- Email: [email protected]
Modeling and Control of the Infection Processes in Industrial Cattlebreeding
Copyright © IFAC Mathematical and Control Applications in Agriculture and Horticulture, Hannover, Germany, 1997
MODELING AND CONTROL OF THE INFECTION PROCESSES IN INDUSTRIAL CATTLEBREEDING. KLESTOVA Z.E. (Ukraine), Ph.D. Department of genetic resistance investigation by infectious diseases. The Institute of Veterinary Medicine, Kiev 151 , Ukrainian Agricultural Academy of Sciences Donetskaja st. 30 MAKARENKO A.S. (Ukraine), Dr. of Sei. Department of mathematical methods of system analysis, the faculty of applied mathematics National Technical University of Ukraine (KPI) 37, Pobeda av., NTUU (KPI), Dept. MMSA (1920)E-mail: [email protected]
1. Introduction. The questions of productive cattelebreeding in particular pigbreeding are very important as for population providing of agricultural production, as for profitable conducting business in agroindustry. On productivity influence many factors, as traditions in productions, natural conditions, feed and other. In present report we concentrated on own problem aspect, which is connected with questions on losses from infective diseases. At practice even in this questions by force of big scales of modern industry of modern industry and many influenced factors preliminarily is difficult to receive some quantitative results. So much is difficult to value the scales of epizooties (mass infection disease of cattle on productive animal farms) and measures on their optimal overcoming. Certainly, the veterinary services have definite recommendation and working plans, but often by meeting a new disease agents and absent larg quantity a new data and can come catastrophically consequences before that these data will be received and will cultured. In such situation perhaps the only acceptable way is mathematical modelling and search optimal on expenditure decisions with model help. By this turn out, that in area of spreading different biological factors exist posing mathematical problem, which are more or less similar. This is epizooties spreading, spreading of plant diseases, spreading of parasites, substitution of own spices of flora and fauna on others etc. It must be take into consideration, that each branch of agroindustry , big animal farms, separate farms etc. have own peculiarity: geographical, technological, economical, cultural, that lead to different their reaction on the same initial influence. That demand to creation "the passport of the object" in connection with such questions. This "passport" must contain different factors - susceptibility of individual animal or plant to ecological, climatic, season, genetic factors . The epizootic and immune problems are one of the limiting factors for cattlebreeding, especially for post-Chernobyl situation in Ukraine. In this brief communication there are firstly described methodological basis for considering the problems above. Than discussed some approaches for their solutions as traditional as new and some realisations. There are also some results on concrete problems for Ukraine condition.
2. Some methodological remarks for considering infection spreading in cattlebreeding. Considering the problems of infection in the industrial cattlebreeding branch it may be useful to take in mind some approximate scheme of region (see Fig. I) : Infection
Economics
Ecology
Other agriculture Branch
Fig. 1
213
Analyses display that separate blocks on the scheme also consists from many subblocks. For example the ecology have influence as on the genotype of animals as on the spreading and survive of viruses. Also the economic block influences on infection block by transportation bounds for example. By considering separate subblocks it is very important to distinguish different space scales and corresponding time scales. In all levels there are specific immunological problems with own mathematical models and approaches. 1. Cell level. On such level important problems are the interaction between viruses and individual cell, mathematical modelling of external factor influence, the effect of drug and so on. Traditionally such problems are solved with the aid of ordinary differential equations (sf. for example [1]). Recently it is found that for prognozing in such problems well adjusted are neuronet methods [2]. On such level we investigated the influence of some specific serum and antiviruses prepares on infected cells [3]. 2. Separate animal or plant. This level was researched by traditional immunology with ordinary differential equations or discrete equations. Yet on this level it may exist the problems on the organism considered as cell union. 3. Separate cattlebreeding complex with many animals. The simplest model for such problems are the nets models for connected discrete or ordinary differential equations [1 ,4] . The Marcov chains also are useful for problems above [5] . The simplest model have the fonn (on analogy with [2,4]), t - time : dn dm ----- = F (n,m) ; ----- = F (n,m) dt I dt 2 where n - the number of health animals, m - number of ill animals. There use some successes in applying such models. But such description became unreal when there are a lot of animals. In such case the aggregation or reducing methods are need [6,7]. If the animals contact stochastically and frequently than it passage to the description with partial differential equations for the density of deceased animals [8]. 4. Geographical region. For such problems supposed that there are many cattlebreeding complexes connected by many bounds. Than it introduced the density U of infected, deceased ,reconvalescent animals and posed partial differential equations of diffusive type: U=DU t
+f(U) xx
Remark, that under the account of memory and nonlocality effects there obtained more correct equations of hyperbolic type [8]. Further the partial differential equations are investigated on the existence of travel waves of epidemic, on
selfoscillation of decease density, on spiral waves etc. by synergetic methods. The characteristic lengths on such level are 50-100 km (geographic region), 100-300 km (province), 300 - 1000 km (country or union of countries). 5. Global (World level). The tools for modelling described problems on this level are the partial differential equations or more aggregated models of ordinary differential equation type for small number of macroparameters (like consideration in global models by lForrester and D.Meadous). Already with described above models there are interesting results. Them have practical importance for planning antiepizooties and antiepidemical action. But there arise usually difficulties in such approaches in mathematical modelling - the search of correct equations for problem description, the sufficient number of equations, definition of coefficients in equations and transition probabilities and so on. In addition there arise also the problems of different risks calculation and the uncompletion of infonnation ( the range for fuzzy sets methods) etc. All above problems lay more or less in the traditional ranges for mathematical modelling specialist (as also for authors of report). Analyses of field of research with the invitation the specialist on immunology, virology, veterinary medicine and genetics and the experience in recent infonnation methods allow to propose entirely new type of mathematical models for the many problems in industrial cattlebreeding. New models are founded on the basic principles described for global soci
3. Immunological models with the associative memory property. The Immunological problems are the subproblems for the global environmental, biological, social and economical brunches. Let us describe the very simple but typical problems. There are many elements (animals in the level of unique catteelbreeding complex or the net of catteelbreeding complexes in the country, or the set of countries) as typical fields for infections. There are some descriptions of element - the parameter sri] for animal for example or some vector of parameters S{i} = {sI , s2, ... sk} , i = I, ...,
214
N, where N is the number of elements. For example for unique animal we can take s[i] = 1 if the animal is immune (after ill for example), s[i] = 0 for healthy animal and s[i] = -1 for ill animal which can spread the infection. There are the bounds between elements. The animals as element have contacts between themselves (usually stochastically in time and space). But for simplicity we can take this contacts as deterministic. The bounds between the complexes may be constant and described the transportation bounds, or tropical chains for infection spreading and so on. Also for the countries as elements the bounds are transportation ways and global tropical chains. Of course, there may be the part of stochastically also in the bounds. Than we may accept that in average the infection are the results of the surrounding influence. Now let us formulate the principles for the dynamically laws for evolution of element states. For this goal its need to consider the series of more and more complex models. Suppose firstly that there is no season and climatic changes and external conditions are constant (temperature, transportation's conditions, number of animals, the type and force of viruses). Suppose also that there isn't immunity to viruses. As adoption consider that the epizootic is deterministic and the same initial conditions (that is the number place of ill animal cause the same epizootic). For simplicity we can consider the discrete moments of time: 0 , t, 2t, 3t, .. . , where t - time step. So the first models may be discrete. Than in any moment of discrete time the elements form some space picture in geographical region with some distribution of ill, healthy and immune and so on animals. Let us call this space pattern as "space pattern of epizootic" (SPE). Father, the series of "space patterns of epizootic" in successive moments of time form the "space- time patterns of epizootic" (STPE). Remark that all notions and models also is suitable for another immunilogical problems (for example for epidemic for human). As the system can have in general the complex behaviour we may suppose the nonlinearity in the nature of problems. So different condition can causes the new STPE. In STPE the elements are connected by space bounds in the same moment of time and by bounds between elements from different SPE. If we take the space and time coordinates as equal in rights than in space- time dimensions we can consider the unique epizootic as one pattern. This pattern is the pattern of element states in all elements space points and in all discrete moments of time. The connections J{ij,k,l} between s[i,k] and s[j,l] (the states of ij elements in the moments k and 1) describe the influence on i- th element in moment k by j element in moment l. If there isn't internal dynamics of elements than the elements behaviour determinate by the external influence by another elements. This influence may be described by mean influence field acting on i-th element. This field h[i,k] (in moment k) is equal to h[i,k] = Sum at ion { J{ij,k,l}
* s[j'!]} (1) i notj, (1)
Than in the simplest variant we can suppose, that the dynamics of i-th element is describes (on analogy with our models [12]) by the law s[i,k,m+ 1] = s[i,k,m] + I if h[i,k,m] > 0 & s[i,k,m] <1 s[i,k,m+ 1] = s[i,k,m] - 1 if h[i,k,m] < 0 & s[i,k,m] > -1 (2) s[i,k,m+l] = s[i,k,m] else In (2) m is some fictive evolutionary parameter and we adopt the dependence s[i,k] and h[i,k] from them. The results of evolution by the law (2) is some stable in space- time dimension STPE. Thus s[i,k] consist the real epizootic pattern. Such algorithm starts when we give arbitrary condition in the sense that we pose the all elements states in all moments of time. If it needs to know the stream of epizootic in time we have only initial condition in first discrete moment oftime (or in some close moments of time). But if we continue the initial conditions in some manner (say by supposing s[i,k] = 0 for k bigger than some kO), than we have precisely the problem above. Of course we can take into consideration also internal dynamics of element i by including to the summation in (1) the case i=j with bounds J{i,i,k,l} (selfinteraction or unique element ill stream). Remark for the topics below that described models may served as the tool for prognoses in the problems with time series. There are some interesting considerations about the bounds J[ij,k,l]. For the goals of dynamics of model we must know there values. There are two main approaches to such problem. In first remark that the epizootic have more or less similar streams. The bounds J[ij,k,l] was created historically and naturally in long period of time. The STPE is stable object and from many initial conditions the space- time distribution tends to the real STPE. So the STPE is the attractor of epizootic problem. The system of elements goes to some attractor and form the real epizootic pattern. This is strict analogy with associative memory type behaviour in the pattern recognition and neuronet methods [13].So the formulas for the evaluation J[ij,k,l] may by familiar with formulas in neuronet models. In the simplest case we can take the formula
215
J[iJ.k,I]
=
Sum a t ion { s[i,k.,p]
* sU,l,p] } (3),
p= 1,2, ... . P where P - number of training patterns (epizootic). In second approach the values of J[ij,k.,l] are evaluated by experts. In both causes if we know bounds we can calculate the epizootic pattern by (1),(2). There are useful to make some remarks. First, if external conditions are changed with time it is easy to take into account this fact by applying STPE in different seasons as different learning epizootic patterns. Second, we can take the bounds which depend on fictive parameter m. (The nonconstant character of bounds on real time already accounting in model). And last remark. How do we must build model for new viruses epizootic? At first its seems that this class of models are insufficient because the situations are also new and there are changes in presumable STPE. Naturally there are some changes but there are very many implicit invariant in the problem. First of all we can apply the expert knowledge on new epizootic. Although at the beginning of epizootic we have only few date on decease the experts can suppose some values for new J[i,i.k,l] (for the element evolution) or calculate by some more simplest models. Next the epizootic has usually the nidus character with some points of initiation epizootic. So the data from this initial nidus may be utilise by experts for evaluation J[ij,k,l] outside the nidus. There also may exist many modifications and complications of the model. Remark only that the model for STPE remember in many features the usual perception [13] for pattern recognition. Also for the epizootic flow prediction may be find by the traditional neuronet methods for the time series of date prediction. For such goal some first SPE was considered as training patters for the neuronet considering them as input data and prognosing SPE as output data with usual beckpropagation algorithm for learning. But the models above are more general because they can in principle take into account the history of previous epizootic.
4. The economic and control problems. In the subdivision of this report below it had discussed some methodology for describing and modelling the epizootic and some new types models. This models can constitute the basis for the considering the cost optimisation problems and stopping epizootic problem. As to the model in subdivision 2 the standard models may be applied for optimisation. Remark that there are a lot publications on such subject. What is need that to adjust the methods for the models. But the difficulties with models transform in difficulties with optimisation methods. The neuronet type models give new possibility for solution economical and control problems [9,11] . Its follows from the models that we can easy calculate the loss from the diseases because we know precisely the real part of ill animals (of infopcate cattelebreeding complexes ets). So we can calculate the cost of given STPE. Next the epizootic is only the subpattern in global models of socio- economical processes [9,10],so we can create a global measure of STPE evaluation. If we consider STPE step by step in real time than we can say about the "trajectory of epizootic" (that is about series of space patterns). So the optimisation consist in searching the "optimal trajectory of epizootic". Of course optimality must understand some qualitative criteria. The control of epizootic consist at first step in searching some measures. There are a lot of controlling influences: the applications of drug, vaccination, isolation of ill animals and complexes, reducing the transport contacts and so on. Such affairs have different cost and there are interesting problem to minimise cost of lost from epizootic. What is surprising that such control is easily described and displayed in proposed models. For example the drug application changes the J[ij,k.,l]. The isolation of animals make J[ij,k,l] = 0 for all j and i from some P (P - the set of isolating animals). The vaccination and others medical actions also change J (and in more developed models change the states s[i,k]). As to the situation in Ukraine there are a lot of date and the problems of local and global characters. First problem is the epizootic on the scale of all Ukraine and on international scale. Now it is well known that in Europe there exist the spreading of animal decease - the pig plague, the cow hydrophobia and so on. Thus we have to calculate the spreading of infection. Also there are many investigations on real situations of the cattelbreeding complexes in Ukraine and [14] on the influence of some prepares on infection in the cell level [3]. The new area consist investigations of date on the subject of reports and connected dates in the scale of Ukraine on the basis of the geoinformatical systems [15].
5. Resume. Thus in proposed report there as are some methodological approaches for the epizootic and immunology problems in agriculture as some new models for their modelling and solution. Proposed models allow to pose and solve the problem of calculation the cost of epizootic and optimisation the epizootic cancelling.
216
UTERATURE 1. Marchuk G.I. Mathematical immunology. M.:Nauka,1992. 2. Kenn~ M.I. Review of Applications of Artificial Neural Netvorks on Biotechnology. The First New Zeland Int. Two- Stream Conf. on Artific. Neural Networks and Expert Syst .. Dunedin, New Zealand, Now.24-26, 1993. p.252253. 3. Klestova Z.E. Swine fever virus influence on chromosome apparatus of swine cells. Third ESVV Symposium on Pestiviruses infection. Lelystad 19-20 Sept. 1996. The Netherlands,. p.84. 4. Rudnichenko G. Y. , Rubin A.B. Mathematical models of productive processes. M.Moscvow University, 1993. 302p. (m Russian). 5. Mathematical Modeling ofPopulations. Novosibirsk., Nauka SO, 1992. 6. Xuang K. Statistical Mechanics, M.: MlR Publish. 1968. (in russisn). 7. Makarenko A Mathematical Modeling of Memory Effects Influence on Fast Hydrodynamic and Heat Conduction Processes. Control and Cybernetics, 1996. Vol.25, n.3 . p. 621-630. 8. Danilenko VA ,Chrischenuk V., Korolevich V, Makarenko A SElforganization in strongly nonequilibria media. Collapses and structures. Kiev,IGF NANU, 1992. 144 p. 9. Makarenko A About the models of Global Socio- Economical processes. Proceed. of Ukrainian of Sci., 1994, n.12, p. 85-87. 10. Makarenko A New class of Global Economic Models of Associative Memory Type as Tool for considering Sustainable Development. Accepted to 14 Conf. WACRA - EUROPE, Madrid, Sept.16-19, 1997. 1l. Makarenko A , Klestova Z. New class of global models of associative memory type as tools for considering environmental change. Accepted to NATO Workshop: Global Environmental Change and Human Security, Budapest, 1-12 Oct. , 1997. 12. Dobronogov A , Levkov S., Makarenko A, Nikshich D., Plostak M. Associative memory approach to modelling geopolitical structures. World Congress on Neural Networks, San Diego, Calif. Sept. 15-18, 1996. p. 749-752. 13. Loskutov S. , Michailov A Introduction to synergetic. M.: Nauka, 1991. 14. Klestova Z., Sobko A. The problems of infection pathology in animals reproduces. Herald of Agrarical Science (Ukraine), 1994. n. 4. IS . Chabanuk V S. Decision Support Systems in Chernobyl GIS Project. UNUIIIST - IORC Expert Group Workshop on: Decision Support System for Sustainable Development, Makau, 25 Feb. - 8 March, 1996. Proceed.
217
MODELING AND CONTROL OF THE INFECTION PROCESSES IN INDUSTRIAL CATTLEBREEDING. KLESTOVA Z.E. (Ukraine), Ph.D. Department of genetic resistance investigation by infectious diseases. The Institute of Veterinary Medicine, Kiev 151 , Ukrainian Agricultural Academy of Sciences Donetskaja st. 30 MAKARENKO A.S. (Ukraine), Dr. of Sei. Department of mathematical methods of system analysis, the faculty of applied mathematics National Technical University of Ukraine (KPI) 37, Pobeda av., NTUU (KPI), Dept. MMSA (1920)E-mail: [email protected]
1. Introduction. The questions of productive cattelebreeding in particular pigbreeding are very important as for population providing of agricultural production, as for profitable conducting business in agroindustry. On productivity influence many factors, as traditions in productions, natural conditions, feed and other. In present report we concentrated on own problem aspect, which is connected with questions on losses from infective diseases. At practice even in this questions by force of big scales of modern industry of modern industry and many influenced factors preliminarily is difficult to receive some quantitative results. So much is difficult to value the scales of epizooties (mass infection disease of cattle on productive animal farms) and measures on their optimal overcoming. Certainly, the veterinary services have definite recommendation and working plans, but often by meeting a new disease agents and absent larg quantity a new data and can come catastrophically consequences before that these data will be received and will cultured. In such situation perhaps the only acceptable way is mathematical modelling and search optimal on expenditure decisions with model help. By this turn out, that in area of spreading different biological factors exist posing mathematical problem, which are more or less similar. This is epizooties spreading, spreading of plant diseases, spreading of parasites, substitution of own spices of flora and fauna on others etc. It must be take into consideration, that each branch of agroindustry , big animal farms, separate farms etc. have own peculiarity: geographical, technological, economical, cultural, that lead to different their reaction on the same initial influence. That demand to creation "the passport of the object" in connection with such questions. This "passport" must contain different factors - susceptibility of individual animal or plant to ecological, climatic, season, genetic factors . The epizootic and immune problems are one of the limiting factors for cattlebreeding, especially for post-Chernobyl situation in Ukraine. In this brief communication there are firstly described methodological basis for considering the problems above. Than discussed some approaches for their solutions as traditional as new and some realisations. There are also some results on concrete problems for Ukraine condition.
2. Some methodological remarks for considering infection spreading in cattlebreeding. Considering the problems of infection in the industrial cattlebreeding branch it may be useful to take in mind some approximate scheme of region (see Fig. I) : Infection
Economics
Ecology
Other agriculture Branch
Fig. 1
213
Analyses display that separate blocks on the scheme also consists from many subblocks. For example the ecology have influence as on the genotype of animals as on the spreading and survive of viruses. Also the economic block influences on infection block by transportation bounds for example. By considering separate subblocks it is very important to distinguish different space scales and corresponding time scales. In all levels there are specific immunological problems with own mathematical models and approaches. 1. Cell level. On such level important problems are the interaction between viruses and individual cell, mathematical modelling of external factor influence, the effect of drug and so on. Traditionally such problems are solved with the aid of ordinary differential equations (sf. for example [1]). Recently it is found that for prognozing in such problems well adjusted are neuronet methods [2]. On such level we investigated the influence of some specific serum and antiviruses prepares on infected cells [3]. 2. Separate animal or plant. This level was researched by traditional immunology with ordinary differential equations or discrete equations. Yet on this level it may exist the problems on the organism considered as cell union. 3. Separate cattlebreeding complex with many animals. The simplest model for such problems are the nets models for connected discrete or ordinary differential equations [1 ,4] . The Marcov chains also are useful for problems above [5] . The simplest model have the fonn (on analogy with [2,4]), t - time : dn dm ----- = F (n,m) ; ----- = F (n,m) dt I dt 2 where n - the number of health animals, m - number of ill animals. There use some successes in applying such models. But such description became unreal when there are a lot of animals. In such case the aggregation or reducing methods are need [6,7]. If the animals contact stochastically and frequently than it passage to the description with partial differential equations for the density of deceased animals [8]. 4. Geographical region. For such problems supposed that there are many cattlebreeding complexes connected by many bounds. Than it introduced the density U of infected, deceased ,reconvalescent animals and posed partial differential equations of diffusive type: U=DU t
+f(U) xx
Remark, that under the account of memory and nonlocality effects there obtained more correct equations of hyperbolic type [8]. Further the partial differential equations are investigated on the existence of travel waves of epidemic, on
selfoscillation of decease density, on spiral waves etc. by synergetic methods. The characteristic lengths on such level are 50-100 km (geographic region), 100-300 km (province), 300 - 1000 km (country or union of countries). 5. Global (World level). The tools for modelling described problems on this level are the partial differential equations or more aggregated models of ordinary differential equation type for small number of macroparameters (like consideration in global models by lForrester and D.Meadous). Already with described above models there are interesting results. Them have practical importance for planning antiepizooties and antiepidemical action. But there arise usually difficulties in such approaches in mathematical modelling - the search of correct equations for problem description, the sufficient number of equations, definition of coefficients in equations and transition probabilities and so on. In addition there arise also the problems of different risks calculation and the uncompletion of infonnation ( the range for fuzzy sets methods) etc. All above problems lay more or less in the traditional ranges for mathematical modelling specialist (as also for authors of report). Analyses of field of research with the invitation the specialist on immunology, virology, veterinary medicine and genetics and the experience in recent infonnation methods allow to propose entirely new type of mathematical models for the many problems in industrial cattlebreeding. New models are founded on the basic principles described for global soci
3. Immunological models with the associative memory property. The Immunological problems are the subproblems for the global environmental, biological, social and economical brunches. Let us describe the very simple but typical problems. There are many elements (animals in the level of unique catteelbreeding complex or the net of catteelbreeding complexes in the country, or the set of countries) as typical fields for infections. There are some descriptions of element - the parameter sri] for animal for example or some vector of parameters S{i} = {sI , s2, ... sk} , i = I, ...,
214
N, where N is the number of elements. For example for unique animal we can take s[i] = 1 if the animal is immune (after ill for example), s[i] = 0 for healthy animal and s[i] = -1 for ill animal which can spread the infection. There are the bounds between elements. The animals as element have contacts between themselves (usually stochastically in time and space). But for simplicity we can take this contacts as deterministic. The bounds between the complexes may be constant and described the transportation bounds, or tropical chains for infection spreading and so on. Also for the countries as elements the bounds are transportation ways and global tropical chains. Of course, there may be the part of stochastically also in the bounds. Than we may accept that in average the infection are the results of the surrounding influence. Now let us formulate the principles for the dynamically laws for evolution of element states. For this goal its need to consider the series of more and more complex models. Suppose firstly that there is no season and climatic changes and external conditions are constant (temperature, transportation's conditions, number of animals, the type and force of viruses). Suppose also that there isn't immunity to viruses. As adoption consider that the epizootic is deterministic and the same initial conditions (that is the number place of ill animal cause the same epizootic). For simplicity we can consider the discrete moments of time: 0 , t, 2t, 3t, .. . , where t - time step. So the first models may be discrete. Than in any moment of discrete time the elements form some space picture in geographical region with some distribution of ill, healthy and immune and so on animals. Let us call this space pattern as "space pattern of epizootic" (SPE). Father, the series of "space patterns of epizootic" in successive moments of time form the "space- time patterns of epizootic" (STPE). Remark that all notions and models also is suitable for another immunilogical problems (for example for epidemic for human). As the system can have in general the complex behaviour we may suppose the nonlinearity in the nature of problems. So different condition can causes the new STPE. In STPE the elements are connected by space bounds in the same moment of time and by bounds between elements from different SPE. If we take the space and time coordinates as equal in rights than in space- time dimensions we can consider the unique epizootic as one pattern. This pattern is the pattern of element states in all elements space points and in all discrete moments of time. The connections J{ij,k,l} between s[i,k] and s[j,l] (the states of ij elements in the moments k and 1) describe the influence on i- th element in moment k by j element in moment l. If there isn't internal dynamics of elements than the elements behaviour determinate by the external influence by another elements. This influence may be described by mean influence field acting on i-th element. This field h[i,k] (in moment k) is equal to h[i,k] = Sum at ion { J{ij,k,l}
* s[j'!]} (1) i notj, (1)
Than in the simplest variant we can suppose, that the dynamics of i-th element is describes (on analogy with our models [12]) by the law s[i,k,m+ 1] = s[i,k,m] + I if h[i,k,m] > 0 & s[i,k,m] <1 s[i,k,m+ 1] = s[i,k,m] - 1 if h[i,k,m] < 0 & s[i,k,m] > -1 (2) s[i,k,m+l] = s[i,k,m] else In (2) m is some fictive evolutionary parameter and we adopt the dependence s[i,k] and h[i,k] from them. The results of evolution by the law (2) is some stable in space- time dimension STPE. Thus s[i,k] consist the real epizootic pattern. Such algorithm starts when we give arbitrary condition in the sense that we pose the all elements states in all moments of time. If it needs to know the stream of epizootic in time we have only initial condition in first discrete moment oftime (or in some close moments of time). But if we continue the initial conditions in some manner (say by supposing s[i,k] = 0 for k bigger than some kO), than we have precisely the problem above. Of course we can take into consideration also internal dynamics of element i by including to the summation in (1) the case i=j with bounds J{i,i,k,l} (selfinteraction or unique element ill stream). Remark for the topics below that described models may served as the tool for prognoses in the problems with time series. There are some interesting considerations about the bounds J[ij,k,l]. For the goals of dynamics of model we must know there values. There are two main approaches to such problem. In first remark that the epizootic have more or less similar streams. The bounds J[ij,k,l] was created historically and naturally in long period of time. The STPE is stable object and from many initial conditions the space- time distribution tends to the real STPE. So the STPE is the attractor of epizootic problem. The system of elements goes to some attractor and form the real epizootic pattern. This is strict analogy with associative memory type behaviour in the pattern recognition and neuronet methods [13].So the formulas for the evaluation J[ij,k,l] may by familiar with formulas in neuronet models. In the simplest case we can take the formula
215
J[iJ.k,I]
=
Sum a t ion { s[i,k.,p]
* sU,l,p] } (3),
p= 1,2, ... . P where P - number of training patterns (epizootic). In second approach the values of J[ij,k.,l] are evaluated by experts. In both causes if we know bounds we can calculate the epizootic pattern by (1),(2). There are useful to make some remarks. First, if external conditions are changed with time it is easy to take into account this fact by applying STPE in different seasons as different learning epizootic patterns. Second, we can take the bounds which depend on fictive parameter m. (The nonconstant character of bounds on real time already accounting in model). And last remark. How do we must build model for new viruses epizootic? At first its seems that this class of models are insufficient because the situations are also new and there are changes in presumable STPE. Naturally there are some changes but there are very many implicit invariant in the problem. First of all we can apply the expert knowledge on new epizootic. Although at the beginning of epizootic we have only few date on decease the experts can suppose some values for new J[i,i.k,l] (for the element evolution) or calculate by some more simplest models. Next the epizootic has usually the nidus character with some points of initiation epizootic. So the data from this initial nidus may be utilise by experts for evaluation J[ij,k,l] outside the nidus. There also may exist many modifications and complications of the model. Remark only that the model for STPE remember in many features the usual perception [13] for pattern recognition. Also for the epizootic flow prediction may be find by the traditional neuronet methods for the time series of date prediction. For such goal some first SPE was considered as training patters for the neuronet considering them as input data and prognosing SPE as output data with usual beckpropagation algorithm for learning. But the models above are more general because they can in principle take into account the history of previous epizootic.
4. The economic and control problems. In the subdivision of this report below it had discussed some methodology for describing and modelling the epizootic and some new types models. This models can constitute the basis for the considering the cost optimisation problems and stopping epizootic problem. As to the model in subdivision 2 the standard models may be applied for optimisation. Remark that there are a lot publications on such subject. What is need that to adjust the methods for the models. But the difficulties with models transform in difficulties with optimisation methods. The neuronet type models give new possibility for solution economical and control problems [9,11] . Its follows from the models that we can easy calculate the loss from the diseases because we know precisely the real part of ill animals (of infopcate cattelebreeding complexes ets). So we can calculate the cost of given STPE. Next the epizootic is only the subpattern in global models of socio- economical processes [9,10],so we can create a global measure of STPE evaluation. If we consider STPE step by step in real time than we can say about the "trajectory of epizootic" (that is about series of space patterns). So the optimisation consist in searching the "optimal trajectory of epizootic". Of course optimality must understand some qualitative criteria. The control of epizootic consist at first step in searching some measures. There are a lot of controlling influences: the applications of drug, vaccination, isolation of ill animals and complexes, reducing the transport contacts and so on. Such affairs have different cost and there are interesting problem to minimise cost of lost from epizootic. What is surprising that such control is easily described and displayed in proposed models. For example the drug application changes the J[ij,k.,l]. The isolation of animals make J[ij,k,l] = 0 for all j and i from some P (P - the set of isolating animals). The vaccination and others medical actions also change J (and in more developed models change the states s[i,k]). As to the situation in Ukraine there are a lot of date and the problems of local and global characters. First problem is the epizootic on the scale of all Ukraine and on international scale. Now it is well known that in Europe there exist the spreading of animal decease - the pig plague, the cow hydrophobia and so on. Thus we have to calculate the spreading of infection. Also there are many investigations on real situations of the cattelbreeding complexes in Ukraine and [14] on the influence of some prepares on infection in the cell level [3]. The new area consist investigations of date on the subject of reports and connected dates in the scale of Ukraine on the basis of the geoinformatical systems [15].
5. Resume. Thus in proposed report there as are some methodological approaches for the epizootic and immunology problems in agriculture as some new models for their modelling and solution. Proposed models allow to pose and solve the problem of calculation the cost of epizootic and optimisation the epizootic cancelling.
216
UTERATURE 1. Marchuk G.I. Mathematical immunology. M.:Nauka,1992. 2. Kenn~ M.I. Review of Applications of Artificial Neural Netvorks on Biotechnology. The First New Zeland Int. Two- Stream Conf. on Artific. Neural Networks and Expert Syst .. Dunedin, New Zealand, Now.24-26, 1993. p.252253. 3. Klestova Z.E. Swine fever virus influence on chromosome apparatus of swine cells. Third ESVV Symposium on Pestiviruses infection. Lelystad 19-20 Sept. 1996. The Netherlands,. p.84. 4. Rudnichenko G. Y. , Rubin A.B. Mathematical models of productive processes. M.Moscvow University, 1993. 302p. (m Russian). 5. Mathematical Modeling ofPopulations. Novosibirsk., Nauka SO, 1992. 6. Xuang K. Statistical Mechanics, M.: MlR Publish. 1968. (in russisn). 7. Makarenko A Mathematical Modeling of Memory Effects Influence on Fast Hydrodynamic and Heat Conduction Processes. Control and Cybernetics, 1996. Vol.25, n.3 . p. 621-630. 8. Danilenko VA ,Chrischenuk V., Korolevich V, Makarenko A SElforganization in strongly nonequilibria media. Collapses and structures. Kiev,IGF NANU, 1992. 144 p. 9. Makarenko A About the models of Global Socio- Economical processes. Proceed. of Ukrainian of Sci., 1994, n.12, p. 85-87. 10. Makarenko A New class of Global Economic Models of Associative Memory Type as Tool for considering Sustainable Development. Accepted to 14 Conf. WACRA - EUROPE, Madrid, Sept.16-19, 1997. 1l. Makarenko A , Klestova Z. New class of global models of associative memory type as tools for considering environmental change. Accepted to NATO Workshop: Global Environmental Change and Human Security, Budapest, 1-12 Oct. , 1997. 12. Dobronogov A , Levkov S., Makarenko A, Nikshich D., Plostak M. Associative memory approach to modelling geopolitical structures. World Congress on Neural Networks, San Diego, Calif. Sept. 15-18, 1996. p. 749-752. 13. Loskutov S. , Michailov A Introduction to synergetic. M.: Nauka, 1991. 14. Klestova Z., Sobko A. The problems of infection pathology in animals reproduces. Herald of Agrarical Science (Ukraine), 1994. n. 4. IS . Chabanuk V S. Decision Support Systems in Chernobyl GIS Project. UNUIIIST - IORC Expert Group Workshop on: Decision Support System for Sustainable Development, Makau, 25 Feb. - 8 March, 1996. Proceed.
217