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mathematical modeling of infectious diseases has become an area of increasing interest in the last decades @xcite . the development and use of mathematical models are a powerful tool to understand the complex problem of infectious disease transmission . after the success of the simple sir compartmental model in the description of the basic and common features of the transmission process @xcite , the models have become more complex and specific for the different infectious diseases in order to help in the evaluation and design of control strategies @xcite . these complexities may include , among others , age structure of the population , immune status of the individuals , structure of the social contacts , spatial heterogeneity , etc .    wherever possible , deterministic compartmental models are usually chosen because they are simple to solve numerically , the interpretation of results is direct , and the level of complexity may be increased gradually by adding new compartments . however , from the beginning of infectious disease modeling , the importance of stochastic effects has been mentioned @xcite . in more recent work , the significance that a stochastic treatment could have in the transmission of some diseases such as pertussis has been highlighted @xcite . in particular , the randomness and heterogeneity of contacts are one of the intrinsically stochastic aspects of the contagion process and knowing the accuracy that can be obtained with a deterministic approach to the problem is not a trivial matter .    in ref.@xcite a sirs spatial stochastic model having only local infective contacts is considered on a lattice and a pa scheme to approach the dynamic is developed ( some details are described in section [ sectionda ] ) . in ref.@xcite another model on a lattice is proposed , where local and global contacts are considered with the aim of emulating the more realistic situation of social relations where some `` fixed '' persons are frequently contacted ( the neighbors on the lattice ) and some other unknown people are met by chance ( the global contacts ) . in their model , a parameter @xmath1 characterize the weight of global contacts and , in this way , the influence of the degree of locality or globality of the contacts on the dynamics of the disease transmission is analyzed by changing a single parameter . other authors have characterized the stochastic fluctuations of this model as a function of @xmath1 for a wide range of model parameters that include several infectious diseases @xcite . @xcite studied the quasi - stationary state ( qss ) of the model for parameters corresponding to pertussis disease and quantified the relation between the effective transmission of the disease and the correlation of susceptible individuals with their infected neighbors .    in the present work we propose a deterministic approximation ( da ) to the stochastic model with local and global contacts ( sm ) mentioned above , where the local contacts on the square lattice are treated here under a pa scheme based on the one used by joo _ we compare the predictions of our proposed da with the results of simulations performed with the stochastic model for parameters corresponding to two infectious diseases : pertussis and rubella . we make this comparison for different situations of the disease transmission dynamics : the epidemic spread , the endemic state of the disease and dynamic perturbations that may arise from control measures implemented against the disease . in all the cases studied , the agreement between the results obtained from the sm simulations and da was very good . we consider an @xmath2 square lattice under periodic conditions where we identify sites with individuals . each one of the @xmath3 individuals may be in one of the three epidemiological states : @xmath4 , @xmath5 or @xmath6 ( susceptible , infected or recovered ) . the state of an individual may change through the following processes : infection ( @xmath7 ) , recovering ( @xmath8 ) , death and birth ( @xmath9 , @xmath10 , @xmath11 ) . infections occur through infective contacts among susceptible and infected individuals . we define an infective contact ( ic ) as a contact between two individuals such that if one individual is susceptible and the other infected , the former becomes infected . we assume that an individual at a given site has an ic with a randomly chosen individual on the lattice with probability rate @xmath12 , and with one of their four nearest neighbors with probability rate @xmath13 . local contacts represent the contacts in the circle of stable relations of an individual , while global random contacts represent people met by chance ( for example , on a bus , at the supermarket , etc . ) . by changing @xmath1 , we may change the degree of  locality " of the contacts in the system . the case @xmath1=1 corresponds to the classical sir model ( uniform mixing ) where an individual may have an ic with any other individual in the system with the same probability rate @xmath14 . recovery from infection in this model is the same for every site and occurs at a probability rate @xmath15 . deaths are assumed to be independent of the individual s state and occur at the same probability rate @xmath16 . when an individual dies at a site , another individual is born simultaneously at this site in order to avoid empty sites during the simulation . the processes defined above are markovian , so , the defined probability transition rates between states for each site at a given time control the dynamical evolution of the system . stochastic simulations are performed using gillespie algorithm @xcite . the algorithm gives a sequence of times and the corresponding states where two consecutive states differ by a single process that occurs at a given site . the process and the time when it takes place are generated from simple rules and two random numbers ( see ref . @xcite for details ) .    for a detailed description of the model and the implementation of the simulation algorithm , see ref.@xcite . we consider an @xmath2 lattice as in section [ sectionsm ] . at each time @xmath17 , the individuals must be in one of the three states : @xmath4 , @xmath5 or @xmath6 ( susceptible , infected or recovered ) . we define @xmath18 , @xmath19 and @xmath20 as the total number of individuals in those states . we denote @xmath21 when the state of the individual @xmath22 is @xmath23 ( for example , @xmath24 ) . each pair @xmath25 of ( horizontal or vertical ) neighboring individuals can be in six possible states : @xmath26 , @xmath27 , @xmath28 , @xmath29 , @xmath30 , @xmath31 . we remark that @xmath27 involves the cases @xmath32 and @xmath33 ; and the same holds for @xmath28 , @xmath30 . let @xmath34 be the total number of each type pair of neighboring individuals . our unknowns , arranged in the vector * x*=(@xmath35 ) , are the preceding quantities normalized by @xmath36 . they are functions of the time @xmath17 and are linked by    @xmath37    @xmath38    @xmath39    @xmath40    ( [ l1 ] ) and  ( [ l2 ] ) hold because there are @xmath36 individuals and @xmath41 pairs of neighboring individuals . to obtain  ( [ l3 ] ) ( and analogously  ( [ l4 ] ) ) observe that each individual in susceptible state @xmath4 belongs to four pairs of neighboring individuals of type @xmath26 , @xmath28 or @xmath27 but in their total contribution @xmath42 , the @xmath26 pairs are counted twice .    in the da equations that we propose , the terms related to the birth - death and recovery rates do not depend on the local structure and are treated as in the classic sir scheme . but in the case of the transfer rates among classes due to infections , @xmath43 , we make a decomposition @xmath44 being @xmath45 and @xmath46 , @xmath47 two different rates that must be defined for global an local infective contacts respectively . the decomposition is motivated by the following argument in sm . consider an individual ( or a pair , according to the case ) in any of the states @xmath4 , @xmath26 , @xmath27 or @xmath28 at time @xmath17 . it can reach , through an infective contact , the new state @xmath5 , @xmath27 , @xmath29 or @xmath30 respectively . denote @xmath48 the event  the individual ( or the pair ) changes the first state by the second one due to an infective contact in the time interval @xmath49 $ ] " . then @xmath50 . but @xmath48 is the disjoint union of @xmath51 and @xmath52 , the events corresponding to one global and one local infective contact in @xmath49 $ ] respectively . assuming that @xmath53 and @xmath54 are known , we have ( according to the sm rules ) @xmath55 and @xmath56 . finally , by the total probability law , @xmath57 . let @xmath25 and @xmath58 be two different pairs of neighboring individuals sharing @xmath59 . the key fact of the pa scheme in ref.@xcite ( with only local infective contacts ) consists in the approximation of the probability that the triplet @xmath60 reaches a given state @xmath61 at time @xmath17 by @xmath62    in order to apply the above approximation to the treatment of the local contacts in da we consider the natural correspondence between fractions ( our unknowns in da ) and probabilities . for example @xmath63 ( the fraction of susceptible individuals at time @xmath17 in da ) can be regarded as the probability that a given individual @xmath22 is of type @xmath4 at time @xmath17 in sm , and @xmath64 ( the fraction of @xmath27 pairs at time @xmath17 in da ) as the probability that a given pair @xmath25 of neighboring individuals is of type @xmath27 at time @xmath17 in sm . we present now a characteristic case to show how the preceding observations are used to define the various rates , @xmath43 , contributing to the equations ( we omit the details of the whole construction of the system because the procedure to be applied to the other terms is similar ) . let @xmath43 be the rate at which @xmath65 grows at the expense of the decrease of @xmath66 . for global contacts @xmath67 . let @xmath25 be a pair of type @xmath27 at a fixed time @xmath17 such that @xmath68 . the rate of local infection for this pair is the product of @xmath14 and the fraction of infected individuals in the four nearest neighbors of @xmath59 . this fraction can be regarded as @xmath69 , where @xmath70 is the probability that a fixed triplet @xmath60 ( @xmath71 being a fixed nearest neighbor of @xmath59 different from @xmath22 ) is in situation @xmath72 , given that @xmath33 . we have    @xmath73    then , using  ( [ lebo ] ) and @xmath74    @xmath75    which corresponds to @xmath76 in da . then we take    @xmath77 and @xmath78 is incorporated as a substracting term in the fifth equation and as an adding term in the seventh equation of the system ( 6 ) , that we present below . @xmath79 @xmath80 @xmath81 @xmath82 @xmath83 @xmath84 @xmath85 @xmath86 @xmath87    for better visualization of the dynamics , we presented the initial system of nine differential equations . by using the equalities  ( [ l1 ] ) to  ( [ l4 ] ) we reduced the system to five ones for the numerical resolution in this section we compare the results obtained from the stochastic model ( sm ) simulations with the results obtained using the deterministic approximation ( da ) . we consider l=800 which mimics a city of @xmath36=640,000 inhabitants . for pertussis we take @xmath88 years ) , @xmath89 days ) and @xmath90 1/day , which are standard parameters for sir description of the disease in the pre - vaccine era @xcite . for rubella we take @xmath88 years ) , @xmath91 days ) and @xmath92 1/day @xcite . we consider values of @xmath93 since in ref.@xcite it was observed that for lower values of @xmath1 the probability of establishment and survival of the steady state ( representing the endemic state of pertussis in prevaccine era ) is very low . in the case of rubella , this probability is very low even for @xmath94 . in this work we are interested in considering values of the parameters that may represent real systems , however , we have included in our study the case @xmath94 for rubella for comparison purposes .    in da differential equations are integrated using euler algorithm with a time step 0.01 days that we have checked to give accurate enough results for the problems studied . we first study the epidemic spread of the disease for the case @xmath1=0.4 . we consider the evolution of the system when a single infected individual is introduced in a population of @xmath36 - 1 susceptible individuals . from this initial condition , 5,000 independent runs were performed for sm with different sets of random numbers . for da we take * x@xmath95*=@xmath96 as initial condition .    in fig.1a we compare the results for three representative samples of sm simulations and da for pertussis . in fig.1b the same comparison is shown for rubella . in both cases da correctly predicts the qualitative spread of the epidemic and also gives a quantitatively good prediction for the maximum fraction of infected individuals reached during the course of the epidemic . in fig.1c and 1d we compare the result for da with the average of the 5,000 runs of sm simulations . the discrepancy in the maximum of the fraction of infected individuals in this case is caused because in the average for the sm runs we included the extinctions . at the very beginning of the epidemic spread there is one infected individual and the total rate of infection is @xmath14 , while the rate of recovery or death of the infected individual is @xmath97 . as @xmath98 for both diseases , da always predicts an initial increase in the fraction of infected individuals . in the sm simulations there is a positive probability that recovery occurs before infection , and since there is only one infected individual at the beginning , recovery implies extinction . the fraction of extinctions is greater in the case of rubella as @xmath99 is greater as well . at the first steps of the dynamics , the value of @xmath1 is irrelevant as the infected individual can only contact susceptible ones regardless of whether the contacts are local or global this can be seen in fig.1e and 1f where sm simulations and da predict an exponential increase in the fraction of infected individuals at a rate @xmath100 . an interesting result predicted by sm simulations ( and very well aproximated by da ) is the approximately exponential growth of the epidemic a few days after its beginning with an exponent that differs from the initial one . this can be concluded from the linear behavior observed in the curve of fig.1e from 5 to 15 days , and in fig.1f from 10 to 30 days . exponential fits to the curves in the mentioned ranges give exponents 0.61 and 0.63 for sm and da for pertussis , and 0.267 and 0.272 for sm and da for rubella .    in order to characterize this behavior as a function of @xmath1 , we notice that in this exponential regime the time dependence of the fraction of infected individuals may be written as @xmath101 $ ] . that is to say , the same as in the classical sir model ( with only global contacts ) but with an effective contact rate , @xmath102 , that is different from @xmath14 because of the presence of local contacts . it is possible to obtain an analytical approximate expression to estimate @xmath102 using da equations as follows . at the beginning of epidemic growth , when @xmath103 , we have that @xmath104 , @xmath66 and @xmath105 are also @xmath106 . so , we drop all the second order terms in @xmath107 and @xmath105 , and also the terms @xmath108 . as @xmath109 , we may write the 2nd equation of system ( 6 ) : @xmath110 if @xmath111 presents an exponential behavior , @xmath112 should be proportional to @xmath113 , so , we assume that in this exponential regime : @xmath114 , @xmath115 being a constant . expression ( [ didtap ] ) may now be written as : @xmath116 where @xmath117 plays the same rol as @xmath14 in the exponential growth when only global contacts are present . so , we define @xmath118 . taking @xmath114 and replacing it in the 2nd and 5th equations of system ( 6 ) , we may obtain the value of @xmath115 and compute : @xmath119    an interesting point here is that , at this level of approximation , da predicts that @xmath120 only depends on @xmath1 and not on the other parameters characterizing the disease transmission . in fig.2 we plot this relation for da ( from eq.([betaexp ] ) ) and for sm ( obtained from exponential fits for different @xmath1-values ) for pertussis and rubella . the agreement between da and sm is good , but it becomes worse for lower values of @xmath1 . however , even for @xmath94 ( when the agreement of sm and da values for @xmath102 is not so good ) @xmath121 accurately verifies the da prediction of being dependent only on @xmath1 for pertussis and rubella . comparison of sm and da predictions for the exponential growth regime in an epidemic as a function of @xmath1 . the computation of @xmath120 is performed through expression [ betaexp ]  for da . for sm , @xmath122 is obtained by fitting the exponential regime of the curve obtained for the average fraction of infected individuals as a function of time to @xmath123 $ ] for each case , where @xmath71 is a constant.,width=257,height=188 ]      for the sm presented in section [ sectionsm ]  and a finite value of @xmath36 , the only fixed point ( stationary equilibrium ) corresponds to the case where all the individuals on the lattice are susceptible . for the stochastic nature of the system , sooner or later a fluctuation leads the number of infected individuals to zero and there is no process that produces new infected people if there are none . but for large values of @xmath36 ( as the one taken in the present study ) the system may fluctuate for a long time around a quasi - stationary state ( qss ) before extinction . the definition and properties of such a state have been addressed in other contexts from a mathematical point of view @xcite or with empirical approaches @xcite . in the present work we adopt the empirical strategy developed in ref.@xcite where the qss of the same sm defined in section [ sectionsm ] is studied . we define a time window ( @xmath124 ) and generate ( for each system studied ) more than 20,000 independent samples keeping those that do not become extinct before @xmath125 . we compute the averages of susceptible and infected individuals over the surviving samples and observe that in the ( @xmath124 ) interval they remain constant in time within a given precision . we take @xmath126=20,000 days and @xmath125=40000 days ; with these values the considered averages are independent of the initial conditions ( as @xmath126 is large enough so that correlations in the system may be established ) and enough statistics is obtained to compute averages with a relative precision of 0.01 . ( for details of the definition of the qss and characterization of the fluctuations for the pertussis case see ref.@xcite . )    in the case of da , stationary values are defined as the values of the variables that make all the derivatives in equations ( 6 ) zero . they were obtained numerically solving the equations from a given initial condition , * x@xmath95 * , until * x*@xmath127 is constant in time with a precision 0.00001 . for * x@xmath95 * we take for @xmath128 and @xmath104 the values corresponding to the stationary solution of the sir model and complete the remainding values of * x@xmath95 * as in the uncorrelated case . for example , @xmath66 is initialized as @xmath129 because it is twice the probability that a given pair is of type @xmath27 whose value is @xmath130 in the uncorrelated case .    in fig.[figstat]a and b we compare the da stationary values obtained for the fractions of susceptible and infected individuals for pertussis and rubella with the values of these observables averaged over the samples in the qss of sm . results for both diseases show an excellent agreement between sm and da . in fig.[figstat]c we make the same comparison for the fraction of @xmath27 pairs on the lattice . the agreement is remarkable considering that this is a correlated quantity directly involved in the da to sm . we here study the ability of da to describe the system response to a sudden change in the globality of the contacts represented in the model by the parameter @xmath1 . we simulate disease transmission at the endemic state in a city where , at a given time , infective contacts among people change and become more local . such a change could occur because of several circumstances that affect social activities . for example , in argentina , during the 2009 pandemic flu , the authorities took preventive measures that included suspension of classes at school in july . the population was advised to stay at home whenever possible and to avoid crowded places . these measures aimed at reducing the possibility of an epidemic outbreak during the winter . our purpose here is to simulate the consequence that this sudden reduction in globality of the contacts may have on the disease transmission of other infectious diseases that are at the endemic state . we perform simulations with sm for parameters corresponding to pertussis in the prevaccine era and @xmath131 . once the qss is stablished , at a given time ( t=20,000 days ) we change the value of @xmath1 to 0.4 , which corresponds to a 20% reduction in the globality of contacts . in fig.4a we compare the dynamic evolution of the system for three samples obtained from sm simulations and da result . to obtain the da curve we take the * x * corresponding to the stationary state for @xmath1=0.5 as initial condition and at @xmath132 we change the value of @xmath1 to 0.4 . we observe that da captures the fall down in the fraction of infected individuals after lowering @xmath1 and the pronounced peaks observed in the following years . the perturbation at @xmath132 days puts in phase the oscillations for the different samples that become out of phase again from the third peak . da also accurately reproduces the time elapsed between peaks . oscilations are exponentially damped out in da until the new stationary value is reached . in the case of sm , fluctuations may be very different from sample to sample @xcite , but as we have seen in section [ subsecsb ] , once the qss is reached , the fraction of infected individuals oscilates around a value very similar to that predicted by da . we repeated this study for the case that the system is initially at the qss corresponding to @xmath133 and at @xmath132 @xmath1 is changed to 0.4 . in this case , although the final state is the same , the change in the globality of contacts is 33% . in fig.4b it can be seen that after the perturbation , the system reaches a deep minimum followed by pronounced maxima much higher than in the previous case when the change in @xmath1 was 20% . again , da gives a good description of the observed effects . in this case , most of the samples obtained from the sm simulations become extinct when the system is at the first and pronounced minimum . this is not surprising and could be inferred from the da result that predicts the number of infected individuals to be below 15 in the time interval ( 20,400 , 20,800 days ) for the population considered here . extinctions in the sm simulations are thus expected , given the fluctuations in the number of infected individuals observed in the dynamic behavior of the system . it is worth mentioning that a dynamic behavior similar to that observed in fig.4 has been found in a recent study of pertussis transmission performed with an age - structured deterministic model with nine epidemiological classes @xcite . in that work , the authors proposed that a sudden change in contact rates among individuals could cause a dynamic effect with the presence of deep valleys followed by sharp maxima such as those recently observed in some us states . here we obtained a similar behavior for the disease transmission as a consequence of having reduced the degree of globality of the contacts in a much more simple epidemiological model . in this work we propose a deterministic approach ( da ) for an sir - type epidemiological model that includes global and local contacts on a square lattice . when comparing our da with simulations performed with the stochastic version of the model ( sm ) , we obtained a very good agreement for qualitatively different scenarios of disease transmission and parameters corresponding to pertussis and rubella in the prevaccine era . it would be very valuable to increase the complexity of the model used in the present work in order to study pertussis transmission in the vaccine era . the introduction of pertussis vaccination in the model forces one to include new epidemiological classes that account for individuals that have partially ( or totally ) lost their immunity @xcite as it is well known that immunity to pertussis is not lifelong @xcite . we expect the good description obtained with da for sm to hold if a model with additional epidemiological classes is considered , since the da to sm involves bassically the description of local contacts that has shown to work very well in different dynamical situations . stochastic simulations for a more complex model could be much more time - 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an epidemic model with births and deaths is considered on a two dimensional @xmath0 lattice . 
 each individual can have global infective contacts according to the standard sir model rules or local infective contacts with its nearest neighbors . 
 we propose a deterministic approach to this model and verified that there is a good agreement with the stochastic simulations for different situations of the disease transmission and parameters corresponding to pertussis and rubella in the prevaccine era . 
 x_s    keywords : epidemics ; sir ; lattice ; deterministic model ; pair approximation ; pertussis .