Available flows

If the flows between locations are already known, one possibility to assess the travel rates is to fix the fraction of individuals who are away at the moment, over individuals who stay at home, i.e. to fix the already introduced commuter ratio (cf. Section 4.4):

ε^m_k = N_k^m

N_k^k. (4.59)

E.g. one can setε^m_n =ε0/kn, whereknis the number of neighbouring locations of the location n(the nodes degree). Next, given the partial fluxesωⁿ_nmN_n^m =ω_mnⁿ N_nⁿ=γ_nmF_nm, we obtain for the backward rate which is now not fixed

ωⁿ_nm=γnm

Fnmkn

ε0N_nⁿ and for the forward rate

ω_mnⁿ =γnm

F_nm N_nⁿ , where from (4.53) and (4.59),N_nⁿ =Nn

(1 +3

m&=nε0/kn

)−1

. For travel rates of individuals belonging to the citym, we should swap the indices and changeγ_nmto1−γ_nm. The simplest possibility is to assume equal partial flowsγ = 1/2. Further assumptions can be made on how ε^m_n andγnmare related to degree and size of the nodes.

The next step should be the comparison of the models with random walk and recurrent movements using the proposed models for the travel rates. The interlocation flows should be kept the same in both models. In the random walk case, the transition rates are given through ω_kn = F_kn/N_n. One of the important questions is whether the network topology influences the epidemics due to recurrent travel pattern in another way than epidemics due to random walk travel pattern under disordered conditions. Two simple extreme cases are obvious. In the all-to-all coupling network, there is almost no difference in the dynamics for the commuter ratio corresponding to the equal time spent in neighboring and home locations. In the case of regular lattice, the difference is most pronounced. One can argue that all other topologies, including random networks, leads to intermediate behavior between these two extreme cases.

4.7 Perspectives

So far, we considered the recurrent transport mechanism in metapopulational epidemic sys-tems, related this to the directly coupled model and analysed the continuum approximation.

However, there is still more to investigate on. Generalizations to two-dimensional support and incorporation of long-range movements is still an open question. We argue, that in the latter case the attenuation (as compared with the random walk mechanism) of the epidemic spread is reduced. In the continuum limit, one can take into account fluctuations using field theoretical methods (Doi, 1976b,a; Kree et al., 1989) or moment closure methods (Parham and Ferguson, 2006). Thus far, we considered recurrent movements which are Markov processes.

One can also introduce memory for individuals, prohibiting visits to locations visited in the immediate past. Even periodic movements corresponding to clear home-work rhythm could be considered. The dynamics on network topologies need to be investigated in detail using the models for travel rates incorporating inhomogeneity and spatial properties as proposed in Sec-tion 4.6.1. It is important to know how network properties such as community structure and modularity (Newman, 2006; Watts et al., 2005) influence epidemiological dynamics with re-current transport. Within our framework, we can easily allow travel not only to next-neighbor locations and returns over remote nodes. The problem of the limited length percolation (López

et al., 2007) is also relevant in this context. Inhomogeneity of the human population (Liljeros et al., 2001; Pastor-Satorras and Vespignani, 2000; May and Lloyd, 2001; Sokolov and Belik, 2003) could also be addressed. More complex kinds of epidemiological dynamics (Grenfell et al., 2001) leading to synchronisation in the course of epidemic among different locations require also inter-patch correlations to be considered (Hagernaas et al., 2004). Note, that re-current movements are not only human specific behavior but could be pertinent to animal systems as well.

The recurrent movements play a role not only in the epidemiological systems. There are also many other situations where this mechanism may be encountered. One of possible appli-cations of the recurrent movements is the transport of some items analogous to the transport of pathogens in epidemiology. It can be e.g. money bills (Brockmann et al., 2006) or — as previously mentioned — trackable items from geocaches (Brockmann and Theiss, 2008b). A large field of of biological research is devoted to seed dispersal (Nathan, 2006). Seeds cannot move by themselves and need to be transported by wind or herbivorous animals. Let us look at the model for items dispersal due to recurrent movements more closely and compare it to the random walk model.

We consider a situation in which items can be left somewhere independently of the indi-viduals who transport them (seeds) and a setting in which items cannot be separated from the individuals (some parasites, money bills). We consider a linear chain of locations with the next-neighbour coupling. The first reaction system with an underlying random walk pattern of individual movements reads:

An+Gn

→α Agn

Agn

→β An+Gn (4.60)

A_n "^ω A_n±1

Agn

"ω Agn±1,

where we denote the individuals without items by An, the individuals with items by Ag_n and the items themselves by Gn. The second model accounts for recurrent movements of individuals:

Aⁿ_n+Gn

→α Ag_nⁿ Aⁿ_n±1+Gn

→α Ag_n±1ⁿ

Ag_nⁿ →^β Aⁿ_n+Gn (4.61)

Ag_n±1ⁿ →^β Aⁿ_n±1+Gn

Aⁿ_n "^ω Aⁿ_n±1 Ag_nⁿ "^ω Ag_nⁿ_±1,

whereAⁿ_m andAgⁿ_m are the numbers of individuals with and without items, belonging to the locationmand being now in the locationn. In Figure 4.18, the concentrations of items initially

situated in the first location at different times are presented. Red lines corresponds to the first random walk model, the blue one — to the second recurrent model. One can see, that as expected, random walk movements pattern leads to stronger dispersal due to the same reasons as in the case of epidemics. The interchange of items or pathogens depends on the probability of being at a particular location and it does not change significantly with increasing travel rate in the recurrent process. To quantitatively assess the model with items dispersal, one can try to use methods which were already successfully applied for various reaction-diffusion processes.

0 20 40 60 80 100 walk (red lines) or recurrent movements (blue lines) at dif-ferent times due to numerical solution of the mean-filed ki-netic equations corresponding to (4.60) and (4.61) on a chain with100sites. Parameters are α = β = ω = 1 and initial conditionsG(x = 0, t= 0) = 1000.

Furthermore, our model with origin-dependent transition rates may be relevant for various dispersal phenomena (Clobert et al., 2003), i.e. spread of species or invasion into new habitats, such as the spread of genes (Novembre et al., 2008), influenza viruses (Russell et al., 2008) or infectious vectors (Tatem et al., 2006). Miscellaneous ecological spatially extended sys-tems with interacting species, such as prey-predator or Lotka-Volterra syssys-tems (Volterra, 1926;

Murray, 1993) are also of interest. Our approach can be applied even to such an exotic phe-nomena as rumour spread (Boccaletti et al., 2006). In this context, various self-organisational, non-equilibrium critical phenomena occurred (Horsthemke and Lefever, 2006; Stollenwerk and Jansen, 2007), which can be described in the game-theoretical framework (Nowak, 2006;

Reichenbach et al., 2007) or by field-theoretical methods (Kuzovkov and Kotomin, 1988). We expect to find similar phenomena also in our framework.

The field research, i.e. the empirical assessment the parameters of the models, is of utmost importance and needs to be extended in future. Alongside with the direct methods such as census surveys or tracking of individuals (González et al., 2008; Cooke et al., 2004) indirect methods based on tracking of auxiliary items, could provide a reasonable alternative (Brock-mann et al., 2006; Brock(Brock-mann and Theiss, 2008b).

We wanted to draw attention to presice consideration of agent movements — mediators of spread. Our approach was an attempt to capture essential features of human-mediated

spreading phenomena. We hope that many other adequate models will be established in future.