment. Tis takes inut from te user wile te simulation is running, allowing vaccination strategies to be tested. II. PRELIMINARIES A. Te Probabilistic Cellular Automata Model (PCA) In tis section, we describe te overall design of our automaton. Te model is a two-dimensional grid of cells, eac cell containing te same or different rules due to te fact tat eac eidemic can be in any one of four stages as in te SEIR model we introduced later (i.e. Suscetible, Exosed, Infected and Recovered). Te suscetible individuals are tose caable of contracting te disease; For many infections tere is a eriod of time during wic te individual as been infected but is not yet infectious imself; during tis latent eiod te individual is said to be exosed. In tis case we ave te SEIR model in wic te new class of exosed individuals (E) must be considered. Te infectious individuals are tose caable of sreading te disease; and te recovered individuals are tose immune from te disease, eiter died from te disase, or aving recovered, are definitely immune to it. So, different diagnosis of disease as different rules. Te neigbors of any cell in te grid are te cell itself lus te four ortogonal adjacent cells (fig. 1) according to von Neumann. Fig. 1. Te two-dimensional cellular automata cell's neigbors Once te automaton was embedded in te grid, te cell as an individual finite state macine, began to follow te rule tat is alied to it. A single cell cannot do muc witout interacting wit oter cells. Te lattice starts out wit an initial configuraion of cells in wic te following old. Eac cell can take on one of te 4 different states: 1) Suscetible (i.e. ealty; designated as 0) 2) Exosed (designated as 1) 3) Infected (designated as -1) 4) Recovered (includes bot survivors and deats; designated as 2) It is suosed tat te way of infection is te contact between te infected individual and te ealty individual. Once te ealty individuals ave contracted te infection and ave recovered from it, tey acquire temorary immunity. B. Matematical Formulation Of Te PCA Two-dimensional CA are discrete dynamical systems formed by a finite number of identical objects called cells, wic are arranged uniformly in a two-dimensional sace. Eac cell is endowed wit a state, belonging to a finite state set, tat canges at every discrete ste of time according to a rule, called local transition function. More recisely, a CA can be defined as a 4-ulet, A = (C, S, N, f), were C is te cellular sace formed by a two-dimensional array of r c cells,were r stands for rows and c for columns (see Fig. 2-(a)): C = {(a, b), 1 a r, 1 b c}. Te state of eac cell is an element of a finite set, S, in suc a way tat te state of te cell (a, b) at time t is denoted by Sa, b; ts. Te matrix C ( t) Si, j; t is called configuration of te CA at time t. Moreover, C(0) is te Fig. 2. (a) Cellular sace (b) Von Neumann neigborood (c) Moore neigborood initial configuration of te CA. Te neigborood of a cell(a,b) is te set of cells wose states at time t determine te state of te cell (a, b) at time t + 1, by means of te local trasition function. Deending on te rocess to be modeled,one can coose an aroriate neigborood. Neverteless, te traditional neigboroods considered are te Von Neumann neigborood (see Fig. 2-(b)), and te Moore neigborood (see Fig. 2-(c)). Note tat te main cell is also considered in its neigborood. A neigborood is defined by means of a finite set of indices x y :1 i m N i, i N: Neigborood of a cell c, suc tat for every cell (a, b), its neigborood, N b a, is te set of m cells given by N a x b y a x b y x y N, a b 1 1,..., m, m : k, k, Note tat for Von Neumann neigborood, we ave N 0,0, 1,0, 0,1, 1,0, 0, 1. Cell's state [10] :. 40 P a g e

3 0, 1,1,2 (1) Sc t1 f :S (t 1) S (t) 1 (2) c c Were S c : State of te cell at iteration t 1 f : Transition function (Determines ow te cell s state can cange). As te cellular sace is considered to be finite, boundary coditions must be taken into account in order to asure te well-defined dynamics of te CA. C. Probabilistic Analysis of Cellular Automata Rules Wit eac time ste, te state of eac individual cell canges according to a set of rules based on te states of te cell s neigbors. Tese rules can be eiter deterministic (certain) or stocastic (robability-based/uncertain). To determine tis exactly we need to define two essential factors, if te decision rules ave te form C D, meaning IF C THEN D, were C is te condition attribute, and D is te decision attribute of te decision rule ten we ave, Te certainty factor of te decision rule C D, denoted by cer ( C,, is defined as : cer( C, ( D/ C) (3) Were cer ( C, [0,1 ]. If cer ( C, 1, ten te given decision rule is a deterministic or certain decision rule. Oterwise, 0 cer ( C, 1, te given decision rule is a stocastic or uncertain decision rule. Te coverage factor of te decision rule C D in S, denoted by cov( C, D ), is defined as: cov( C, ( C/ (4) In tis work we consider a non-uniform case of cellular a tomata were different cells may contain different rules. At a given moment, only one rule is active for a cell and determines te cell's function. A non-active rules may be activated in next time stes. Let any cell in te lattice be labeled by its osition c i, j were i and j are te row and column indices. A function S t c is te state of cell c at time ste t. Te rules of te model secify ow te state S c t1 is to be comuted from te states at time ste t of it's neigbors ( i 1, j),( i 1, j),( i, j 1),( i, j 1) as in fig. 3: Fig. 3. Parameters of te transition rules. Here we assume tat tere are two ossible structures eiter orizontal or vertical interaction between cells eac of tem as different robability of infection resectively. A small robability of sontaneous infection is assumed. Tis reresents te ossibility of external infection e.g. infection due to traveling or imorted objects [1]. In te orizontal case, te state of cell i, j deends only on te states of cells i, j1, i, j1 wile during vertical i 1, j, i1, j interaction it's state deends on states of cells. So, te automata rules must be formalized for te two directions as follows: 5) For Horizontal interaction (wit robability of infection ) : a) If Si t0 and Si, j1; t 1 or Si, j1; t 1 ten Si t1 1 wit robability (5).. (7). b) If Si t 1 ten Si t1 2 c) If Si t 1 d) Si t1 1 (6) ten Si t1 2 (indeendent of it's interaction neigbors) wit robability (8). 6) For Vertical interaction (wit robability of infection v ) : 41 P a g e

4 a) If Si t 0 and Si, j1; t 1 or Si, j1; t 1 ten Si t1 1 wit robability v (9). b) If Si t 1 ten Si t1 2 (10). c) If Si t 1 ten Si t1 2 (11). d) Si t1 1 (indeendent of it's interaction neigbors) wit robability (12). If a cell is suscetible (0), te simulation module counts te number of infected cells tat are its nearest neigbors. Te simulation ten calculates te robability te cell can avoid being infected 1 m,were β is te infection rate (te robability a contact wit an infected erson is actually infectious) and m is te number of infected cells. Subtracting tis value from 1 yields te robability of te cell being infected [10]. III. THE OPTIMIZATION PROBLEM Te otimization roblem is as follows: Given a limited quantity of influenza vaccine and a articular oulation structure and infection rate attern for a single wave of andemic influenza, wat roortion of eac stage sould be vaccinated to minimize te imact of te eidemic? [ see :9] We divide te oulation into four different cases: Suscetible, Exosed, Infected and Recovered tat are indexed as i 1,.., 4.We let ni be te number of individuals in 4 n class i n i, and te total oulation size is i1. We let be te infection rate, i.e. (te robability a contact wit an infected erson is actually infectious), were eiter takes te value or V deending on te movement direction. We let V be te total number of vaccine doses available before day one of te eidemic, and i te roortion of individuals in case i tat is vaccinated. Tus, te total number of doses distributed is, 4 Q ni i i1 Q V were, since we cannot use more vaccine tan tere is available. We assume tat eac erson vaccinated receives one dose of vaccine. Four different values of te vaccination rate are considered: V = 0, 0.2, 0.3, 0.4. Note tat as V increases, te number of infected individual decreases. To reflect te imact w of a single illness, we let i be te weigt assigned to an illness in eac case for minimization of te loss function. Ten, te otimization roblem is exressed as 13 4 min n i w i i 1 Suc tat 4 n i i V i 1 We concentrate on minimizing overall illness in te oulation as well as number of lives lost given a redetermined number of doses V of vaccine. We use weigts of one, w 1, i 1,..., 4 i.e. i for minimizing illness objective function. Te rules ave been alied based on an otimization algoritm as illustrated in Fig. 4 sowing te beavior of te eidemic troug different robabilities. Pseudocode Overall structure Main function Get information from user and initialize variablesvisualise state Loo troug generations: 1) Udate count of infected neigbours for eac cell 2) Udate state of eac cell based on number of infected neigbours 3) Visualize state 4) Sto In tis work we consider ow we can model te sread of an eidemic using a two-dimensional cellular automaton. We will use rogramming features of te matrix algebra ackage MATLAB to develo an imlementation tat simulates a simle model [3] as sown in fig.4. We want to construct a model of an eidemic tat will develo over a fixed N by N grid in a given number of generations. We aly a set of rules to eac cell tat will determine its fate in te next generation, for examle weter a given cell will become infected or not. Te robabilities of state canges are a set of redefined arameters. By studying te effects of varying tese arameters we try to redict ow te eidemic will develo over time will it eventually die out. We aly te same strategy of vaccination used by [3] were te user ave te ability to vaccinate secific regions of te environment. Tis takes inut from te user wile te simulation is running, allowing vaccination strategies to be tested. Since vaccination is not usually ermanent but lasts for a eriod of time, it is imlemented in muc te same way as te temorary immunity tat follows organism recovery. Wile te rogram is running, te user can click on te grid causing vaccination to be simulated at te clicked oint P a g e

5 Pseudo code for main rogram Initialize constants to reresent cell states (Suscetible, Exosed, Infected, and Recovered) Inut N, te array size to use Inut and v, te robabilities of becoming infected during orizontal and vertical interaction Inut q, te robability of becoming recovered. Otionally inut ca_state, te state matrix, from given initial configuration If no initial configuration was sulied: Initialize ca_state wit all cells Suscetible Set randomly-selected cells near centre to Infected Add a border of immune cells to ca_state Visualize ca_state Initialize infected_neigbours, te infected neigbour count matrix Loo troug generations y) N+1) Loo troug valid x-coordinates (2 to N+1) Loo troug valid y-coordinates (2 to N+1) Udate count of infected_neigbours(x, End loo troug y-coordinates End loo troug x-coordinates Loo troug valid x-coordinates (2 to N+1) Loo troug valid y-coordinates (2 to Udate ca_state(x, y) based on infected_neigbours(x, y) End loo troug y-coordinates End loo troug x-coordinates Visualize ca_state End loo troug generations even wit fixed, v and q ) it is necessary to run several simulations to obtain an aroximate understanding of wat sould aen to te sread of te eidemic. We also need to test several different values of and v. Initially, we will set q as a constant 1 and vary, v. Tus an Infected cell will immediately become Recovered. We will test for, v 0.1,0.3,0.5,0.7,0.9. Case 1: 0.1, {0.1,0.3,0.5,0.7,0.9 }. v Case2: 0.3, {0.1, 0.3, 0.5, 0.7}. v Fig. 4. Pseudo code for main rogram IV. SIMULATION AND ANALYSIS A. Different scenarios based on varying -orizontal ( ) and -vertical ( v ) Because our cellular automaton is robabilistic (i.e. random numbers affect te cances of different scenarios arising, Case 3: 0.5, {0.1,0.3,0.5 }. v 43 P a g e

6 Case 4: 0.7, {0.1,0.3}. v B. We run te simulations for 100 generations and observe te beavior of te disease in a grid of size TABLE I. Descrition of observed beaviour for a range of values of v, and q (green is Suscetible, red is Infected, and black is Recovered) q , v 0.1, 0.1 raidly. Almost all cells Infected all cells Suscetible witin arox 66 generations. cells Suscetible witin arox 5 generations. 0.1, 0.3 raidly. Almost all cells Infected all cells Suscetible witin arox 16 generations. cells Suscetible witin arox 4 generations. 44 P a g e

7 0.1, 0.5 raidly. Almost all cells Infected all cells Suscetible witin arox 30 generations. cells Suscetible witin arox 4 generations 0.1, 0.7 raidly. Almost all cells Infected all cells Suscetible witin arox 66 generations cells Suscetible witin arox 9 generations 0.1, 0.9 raidly. Almost all cells Infected all cells Suscetible witin arox 14 generations cells Suscetible witin arox 9 generations 0.3, 0.3 Sreads more quickly. Sreads slowly. Sreads more slowly. 0.3, 0.5 Sreads more quickly. Sreads raidly. Sreads more slowly. 45 P a g e

8 0.3, 0.7 Sreads more quickly. Sreads raidly. Sreads more slowly. 0.5, 0.5 Sreads more quickly. Sreads raidly. Sreads slowly. Note tat: te figures sown in table1 will be a bit different every time we execute te rogram as te vaccine is given randomly by eac user. V. CONCLUSION Tese eidemic scenarios resented above rovide an oortunity to demonstrate te visualization caabilities of a graical CA model. Wen [3] use a single robability of infection, tey got only two results in wic all cells are in suscetible state. But, ere wen dividing te cellular sace into two directions of motion wit two different robabilities of infection, tere are more otional values for, v differ at wic generation it is obtained as sown above in table1. We introduced a teoretical model to simulate te sreading of an eidemic. It is based on transferring te roblem into arametric one and te rules into restrictions, obtaining te otimization roblem (14). It is solved wit te cosen values of, v and q get te minimum value of te function wic minimize te imact of te eidemic. VI. FUTURE WORK PCA are lattice model of satially extended systems wit robabilistic local dynamical rules of evolution. It is difficult to analyze rigorously, so te comutational simulation rovide an alternative tool. So te future work is to develo an iterative Probabilistic Neural Networks wit fully arallel robabilistic feedback dynamic. In addition, Parallel Genetic Algoritms can be incororated by modifying te robabilities. PGA searc troug te sace of arameters to calibrate te model to observe data. Wilst, PNN syntesis of aroaces based on rior analysis and contextual information. ACKNOWLEDGMENT We tank Prof. E. Amed, Matematics Det., Faculty of Science, Mansoura University, Egyt, for is comments. Also, secial tanks for MATLAB Develoers, were matrix algebra ackage is used for imlementing tat simulation model. REFERENCES [1] Amed E., Agiza H.N., (1998) On modeling eidemics. Including latency, incubation and variable suscetibility, Pysica A [2] Anderson, R., May, R., Infectious Diseases of Humans:Dynamics and Control. Oxford University Press, New York. [3] Andrew Hawkins, Danny Roff, Adam Gundry, Cellular Automata and Satial Eidemics. November 25. tt://citeseerx.ist.su.edu/viewdoc/download?doi= &re =re1&tye=df. [4] Greenalg, D., Control of an eidemic sreading in a eterogeneously mixing oulation. Mat. Biosci. 80, [5] Hetcote, H., Waltman, P., Otimal vaccine scedules in deterministic eidemic models. Mat. Biosci. 18, [6] Hoya Wite S., Martı n del Rey A., Rodrı guez Sa ncez G., (2007). Modeling eidemics using cellular automata. Alied Matematics and Comutation [7] Kermack W.O., McKendrick A.G., Proc. Roy. Soc. Edin. (1927) A [8] Robbins, H., Munro, S., A stocastic aroximation metod. Ann. Mat. Statist. 22, [9] Patel Rajan, Ira M. Longini Jr., M. Elizabet Halloran,(2005) Finding otimal vaccination strategies for andemic influenza using genetic algoritms. Journal of Teoretical Biology [10] Wang Jeffrey B., Scool s Out? Designing Eidemic Containment Strategies wit a satial stocastic metod. tt:// [11] Weisstein, E., Robbins munro stocastic aroximation.matworld;tt://matworld.wolfram.com/robbins MonroStocasticAroximation.tml. [12] Wickwire, K.H., Guest, D., Otimal control olicies for reducing te maximum source of a closed eidemic. Mat. Biosci. 32, AUTHORS PROFILE Wesam Elsayed, Deartment of Matematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egyt 46 P a g e

9 address: Amed El-bassiouny, received is B.Sc. and M.Sc. from Matematics Det. Faculty of Science, Mansoura University, Egyt. He received is P.D. from University of Kiev, Soviet Union. He became a Lecturer at Matematics Det., Faculty of Science, Mansoura University, Egyt on address: el_bassiouny@mans.edu.eg Elsayed Radwan, received is B.Sc. and M.Sc. from Matematics Det. Faculty of Science, Mansoura University, Egyt. He received is P.D. from Intelligent Lab., Graduate Scool of Engineering, Toin University of Yokoama, Jaan. He became a Lecturer at Comuter Science Det., Faculty of Comuter and Information Sciences, Mansoura University, Egyt on His areas of interest are Soft Comuting, Evolutionary Comuting, Pattern Recognition, and Cellular Neural Networks. address: elsfradwan@yaoo.com 47 P a g e