You are an expert at summarizing long articles. Proceed to summarize the following text: 
mathematical models are used throughout infectious disease epidemiology to understand the dynamical processes that shape patterns of disease  @xcite . while early models did not include complex population structure , modeling approaches now frequently let the epidemic spread take place on a network , which enables greater realism than a model in which all individuals mix homogeneously . it does , however , pose many technical problems for model analysis , particularly the question of how heterogeneity in the number of links each individual participates in  their degree  influences the epidemic  @xcite .    similarly , even though some of the earliest mathematical models of infectious diseases were stochastic , accounting for the chance nature of transmission of infection  @xcite , much of the applied modeling that followed was deterministic and based on non - linear differential equations  @xcite . more recent applied work has , however , recognized the importance of using stochastic epidemic models  @xcite and also of the development of associated methodology  @xcite . the difficulty in mathematically analyzing models which include both stochastic elements and network structure can be a reason for not including these factors , but we prefer to include them , and subsequently to systematically reduce the complexity of the resulting model . this is the approach we adopt in this paper ; the reduction process being made possible because of the existence of a separation of time - scales : many variables `` decaying '' away rapidly , leaving a few slow variables which govern the medium- to long - term dynamics .    compartmental models of epidemics typically assume that the majority of the population starts susceptible to infection , with a small number infectious , who then spread infection to others before recovering . a key distinction is between susceptible - infectious - susceptible ( sis ) dynamics in which recovery returns an individual to the susceptible compartment and susceptible - infectious - recovered ( sir ) dynamics in which recovery removes an individual from a further role in the next epidemic  @xcite , with the former being used to model e.g.  sexually transmitted infections other than hiv and the latter e.g.pandemic influenza  @xcite .    in the theoretical analysis of epidemic models , a crucial quantity corresponds to the basic reproductive ratio @xmath0verbally described as the expected number of secondary cases infected per primary case early in the epidemic  @xcite . depending on the value of @xmath0 , either @xmath1 individuals will experience infection in a population of size @xmath2the model is then said to be supercritical  or @xmath3 individuals will experience infection in a population of size @xmath2the model is then subcritical  thus defining an epidemic threshold .    in reality , the contact network on which the disease spreads will often change over the course of the epidemic ; while approaches exist in which the dynamics of both the disease and the network are considered  @xcite , it is more common to consider two limiting cases . the first of these is a _ static _ approach ( also called ` quenched ' ) in which the network is assumed to evolve much more slowly than the disease and which is typically approached analytically through the use of pair approximation and related techniques  @xcite . the second is a _ dynamic _ limit ( also called ` annealed ' or ` discrete heterogeneous ' ) in which the network is assumed to evolve much more quickly than the epidemic , and can be described by an effective network characterized by its degree distribution , in which all individuals sharing the same degree are considered equivalent . this case can be analyzed through use of a set of degree - indexed differential equations ( often called the ` heterogeneous mean field ' approach ) provided the maximum degree is not too large  @xcite .    when the distribution of degrees in the population is highly variable  a situation that appears to be supported empirically  @xcite  it was recognized that the epidemic may not exhibit straightforward critical behavior . this happens because as the population size @xmath2 becomes large , extremely small levels of infectiousness can lead to large epidemic sizes  @xcite or more accurately speaking the critical level of infectiousness can depend very sensitively on the largest degree , @xmath4 , in the network  @xcite . the behavior of highly heterogeneous network epidemics near criticality continues to generate interest in both the physics and mathematics literature  @xcite .    in this paper we investigate the stochastic behavior of heterogeneous network epidemics over time . we study a network in the dynamic limit , characterized by a power - law degree distribution such that the probability of an individual having degree @xmath5 is given by @xmath6 , with @xmath7 , although the method of analysis is applicable to other distributions . this type of network has the property that the basic reproductive ratio , which is found to be proportional to the second moment of the degree distribution @xmath8 , diverges in the limit of infinite populations , leading to the absence of an epidemic threshold . this is evidently not the case when the population  and thus the degree cutoff  is finite , but the second moment of the distribution can still be extremely large for sufficiently large @xmath2 , and heterogeneity can play an important role in the dynamics of the system .    for the case of large but finite population size , we derive a two - dimensional stochastic differential equation ( sde ) approximation to full sir dynamics , which reduces to an analytically solvable one - dimensional system early in the epidemic . we perform simulations using a power - law degree distribution with a maximum degree cutoff @xmath4 , which show that our approach provides a good approximation provided @xmath4 is not too close to the population size . there have , to our knowledge , been no directly comparable studies of this kind . perhaps those closest have been , first the study of finite - size scaling of the sis model near the critical point to produce a phenomenological one - dimensional sde  @xcite , second , the derivation of a one - dimensional sde for the sis model from first principles , but using an adiabatic approximation that would only be expected to hold near the critical point  @xcite , and finally the study of a four - dimensional sde for the sir model on a static network that is much more complex than ours and less amenable to analysis  @xcite . the outline of the paper is as follows . in sec . [ sec : formulation ] we introduce the model , and formulate it first at the level of individuals ( the microscale ) and then at the level of populations , but retaining a stochastic element ( the mesoscale ) . we begin to explore the reduction of the mesoscopic - level sdes in sec .  [ sec : det ] and complete the process in sec . [ sec : fastvarelim ] . in sec . [ sec : cir ] we perform a further reduction which allows us to make contact with a model which has already been discussed in the literature . in sec . [ sec : size ] these reduced models are used to find the distribution of epidemic sizes , and we conclude with a discussion of our results in sec .  [ sec : discussion ] . there are two appendices where technical details relevant to secs . [ sec : det ] and [ sec : fastvarelim ] have been relegated . in this section we formulate the model first at the microscale , that is , in terms of individuals , and from this derive a mesoscopic version of the model where the variables are continuous , and represent the fractions of types of individuals who are infected or who are susceptible to infection . the general procedure to do this is reviewed in ref . @xcite , and has been previously applied to models of epidemics on networks  @xcite , albeit in situations where the nodes of the network contained many individuals , rather than just one , as in the present context . as discussed in the introduction , individuals are located at the nodes of a network through which they are exposed to infection from individuals at neighboring nodes . since we work in the dynamic limit , individuals do not alter their number of contacts when acquiring the infection or recovering from it , and the network plays no role other than encoding the information about how many connections to other individuals a given individual possesses . an individual is labeled according to whether it is ( i ) susceptible , infected or recovered , and ( ii ) the degree of the node where it is located . thus the variables at the microscale are @xmath9 , @xmath10 and @xmath11 , @xmath12 . we will assume that the population is closed , i.e. , that @xmath13 , with total population @xmath14 , at all times , so that we can remove one of the variables : @xmath15 . two types of events can occur : the infection of a susceptible individual , whenever an individual of type @xmath16 comes in contact with one of type @xmath17contacts between individuals @xmath18 and @xmath19 having degrees @xmath20 and @xmath21 , respectively , are governed by a poisson process with rate @xmath22 ; and the recovery of an infectious individual with rate @xmath23 . in this paper we will be interested in the properties of an epidemic that takes place over a much shorter timeframe than demographic processes and hence will ignore the birth and death of individuals , leaving the only interactions between the individuals to be :    1 . infection of an individual at a node of degree @xmath5 . the transition rate for this process is given by @xmath24 where the summation is over infected individuals labeled according to the degree of the node on which they are located . the only arguments of @xmath25 which are shown explicitly are those involving individuals on nodes of degree @xmath5 , since only these change in this process ; those on the right represent the initial state , and those on the left represent the final state . 2 .   recovery of a infected individual at a node of degree @xmath5 . this proceeds at a rate given by @xmath23 , and so the transition rate for this process is given by @xmath26    the dynamics of the process can be described by writing down an equation for the probability that the system is in a state @xmath27 at time @xmath28 , which will be denoted by @xmath29 . this is the master equation  @xcite , which equates @xmath30 to the net rate of increase of @xmath31 due to the processes given by eqs . ( [ eqn : rate_1 ] ) and ( [ eqn : rate_2 ] ) , and is given by @xmath32   \nonumber \\ & \quad - \sum^k_{k=1 } \left[t_1(s_k-1,i_k+1\vert s_k , i_k)+t_2(s_k , i_k-1\vert s_k , i_k)\right]p(s_k , i_k , t ) . \label{eqn : master}\end{aligned}\ ] ] once again , the dependence of the probability distribution , @xmath31 , on the state variables @xmath33 has been suppressed in the sum over @xmath5 on the right - hand side of this equation . we will refer to the discrete model defined by eqs . as the full microscopic model , since we we shortly derive a mesoscopic model from it , and subsequently further reduce this model . once an initial state has been specified , the model is completely determined , and in principle @xmath34 can be found for any state @xmath27 at any time @xmath28 . this behavior can be explored numerically _ via _ monte carlo simulations ( e.g.  using gillespie s method  @xcite or other approaches  @xcite ) however our main interest in this paper is in approximating the master equation to obtain models which are more amenable to analysis .    a mesoscopic description of the process involves going over to new variables @xmath35 and @xmath36 , @xmath12 , which are assumed continuous for large @xmath2 , but retaining the stochastic nature of the system by keeping @xmath2 large , but finite . a macroscopic description , on the other hand , would involve taking the limit @xmath37 , which renders the process deterministic , and which eliminates all effects of the original discrete nature of the system . the construction of the mesoscopic model involves expanding the master equation , eq . , in powers of @xmath38 and truncating this expansion at order @xmath39 . this is discussed in many places in the literature , but here we follow ref . @xcite , which gives explicit forms for the functions @xmath40 and @xmath41 , @xmath42 , which define the mesoscopic model . carrying out this procedure , the master equation becomes a fokker - planck equation  @xcite of the form @xmath43+\frac{1}{2n^2}\sum^k_{k=1}\sum\limits_{\mu,\nu = 1}^2\frac{\partial ^2}{\partial x^{(k)}_\mu \partial x^{(k)}_\nu}\left[b^{(k)}_{\mu \nu}(\bm{x } ) p\left(\bm{x } , t\right)\right ] , \label{eqn : fpe}\end{aligned}\ ] ] where @xmath44 ; @xmath45 , @xmath46 . we have also introduced @xmath47 . the explicit form of the functions @xmath40 and @xmath41 are  @xcite @xmath48 where @xmath49 are the original transition rates ( scaled by @xmath2 ) given in eqs . ( [ eqn : rate_1 ] ) and ( [ eqn : rate_2 ] ) , but written in terms of the continuous state variables @xmath50 and @xmath51 . the final stage of the construction of the mesoscopic form of the model is to adopt a coarser time scale by introducing a new time variable @xmath52 . from eq . ( [ eqn : fpe ] ) it can be see that without this change of variable , the macroscopic ( @xmath37 ) limit eliminates the right - hand side of the equation , giving a trivial macroscopic limit . with the change of variable , the first term on the right - hand side survives ( but without the factor @xmath53 ) , giving a liouville - like equation for @xmath31 , which implies a non - trivial deterministic dynamics . a clearer way to see this , and in fact a more intuitive formulation of the mesoscopic dynamics to that of the fokker - planck equation , is to use the sde which is equivalent to eq . ( [ eqn : fpe ] ) . we utilize the general result that a fokker - planck equation of the form given by eq . ( [ eqn : fpe ] ) is equivalent to the sde @xmath54 , where the @xmath55 are gaussian random variables with zero mean and a correlation matrix given by the matrix @xmath56 . in terms of the variables @xmath50 and @xmath51 , the sdes take the form @xmath57 where @xmath58 here the noise is to be interpreted in the sense of it  @xcite . it is clear from eqs . ( [ eqn : sk ] ) and ( [ eqn : ik ] ) that the limit @xmath59 eliminates the noise terms leaving deterministic equations , and we recover eq . ( 2.6 ) from ref . @xcite . it is convenient to make the noise in the sdes explicitly multiplicative . since the rates @xmath60 are non - negative we may introduce new functions @xmath61 , @xmath62 , and then transform to new noise variables @xmath63 it is straightforward to check that @xmath64 , and therefore that all the state dependence has been transferred from the correlator to the sdes . equations  correspond to the full mesoscopic model .    in order to describe the evolution of the entire population , we need to consider the sdes for all @xmath65 ; that is , we have to work with @xmath66 equations . in this paper , we will be interested in networks where individuals can pick degrees from a truncated zipf distribution of the form @xmath67 and we let @xmath68 . since we are interested in the case when @xmath4 is large , the number of equations making up the full mesoscopic model are so large that it can make any direct treatment impractical . therefore , in the following sections we will be concerned with reducing the number of equations , and we start by looking at the deterministic limit of the system . this will also have a crucial role to play in the reduction of the stochastic model , described in sec . [ sec : fastvarelim ] . the deterministic limit which controls the dynamics of the macroscopic version of the model is found by taking the @xmath37 limit of eqs . , and so eliminating the noise terms . for notational convenience we reorder the entries of the state vector , so that @xmath69 , and write eqs .   in the form @xmath70 . to find the fixed points of the dynamics we set the time derivatives on the left - hand side of these equations to zero ; these fixed points will be denoted by an asterisk and obey @xmath71 . it is immediately clear that there is a set of stable fixed points with @xmath72 and @xmath73 undetermined , for all @xmath5 , corresponding to a disease - free state .    for a given initial condition , the fixed point is uniquely determined , however it may not be stable . to investigate the stability of the fixed point @xmath74 we perform a linear stability analysis . the @xmath75 stability matrix @xmath76 is defined so that near @xmath77 , the dynamics are @xmath78 , and is found to have the form @xmath79 where @xmath80 is a @xmath81 matrix with entries @xmath82 and @xmath83 is the @xmath81 unit matrix . the eigenvalues and eigenvectors of @xmath84 are straightforward to determine . any vector which has the last @xmath4 entries equal to zero is an eigenvector with eigenvalue zero , reflecting the degeneracies of the system . similarly a vector with the first @xmath4 entries set equal to zero and the last @xmath4 entries equal to @xmath85 , @xmath12 , is an eigenvector with eigenvalue @xmath86 if @xmath87 , that is , if @xmath88 . these then yield another degenerate set of @xmath89 eigenvalues @xmath86 . the final eigenvalue can be found by equating the trace of @xmath84 to the sum of its eigenvalues . in summary , the @xmath90 eigenvalues are @xmath91 while a suitable set of corresponding eigenvectors are listed in appendix  [ sec : eve ] . from eq . ( [ eqn : eva_full ] ) we see that the condition for the fixed points to be ( marginally ) stable is @xmath92    as mentioned in the introduction , there are two possible outcomes for the deterministic dynamics depending on the parameter values : either a negligible or a significant number of individuals in the population will get infected during the course of the epidemic . the quantity that determines which of these cases we are in is often referred to as the basic reproductive ratio ; the final size of the epidemic , in turn , can be obtained from the @xmath93 coordinates of the fixed point , @xmath94 . in order to determine these in the general heterogeneous case , we look first at the homogeneous case , where @xmath95 . if we set an initial condition with only a few infective individuals in an almost completely susceptible population , we will have @xmath96 and @xmath97 , where @xmath98 and @xmath99 ; with this , the @xmath37 limit of eq . ( [ eqn : ik ] ) takes the form @xmath100 , early in the epidemic . from this we obtain @xmath101 where @xmath102 corresponds to the basic reproductive ratio . this defines an epidemic threshold , such that for @xmath103 the number of infectives grows exponentially in the early stages of the epidemic , while for @xmath104 the epidemic does not take off and its final state corresponds to @xmath105 .    for the case when @xmath103 , we can find the @xmath93 coordinate of the fixed point by dividing eq .   by eq .   with @xmath95 ; this yields @xmath106 if we set again an initial condition @xmath107 , with @xmath108 , we can integrate the equation above to find @xmath109 where we have used that the final state of the epidemic corresponds to @xmath110 . therefore , to a very good approximation , we can find @xmath93 coordinate of the fixed point as the smallest solution to @xmath111 note that @xmath112 is always a solution , and the only one for @xmath104 . the total epidemic size will be given by @xmath113 , which corresponds to the fraction of recovered individuals at the end of the outbreak . this is due to the fact that the initial number of recovered individuals is zero , and that every individual that gets infected during the course of the epidemic will eventually recover and remain permanently recovered . now , a significant simplification can be made in the deterministic limit of the heterogeneous case , which reduces the @xmath90 dynamical equations to @xmath114 equations ( see refs . @xcite , which consider the static case ) . this is achieved by use of the ansatz @xmath115 ; we also define @xmath116 such that @xmath117 . with these two variables , eqs . , in the case when @xmath118 , can be rewritten independently of @xmath5 as @xmath119 where we have introduced @xmath120 . if we also introduce the probability generating function of the degree distribution @xmath121 , then @xmath122 . in this reduced model , there is a line of stable fixed points , uniquely determined by the initial conditions , which satisfy @xmath123 . the stability matrix for these fixed points is the @xmath124 matrix @xmath125 which has a similar form to the stability matrix for the full system given by eq . ( [ eqn : j_2k_defn ] ) . once again , an eigenvector exists which has the last entry equal to zero . this corresponds to an eigenvalue zero , reflecting the existence of a line of fixed points . the trace of @xmath126 then gives the second eigenvalue , and the two eigenvalues are thus @xmath127 with the eigenvectors given in appendix  [ sec : eve ] . the stability condition is now @xmath128    now we can find analogous expressions to those in eqs .   and for the heterogeneous case . we start by noting that , in terms of @xmath129 , the total number of susceptible individuals will be given by @xmath130 it is clear that if @xmath131 , then @xmath132 ; conversely , if @xmath133 then @xmath134 , and so @xmath135 . this means that a completely susceptible population will correspond to @xmath129 being equal to unity . a small initial number of infectives will , in turn , correspond to @xmath136 . in addition from eq . ( [ eqn : lambda_det ] ) , @xmath137 , where @xmath138 , at early times . therefore @xmath139 from eq .  ( [ eqn : lambda_early ] ) it is clear that there is a critical value of @xmath140 equal to @xmath141 ; if @xmath140 is larger than @xmath141 the infection grows , and if it is below @xmath141 it does not . thus @xmath140 is the heterogeneous basic reproductive ratio , generalizing @xmath142 in the homogeneous case . central to our investigation is the fact that this quantity does not have a deterministic threshold for networks characterized by power - law degree distributions in the case of interest , i.e. , when @xmath7 .    in order to find the heterogeneous final size , we proceed again as in the homogeneous case ; dividing eq .   by eq . , we find @xmath143 from this , the heterogeneous analog of eq . is given by @xmath144 where we again set an initial condition with a very small number of infectives , @xmath145 , in an almost completely susceptible population , @xmath146 , and the final state is given by @xmath123 . the @xmath129 component of the fixed point will then correspond to the smallest solution to @xmath147\right\rbrace , \label{eqn : theta_fix}\end{aligned}\ ] ] with @xmath140 the heterogeneous basic reproductive ratio defined above . the epidemic size , by definition , will be given by @xmath148 . furthermore , the @xmath50 components of the fixed point correspond to @xmath149 . our interest in this paper is in the ( finite @xmath2 ) stochastic dynamics of the model , where a complete reduction of the kind that led to eqs . ( [ eqn : theta_det ] ) and ( [ eqn : lambda_det ] ) is not possible . in order to see this , we proceed in the same manner as in the deterministic case : we start from eq . , this time without taking the @xmath150 limit , and use the ansatz @xmath151 ; summing over @xmath5 and dividing by @xmath152 , we arrive at @xmath153 where we have introduced @xmath154^{-1}\sum_k \sigma^{(k)}_1 \rho^{(k)}_1(\tau)$ ] . since the noises @xmath155 have zero mean and are independent of each other , we can rewrite @xmath156 as @xmath157 where @xmath158 has zero mean and unit variance , and @xmath159^{-2}\sum_k \left[\sigma^{(k)}_1\right]^2=\left[g^\prime ( \theta)\right]^{-1}\beta \lambda \theta$ ] . with this , we arrive at a reduced equation for @xmath129 given by @xmath160    as in the deterministic case , then , we can express the @xmath4 equations for the @xmath50 variables , described by eq . , entirely in terms of @xmath129 and @xmath161 _ via _ eq .  . however , we will see that the same does not hold for the dynamics of the @xmath51 variables , as we have anticipated .    in a similar fashion as with the @xmath50 variables , we take the equation for the @xmath51 variables , eq .  , multiply it by @xmath5 and sum over @xmath5 , to obtain @xmath162    let us now denote the noise terms appearing in the equation for @xmath161 above as @xmath163 and @xmath164 . these two noise terms , like @xmath156 , have zero mean ; also , it can be readily verified that the correlations between all three of them are given by @xmath165 , where now @xmath166 , with @xmath167,\qquad \bar{b}_{33}=\gamma \sum_k k^2 i_k,\end{aligned}\ ] ] and @xmath168 , where @xmath169 . we can now , as in sec . [ sec : formulation ] , express @xmath170 and @xmath171 in terms of three independent normally - distributed random variables , @xmath172 and @xmath173 , with zero mean and @xmath174 , @xmath166 . the definition of @xmath158 in terms of @xmath156 has already been given by eq . ( [ eqn : xi_to_zeta ] ) ; the other two are : @xmath175 where @xmath176 and @xmath177 . substituting all this back into eq . yields @xmath178 . \label{eqn : lambda_stoch}\end{aligned}\ ] ]    the main feature we note from this is that the two - dimensional system of equations can not be closed . this is so because @xmath179 is still dependent on @xmath51 for all values of @xmath5 , and so it is not possible to achieve a complete reduction of the stochastic system . therefore , when we take the intrinsic noise present in the system into account , a partially reduced model of @xmath180 equations  eq .   for @xmath129 , with @xmath181 , and eq .   for @xmath51is the best we can achieve .    due to this , the exploration of the epidemic dynamics for extremely heterogeneous cases , in which @xmath4 can take very large values , is rendered impractical : even after reducing the number of equations , we will still be left with a very - high - dimensional system . however , progress can be made by observing that , under certain conditions , the deterministic limit of the model presents a separation of time - scales , as we shall see in the following section . we will end this section by carrying out this partial reduction in the deterministic case , as a preparation for the stochastic treatment . this partial reduction involves applying the above reduction to the @xmath50 variables only , by writing @xmath115 , but keeping the @xmath4 @xmath51 variables . there are now @xmath180 dynamical equations which take the form @xmath182 there is again a line of fixed points corresponding to disease - free states , @xmath183 , @xmath12 , but with @xmath129 undetermined . the @xmath184 stability matrix is now @xmath185 where @xmath186 is a @xmath4-dimensional vector whose @xmath187-th entry is @xmath188 , @xmath80 is a @xmath81 matrix with entries @xmath189 and @xmath83 is the @xmath81 unit matrix . the eigenvalues and eigenvectors can be found in the same way as before , the eigenvalues having the form @xmath190 with the eigenvectors given in appendix  [ sec : eve ] . the stability condition is once again given by eq . ( [ eqn : stability_cond_total ] ) . as we have discussed , the deterministic limit of eqs .   can be greatly simplified by performing a change of variables and describing the evolution of the system in terms of @xmath129 and @xmath161 . the inclusion of demographic noise , however , hinders such a reduction of dimensionality . it is not possible to obtain a closed two - dimensional system of equations , thus enormously complicating the analysis of the stochastic epidemic , compared to its deterministic limit .    here , we attempt to overcome this issue by exploiting the properties of the deterministic limit of the model , and by finding a way to close the equations for @xmath129 and @xmath161 in the finite-@xmath2 case , while still retaining the essential features of the full model . in order to do this , we start from eqs . ([eqn : semi_2 ] ) for the partially reduced @xmath180-dimensional deterministic system for @xmath129 and @xmath51 . examination of the eigenvalues appearing in eq .  , shows that @xmath191 . if the ratio @xmath192 is small , then there is the potential to carry out a reduction of the system . in such case , the deterministic dynamics would quickly evolve along the directions corresponding to the set of most negative eigenvalues towards a lower - dimensional region of the state - space , in which the system as a whole evolves in a much slower time - scale , and which contains the fixed point @xmath193 . as illustrated in figure  [ fig : l_r0 ] , showing the value of @xmath194 for different choices of @xmath4 , we would expect this time - scale separation to occur for combinations of parameters for which the final size of the epidemic is small , i.e. , for large @xmath195 .    since there is one most negative eigenvalue , with a degeneracy equal to @xmath89 , after sufficient time has passed the evolution of the system will then be constrained to a two - dimensional surface , which we will refer to as the slow subspace . with the aim of reducing the dimensionality of the problem , then , we ignore the fast initial behavior of the system and focus the analysis on the slow surface we do so by imposing the condition that no dynamics exists along the @xmath89 fast directions , i.e. , @xmath196 where @xmath197 , and @xmath198 is the @xmath19-th left - eigenvector of the jacobian @xmath199 evaluated at the fixed point @xmath193 , with @xmath195 given by eq .  . this procedure goes under the name of adiabatic elimination , or fast - variable elimination  @xcite , and it is a common practice in the simplification of deterministic non - linear dynamical systems  e.g . , systems of oscillating chemical reactions  @xcite .     and @xmath200 , @xmath201 , as a function of @xmath195 for different values of @xmath4 ( @xmath202 ) . ]    for ease of application of the method , the left- and right - eigenvectors of the jacobian are normalized in such a way that @xmath203 these are given , respectively , by eqs .   and . imposing the condition given by eq .  , one finds that @xmath204 , where @xmath205 and @xmath206 is given by eq . ( [ xandy ] ) . in terms of the @xmath129 and @xmath51 variables this reads @xmath207\,\sum^k_{l=1 } l i_l , \ \ k=2,\ldots , k , \label{eqn : slow_ik_intermediate}\end{aligned}\ ] ] and this gives the behavior of @xmath51 which limits the dynamics of the deterministic system to a slow subspace . a more convenient form can be found by multiplying eq . ( [ eqn : slow_ik_intermediate ] ) by @xmath5 and summing from @xmath208 to @xmath209 . this allows us to relate @xmath210 to @xmath211 to find the following @xmath89 expressions for @xmath51 : @xmath212 } k d_k \left[\gamma   { \theta^*}^k-\beta   \left(\phi(\theta){\theta^*}^k-\phi(\theta^*)\theta^k\right)\right ] , \ \ k=2,\ldots , k.\label{eqn : slow_ik}\end{aligned}\ ] ]    it should be stressed that the @xmath89 equations ( [ eqn : slow_ik ] ) , which we may write in the form @xmath213 , @xmath205 , should be interpreted as the time - evolution that the @xmath51 need to follow , so that @xmath214 in the directions @xmath215 , @xmath205 . this follows from the deterministic equations of motion @xmath216 and from eq . ( [ eqn : kill_fast ] ) . thus the only temporal evolution is in the @xmath217 and @xmath218 directions . the functional forms @xmath213 , with @xmath219 defined by eq . ( [ eqn : slow_ik ] ) , formally only hold sufficiently near the fixed point that linearization is accurate . however in fig . [ fig : slow_km10 ] we show three - dimensional projections of the deterministic dynamics from eqs . for an epidemic on a network characterized by a power - law degree distribution with @xmath220 and maximum degree @xmath221 , with heterogeneous basic reproductive ratio @xmath222 , which gives @xmath223 . it appears that the solid red line , which represents the deterministic dynamics , lies in the slow subspace for most of the time , not just at late times when the system approaches the fixed point . that this is the case can be seen from a consideration of the jacobian of the system at an arbitrary time , and not just at the fixed point : @xmath224    it is straightforward to check that @xmath225 is a right - eigenvector of the matrix in eq . ( [ eqn : j_k_plus_one_arb_time ] ) with eigenvalue @xmath86 , if @xmath226 . therefore , remarkably , the eigenvectors @xmath227 of the jacobian at the fixed point , defined in eq . ( [ eqn : eve_semi ] ) , are eigenvectors of the jacobian at all times . the existence of this negative eigenvalue with degeneracy @xmath89 means that the system remains in the plane described by the vectors @xmath217 and @xmath218 , since any small deviations out of the plane will tend to be pushed back again . normally , when carrying out a fast - variable elimination , one would find that the system possesses a line of fixed points which is quickly approached at early times , dominated by the deterministic dynamics , and along which the slow , stochastic dynamics takes place . in this case , however , the line of fixed points corresponds to an absorbing boundary of the system : the evolution of the epidemic stops altogether once it reaches the disease - free state . therefore , it is important that the projection onto the slow variables constitutes a good approximation to the dynamics also far from the line of fixed points . we parameterize the slow subspace using the coordinates @xmath228 and @xmath229 . using eq . ( [ eqn : leve_semi ] ) and @xmath230 , we obtain @xmath231    a linear transformation of eqs . ( [ eqn : slow_z1 ] ) and ( [ eqn : slow_z2 ] ) shows that an equivalent set of coordinates are @xmath129 and @xmath232 , previously called @xmath161 . so , in summary , in sec . [ sec : det ] we showed the system of @xmath90 equations leads to a closed set of equations in @xmath129 and @xmath161 . here , through a consideration of the slow modes of the deterministic equations , we have shown that not only can we describe the system in terms of @xmath129 and @xmath161 , but that it remains in the vicinity of a subspace which can be parameterized by these variables . ( left ) and @xmath233 ( right ) for @xmath221 , @xmath220 , @xmath222 , @xmath234 . red , dashed line : deterministic limit , eqs .   ; blue , yellow , green lines : stochastic differential equations and . the initial condition corresponds to @xmath235 , @xmath236 for @xmath205 , and @xmath237 . in both cases , the system is pushed back from the initial condition ( lower right in both panels ) towards the slow subspace described by eq . , on which it seems to stay during the rest of the course of the epidemic . note the different scales in the @xmath238 axes of both panels , so the fluctuations away from the slow subspace are small in both cases ; the fluctuations on the slow subspace can , however , be large , as seen from the departure of the stochastic trajectories from the deterministic dynamics.,title="fig : " ] ( left ) and @xmath233 ( right ) for @xmath221 , @xmath220 , @xmath222 , @xmath234 . red , dashed line : deterministic limit , eqs .   ; blue , yellow , green lines : stochastic differential equations and . the initial condition corresponds to @xmath235 , @xmath236 for @xmath205 , and @xmath237 . in both cases , the system is pushed back from the initial condition ( lower right in both panels ) towards the slow subspace described by eq . , on which it seems to stay during the rest of the course of the epidemic . note the different scales in the @xmath238 axes of both panels , so the fluctuations away from the slow subspace are small in both cases ; the fluctuations on the slow subspace can , however , be large , as seen from the departure of the stochastic trajectories from the deterministic dynamics.,title="fig : " ]    we would like to carry this reasoning over to the stochastic case , to find a reduced set of sdes , but which are closed , i.e.  which can be expressed entirely in terms of the slow variables @xmath239 and @xmath240 ( or @xmath129 and @xmath161 ) . figure [ fig : slow_km10 ] also shows three - dimensional projections of stochastic trajectories from eqs .   and , for a total population of @xmath234 . the behavior is found to be similar to that of the deterministic case : the dynamics seems to quickly reach , and then fluctuate around , the deterministic slow subspace , with the early behavior dominated by the deterministic dynamics . therefore we expect that we should be able to effectively reduce the stochastic system from @xmath180 dimensions to two dimensions , by neglecting the fast initial behavior of the epidemic . we will do this by projecting the stochastic differential equations , eqs .   and , onto the slow degrees of freedom of the deterministic limit of the model ; that is , we will neglect the fluctuations away from the slow subspace , along the @xmath89 fast directions corresponding to the most negative eigenvalue . this can be formulated mathematically through the application of a projection operator that only picks the components of @xmath241 , @xmath242 , and @xmath243 along the slow eigenvectors of the deterministic limit , @xmath244 and @xmath245@xcite : @xmath246^\top + \bm{v}^{\{k+1\}}[\bm{u}^{\{k+1\}}]^\top . \label{eqn : projector}\end{aligned}\ ] ] the details of the derivation are given in appendix  [ sec : projection ] , where it is shown that the sdes given by eqs . ( [ eqn : theta_stoch ] ) and ( [ eqn : lambda_stoch ] ) of sec . [ sec : det ] are recovered , except that now @xmath177 is determined in terms of @xmath129 and @xmath161 . this closed system of stochastic differential equations for @xmath129 and @xmath161 , which we call the _ reduced mesoscopic model _ , constitutes the main result of this paper and so we reproduce it here : @xmath247 , \label{eqn : slow_lambda}\end{aligned}\ ] ] where @xmath248^{-1}\beta \lambda \theta$ ] , @xmath249 - \bar{\sigma}^2_1 $ ] and @xmath250\right ) . \label{sigma_3_explicit}\end{aligned}\ ] ]    we now compare the dynamics of the full microscopic model , defined by eqs . , to that of the reduced mesoscopic model above . figures  [ fig : ts_full_red_3 ] and [ fig : ts_full_red_4 ] show the time series of the full microscopic model in terms of the variables @xmath129 and @xmath161 , and that of the reduced mesoscopic model , for @xmath251 and different values of @xmath4 having the same deterministic final size @xmath195 , obtained from eq .  ; figure  [ fig : phase_full_red_2 ] , in turn , compares the phase diagrams . we see that , in the case of small @xmath4 , the time series of both variables in the full microscopic model are rather smooth , and are dominated by a horizontal spread due to the fact that the epidemic takes off at different moments . as heterogeneity increases , the dynamics of the system becomes noisier and noisier , as seen in the phase diagrams ; however , the time series of @xmath129 continues to be smooth . this observation will be useful in the following section .    from the time series of the reduced mesoscopic model we see that , except for small @xmath4 , the temporal behavior of @xmath129 and @xmath161 does not present the horizontal spread observed in the full microscopic model . however , once the epidemic has taken off , the dynamics of the full microscopic model is correctly captured when @xmath4 is not so large ; this can be seen from the phase diagrams . we note that the relative magnitude of the eigenvalues for this choice of parameters takes the values @xmath252 , and that the reduced mesoscopic model fares well despite @xmath194 being rather large . we find slightly worse agreement between the full and reduced models for smaller values of @xmath195 , corresponding to larger @xmath194 . one further detail to add is that in the case of the reduced model there are far fewer trajectories in which the epidemic does take off , compared to the full microscopic model ; that is , the reduced model results in an over - representation of very early extinctions . note that , for the value of @xmath2 employed in these figures , @xmath253 is the maximum possible degree cutoff . in this case , we would not expect the reduced model to yield accurate results . this is due to the fact that not even the full @xmath90-dimensional mesoscopic model ( eqs . ) provides a good approximation to the microscopic dynamics in this case , since @xmath4 is of the order of @xmath2 and this would need to be explicitly taken into account when performing the mesoscopic expansion leading to the fokker - planck equation  . we end this section with some further observations regarding the reduced stochastic system . first , the existence of a @xmath89-fold degenerate eigenvalue equal to @xmath86 throughout the time evolution also means that stochastic trajectories , as well as deterministic ones , which venture out of the slow subspace will tend to be pushed back into the plane . thus we would expect the reduced set of sdes to be a good approximation throughout the motion , and not just at late times . this may be the reason why the approximation works well even if @xmath194 is not particularly small . second , in retrospect , the elimination of the fast variables in the stochastic system is not strictly necessary , since the sdes previously obtained in sec . [ sec : det ] are recovered , and @xmath179 could be found in terms of @xmath129 and @xmath161 using eq . ( [ eqn : slow_ik ] ) obtained from the elimination of fast variables in the deterministic system . one can compare the value of @xmath179 in terms of the @xmath51 to the one in terms of @xmath161 in the deterministic limit and see that , indeed , the approximation is not very good for moderate to large values of @xmath194 . this does not seem to play a huge role when considered as one of many terms in the stochastic evolution of @xmath161 . finally , we can make a general comment about the nature of the fluctuations as the epidemic starts to grow from its initial state . in this case the relevant eigenvalues will be those of the jacobian eq . ( [ eqn : j_k_plus_one_arb_time ] ) , but with @xmath254 and @xmath255 , and will therefore be given by those in eq . ( [ eqn : eva_semi ] ) , but with @xmath195 set equal to @xmath141 . therefore the largest eigenvalue is now @xmath256 . in this case , if @xmath257 , so that an epidemic starts to grow , the dynamics is pushed away from the line @xmath258 in its early stages , as can be seen from fig . [ fig : slow_km10 ] where the trajectories leave the vicinity of the absorbing state after reaching the slow subspace . +   +   +   +     +   +   +   +    , @xmath259 and @xmath260 . colored lines correspond realizations of the full microscopic model ( top ) and the reduced mesoscopic model   ( bottom ) started with a total number of 5 infectives with degrees sampled from a truncated zipf distribution with @xmath220 and @xmath221 ( left ) , @xmath261 ( center ) , and @xmath253 ( right ) . the black , dashed line corresponds to the deterministic solution.,title="fig:",scaledwidth=26.0% ] , @xmath259 and @xmath260 . colored lines correspond realizations of the full microscopic model ( top ) and the reduced mesoscopic model   ( bottom ) started with a total number of 5 infectives with degrees sampled from a truncated zipf distribution with @xmath220 and @xmath221 ( left ) , @xmath261 ( center ) , and @xmath253 ( right ) . the black , dashed line corresponds to the deterministic solution.,title="fig:",scaledwidth=26.0% ] , @xmath259 and @xmath260 . colored lines correspond realizations of the full microscopic model ( top ) and the reduced mesoscopic model   ( bottom ) started with a total number of 5 infectives with degrees sampled from a truncated zipf distribution with @xmath220 and @xmath221 ( left ) , @xmath261 ( center ) , and @xmath253 ( right ) . the black , dashed line corresponds to the deterministic solution.,title="fig:",scaledwidth=26.0% ] +   + , @xmath259 and @xmath260 . colored lines correspond realizations of the full microscopic model ( top ) and the reduced mesoscopic model   ( bottom ) started with a total number of 5 infectives with degrees sampled from a truncated zipf distribution with @xmath220 and @xmath221 ( left ) , @xmath261 ( center ) , and @xmath253 ( right ) . the black , dashed line corresponds to the deterministic solution.,title="fig:",scaledwidth=26.0% ] , @xmath259 and @xmath260 . colored lines correspond realizations of the full microscopic model ( top ) and the reduced mesoscopic model   ( bottom ) started with a total number of 5 infectives with degrees sampled from a truncated zipf distribution with @xmath220 and @xmath221 ( left ) , @xmath261 ( center ) , and @xmath253 ( right ) . the black , dashed line corresponds to the deterministic solution.,title="fig:",scaledwidth=26.0% ] , @xmath259 and @xmath260 . colored lines correspond realizations of the full microscopic model ( top ) and the reduced mesoscopic model   ( bottom ) started with a total number of 5 infectives with degrees sampled from a truncated zipf distribution with @xmath220 and @xmath221 ( left ) , @xmath261 ( center ) , and @xmath253 ( right ) . the black , dashed line corresponds to the deterministic solution.,title="fig:",scaledwidth=26.0% ] in the previous section , we have derived a reduced two - dimensional system with three white - noise processes , the reduced mesoscopic model eqs .   , which provides a good approximation to the dynamics of the full microscopic model , provided the maximum degree individuals can pick is not so large . this is true even for relatively large values of @xmath194 , corresponding to small separation between the eigenvalues of the system . we have also noted that the time series of @xmath129 , obtained from the full microscopic model , presents very smooth behavior over the range of parameters explored . inspired by this observation , we attempt to further simplify the model by noting that eqs . have the form @xmath262     +   +   +   +    taking into account the fact that , in any given realization of the process , the lowest value reached by @xmath129 will be roughly around @xmath195 , we compare the magnitudes of the noise intensities @xmath263 between @xmath264 and @xmath265 for fixed @xmath195 and different values of @xmath4 . we find that , in general , and especially during the early stages of the epidemic when @xmath266 , @xmath267 is small compared to @xmath268 and @xmath269see figure [ fig : noises ] ; the effect is more pronounced for more heterogeneous distributions with smaller deterministic final sizes , i.e. , larger @xmath195 . therefore , the system may be described simply by a deterministic ordinary differential equation ( ode ) for @xmath129 , and eqs   become @xmath270 with @xmath271 a gaussian noise with zero mean and @xmath272 , with the noise intensity , @xmath273 , given by @xmath274^{1/2}.\end{aligned}\ ] ] we call this system the _ semi - deterministic mesoscopic model _ , which is two - dimensional like the reduced mesoscopic model , but has only one white - noise process in place of the latter s three . here we have used that @xmath275 , and @xmath276 , @xmath166 . now , we note that eq .   corresponds to a cox - ingersoll - ross ( cir ) model  @xcite @xmath277 where @xmath278 is a time - dependent parameter which is the solution of the deterministic differential equation ( [ eqn : theta_simple ] ) , @xmath279 , and @xmath280+\gamma \frac{\phi(\theta^*)+\psi(\theta^*)}{\phi(\theta^*)}}.\end{aligned}\ ] ] for fixed @xmath281 and @xmath282 , the conditional probability distribution of @xmath283 , given @xmath284 , is known in closed form , and corresponds to a non - central chi - square distribution given by  @xcite @xmath285 where @xmath286 and where @xmath287 is the modified bessel function of the first kind of order @xmath288 .    therefore , if we assume that @xmath129 evolves discretely , staying constant from time @xmath289 to time @xmath290 for reasonably small @xmath291 , we can use the result above to simulate eqs . by sampling the values of @xmath161 directly from the distribution in eq .   with fixed parameters @xmath292 , @xmath293 at every time - step . the case @xmath279 corresponds to the special case of a non - central chi - square distribution with zero degrees of freedom ; thus , in order to simulate eq . we follow the method described in refs . @xcite and sample from a central chi - square distribution with poisson - distributed degrees of freedom , with the convention that the central chi - square distribution with zero degrees of freedom is identically zero  @xcite . figure  [ fig : phase_cir ] shows the dynamics obtained in this way . while we note an additional over - representation of early extinctions , the other results are in very good agreement with the reduced mesoscopic model , eqs .  .    , @xmath259 and @xmath260 . colored lines correspond to realizations of eqs . , started with a total number of 5 infectives with degrees sampled from a truncated zipf distribution with @xmath220 and @xmath221 ( left ) , @xmath261 ( center ) , and @xmath253 ( right ) . the black , dashed line corresponds to the deterministic solution.,title="fig:",scaledwidth=26.0% ] , @xmath259 and @xmath260 . colored lines correspond to realizations of eqs . , started with a total number of 5 infectives with degrees sampled from a truncated zipf distribution with @xmath220 and @xmath221 ( left ) , @xmath261 ( center ) , and @xmath253 ( right ) . the black , dashed line corresponds to the deterministic solution.,title="fig:",scaledwidth=26.0% ] , @xmath259 and @xmath260 . colored lines correspond to realizations of eqs . , started with a total number of 5 infectives with degrees sampled from a truncated zipf distribution with @xmath220 and @xmath221 ( left ) , @xmath261 ( center ) , and @xmath253 ( right ) . the black , dashed line corresponds to the deterministic solution.,title="fig:",scaledwidth=26.0% ] in the previous sections , we have derived a two - dimensional reduction of the @xmath90-dimensional sir dynamics described by eqs .   , by exploiting a separation of time - scales in the deterministic dynamics of the system . we have found reasonably good agreement between the temporal behaviors of the full and reduced models . furthermore , we have also discussed an extra simplification , consisting in neglecting the noise acting on the susceptible variable @xmath129 , and describing the system in terms of one ode and one sde instead of two sdes . now we explore how accurately these reductions capture the distribution of epidemic sizes , given by the number of recovered individuals present in the system at the end of the epidemic , @xmath294 , which can be obtained as @xmath295 and @xmath296 in the full and reduced models , respectively , where @xmath297 and @xmath298 are the final values of @xmath50 and @xmath129not to be confused with the fixed points in the deterministic limit , @xmath73 and @xmath195 . formal mathematical work has considered central limit theorems for models that generalize ours  @xcite ; we are interested here in explicit calculation and simulation approaches . we obtain the distribution of @xmath294 from the full model _ via _ a sellke construction  @xcite , and note that it is characterized by two regions : one to the left of the interval , corresponding to early extinctions , and one to the right corresponding to the epidemic taking off . the first thing we note when comparing this to results from the two versions of the reduced model eqs. , and eqs . is that , as discussed previously , there is an over - representation of very early extinctions in the case of the latter , and therefore the relative sizes of the left- and right - most regions of the distribution are not well captured . we find that the agreement improves significantly if we condition the epidemic to take off , and only consider values of @xmath294 larger than some prescribed limit , say @xmath299that is , we only take into account realizations in which at least two individuals become infected , apart from the initial infective .    with this idea in mind , we can go a bit further by obtaining an approximation of the height of the left - most portion of the distribution of @xmath294 . if we denote by @xmath300 the probability that the initial infective , with degree @xmath5 , does not spread the disease to any other individuals during its _ lifetime_before recovering  and , similarly , by @xmath301 the probability that it spreads the disease to one , and only one other individual with degree @xmath18 , we can write @xmath302 where we have again used the ansatz @xmath303 . the expressions are obtained by interpreting the initial behavior of the epidemic as a birth - death process , where a ` birth ' corresponds to the infection of an individual with degree @xmath18 with rate @xmath304 ; the ` death ' of the initial infective , in turn , corresponds to its recovery , which occurs with rate @xmath23  @xcite . with this , we obtain for the total probability @xmath305 , @xmath306 where we take the first infective to have degree @xmath5 with probability @xmath307 . since , initially , @xmath308 , we have @xmath309 , i.e. , the first moment of the degree distribution , which we denote by @xmath310 . therefore , @xmath311.\label{eqn : p2}\end{aligned}\ ] ]    figure  [ fig : r_inf ] shows the distribution of final sizes from the reduced mesoscopic model and the cir - like model , renormalized so that the fraction of trajectories with @xmath312 is equal to @xmath313 , and compares these to the distribution from the full model . this shows that the birth - death chain approximation we have introduced provides a simple method to adjust for the inaccuracies in early extinctions we have previously noted , yielding an accurate approximation overall .     +   + stochastic epidemics on heterogeneous networks involve two potentially large parameters . the first of these is the size of the population , @xmath2 , which is proportional to the dimensionality of the microscopic description . for large populations , this motivates the derivation of a mesoscopic model based on a diffusion approximation whose errors are controlled by inverse powers of @xmath2 , and which we have formulated in this paper . the second of these is the maximum degree @xmath4 , which is proportional to the dimensionality of the full mesoscopic model . we have been able to use fast - mode elimination techniques to derive two - dimensional approximations to this scheme : the reduced mesoscopic model , which involves three white - noise processes ; and the semi - deterministic mesoscopic model , which involves only one white - noise process .    our numerical results back up our theoretical understanding that these low - dimensional systems are accurate approximations to the microscopic model , except in the case where @xmath4 approaches @xmath2 , although this is to be expected since our derivation of the diffusion limit implicitly assumes that @xmath314 . nevertheless , the reduction can be applied to cases with extremely large values of @xmath4 , as long as they are not too close to @xmath2 , thus significantly simplifying the original model . we have observed that the elimination of fast variables in the system provides a good approximation to the full dynamics even for relatively large values of @xmath194 , corresponding to a small separation between time - scales . we have also shown that the main inaccuracy of the reduction , relating to very early extinction probabilities of the epidemic , can be dealt with using a simple birth - death process argument . in this way , we hope to have provided a useful tool in the study of both infectious diseases and spreading processes on heterogeneous networks . cp - r was funded by conicyt becas chile scholarship no . th was supported by the uk engineering and physical sciences research council ( epsrc ) . a set of right - eigenvectors corresponding to the eigenvalues from eq .   are given by @xmath315 where @xmath316 the @xmath18-th unit vector and @xmath317      a set of right - eigenvectors corresponding to the eigenvalues from eq . are given by @xmath318      the partially reduced system with eigenvalues given by eq . has a set of right - eigenvectors given by @xmath319 where now @xmath320    in eqs . ( [ eqn : eve_full ] ) and ( [ eqn : eve_total ] ) we did not normalize the eigenvectors . however in this case we do need to eigenvectors to be normalized to simplify the application of the reduction technique described in sec . [ sec : fastvarelim ] . this requires finding the corresponding left - eigenvectors . when finding these , we have a complication given by the fact that there is one eigenvalue which is @xmath89-fold degenerate , and the condition   will not necessarily hold in general . however , if we construct a matrix @xmath321 in such a way that its @xmath18-th column corresponds to @xmath322 , we can choose @xmath198 to be the @xmath19-th row of the inverse matrix @xmath323 , if such an inverse exists . we find that indeed this is the case , and the left - eigenvectors in the partially reduced model are then given by @xmath324    we note that @xmath325    using this property , we can verify that the orthogonality condition   is satisfied : @xmath326 in this appendix we derive the two - dimensional reduced mesoscopic model , by projecting the dynamics of the @xmath327-dimensional partially reduced mesoscopic model onto the slow degrees of freedom of its corresponding deterministic limit .    using the projector defined in eq . ( [ eqn : projector ] ) on @xmath328 gives the slow variables defined by eqs . ( [ eqn : slow_z1 ] ) and ( [ eqn : slow_z2 ] ) . later , the inverse transformation , back to the more familiar variables @xmath129 and @xmath329 , will be required . it is @xmath330},\label{eqn : theta_slow}\\ \lambda(\bm{z } ) & = \frac{\phi(\theta^*)}{k d_k{\theta^*}^{k}}\,z_2 , \label{eqn : lambda_slow}\end{aligned}\ ] ] however for the moment we will use the @xmath239 and @xmath240 variables , and will denote the right - hand side of eq . ( [ eqn : theta_slow ] ) as @xmath331 . the deterministic slow dynamics , obtained by applying the projector to @xmath332 , will be given by @xmath333,\label{eqn : slow_dz1}\\ \dot{z}_2 & = z_2\left(\beta \phi ( \chi(\bm{z}))-\gamma \right),\label{eqn : slow_dz2}\end{aligned}\ ] ] where the dot represents the time derivative . applying now the projector to the noise terms in the finite-@xmath2 case we find , for the component along @xmath244 , @xmath334\label{rhs_z2dot } \\ & = \frac{1}{\sqrt{n}}\left[\left(1-\frac{\phi(\chi(\bm z))}{\chi(\bm z)}\frac{\beta \theta^*}{-\gamma + \beta \phi(\theta^*)}\right)\bar{\sigma}_1(\bm z)\zeta_1(\tau ) + \frac{\beta \theta^*}{-\gamma + \beta \phi(\theta^*)}\left(\bar{\sigma}_2(\bm z)\zeta_2(\tau)+\bar{\sigma}_3(\bm z)\zeta_3(\tau)\right)\right ] , \label{eqn : eta_z1}\end{aligned}\ ] ] where we obtain , from eqs .  , and , @xmath335^{1/2},\\ \bar{\sigma}_2(\bm{z})&= \left[\frac{\beta \phi(\theta^*)z_2}{kd_k{\theta^*}^k}\left(\phi(\chi(\bm z))+\psi(\chi(\bm z))-\frac{\phi(\chi(\bm z))^2}{\chi(\bm z)g^\prime ( \chi(\bm z))}\right)\right]^{1/2},\\ \bar{\sigma}_3(\bm z ) & = \left[\frac{z_2}{k d_{k}{\theta^*}^{k}}\left(\gamma \left[\phi(\theta^*)+\psi(\theta^*)\right]+\beta \left[\phi(\theta^*)\psi(\chi(\bm{z}))-\phi(\chi(\bm{z}))\psi(\theta^*)\right]\right)\right]^{1/2}. \label{eqn : sigma_3_bar_determined}\end{aligned}\ ] ] similarly , for the component of the fluctuations along @xmath336 : @xmath337    we now form @xmath338 and @xmath339 as linear combinations of @xmath340 and @xmath341 , using eqs . ( [ eqn : theta_slow ] ) and ( [ eqn : lambda_slow ] ) . these are equal to the appropriate linear combinations of eqs . ( [ eqn : slow_dz1 ] ) , ( [ eqn : slow_dz2 ] ) , ( [ eqn : eta_z1 ] ) and ( [ eqn : eta_z2 ] ) . one finds that eqs . ( [ eqn : theta_stoch ] ) and ( [ eqn : lambda_stoch ] ) are exactly recovered , except that now @xmath177 is determined , given by eq . ( [ eqn : sigma_3_bar_determined ] ) .    47ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ]  + 12$12  & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ( ) , @noop * * ,   ( ) @noop * * ,   ( ) @noop _ _  ( , ,  ) @noop _ _  ( ,  ,  ) @noop * * ,   ( ) @noop * * ,   ( ) ,   @noop _ _  ( ,  ,  ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * , ( ) @noop * * , ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) in @noop _ _  ( )  pp . @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop _ _ ,  ed . ( ,  ) @noop * * ,   ( ) @noop _ _  ( ,  ) @noop _ _ ,  ed . ( ,  ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop _ _  ( ,  ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) @noop _ _ ,  vol . ( ,  )   `` , ''  in  @noop _ _  ( ,  ) @noop * * ,   ( ) @noop * * ,   ( ) @noop * * ,   ( ) ,
networks of contacts capable of spreading infectious diseases are often observed to be highly heterogeneous , with the majority of individuals having fewer contacts than the mean , and a significant minority having relatively very many contacts . 
 we derive a two - dimensional diffusion model for the full temporal behavior of the stochastic susceptible - infectious - recovered ( sir ) model on such a network , by making use of a time - scale separation in the deterministic limit of the dynamics . 
 this low - dimensional process is an accurate approximation to the full model in the limit of large populations , even for cases when the time - scale separation is not too pronounced , provided the maximum degree is not of the order of the population size .