1 Introduction

The importance of “place”, “location” or “venue” in the study of disease transmission has not been recognized widely. Some studies have been undertaken on the probabilities of infection at particular “places” such as within an airplane (Mangili and Gendreau 2005), but generally, few studies have considered the roles that “place” plays in the disease spreading process (i.e., transmission between “places” by the movements of humans). Klovdahl et al. (2001) discussed the importance of “place” in a study of a community-level tuberculosis outbreak. They suggested that “person-oriented” methods should be supplemented by “place-oriented” methods. Koopman (2004) stated that “Data on how often and when individuals go to specific places where they might get infected from other individuals can be used to describe contact patterns by making assumptions or by gathering data regarding the nature of contact at these different places.” It is well known that commonly people repeat similar daily activities and certain activities occur at certain “places”. Based on the simulation model TRANSIMS, Eubank et al. (2004) displayed the changing number of people present within locations such as the home, workplace, school and so on during a whole day.

Although it is well known that the transmission probabilities in different types of places may be different, factors such as environmental conditions and the spatial locations of individuals that affect transmission probabilities at particular places are poorly understood (Koopman 2004). As an example, consider infection in confined spaces. The work of Wells (1955), which assumes complete air mixing, is the basis for models such as the Mass Action model (Riley 1974), Riley, Murphy and Riley’s model (Riley et al. 1978) and Gammaitoni and Nucci’s model (Gammaitoni and Nucci 1997; Beggs et al. 2003). In the above models, the infectious substances are evenly distributed throughout the place so that the proximity of susceptible individuals to infectious individuals, and the duration of exposure are neglected. In reality, the air in confined spaces is seldom mixed completely. Also most places are only partly confined or not confined at all.

At fine space–time scales, research on the dynamics of humans inside a small space can be traced to “room geography” (Jackle et al. 1976) which studied how individuals distribute across a specific small area. However, research progress has been slow due to the lack of systematic data and corresponding tools and methods (Batty et al. 2003). To model humans’ contacts which could cause infections at fine spatial scales, the concept of activity bundle (AB) is defined as a space where contact probability varies as a function of the dynamics of humans inside (Yang and Atkinson 2008a). To avoid confusion, this paper uses “activity bundle” to replace “place”, “location” or “venue”.

In approximate terms, the infection process of airborne infectious disease is as follows: infectious individuals shed infectious substance to the air, the substance stays and/or diffuses up to a certain distance and until a certain duration according to the AB’s physical condition (such as ventilation and sanitation conditions) and then, if susceptibles happen to be within the “contaminated” air and absorb the infectious substance, then infection is possible. It is accepted that the possibility of infection for susceptible individuals increases with proximity to infectious individuals (Hutton et al. 1990; Noakes et al. 2006; Oppong et al. 2006). In the above process, the space–time dynamics of the infectious substance depends on airflow rates, heating and cooling, and the architectural properties of the AB. The more challenging question here is, at the within-AB level, how best to model humans’ space–time interactions (or to be more precise, the space–time relations between susceptibles and infectious individuals or the infectious substance) which are crucial for infection. In this study, the infectious substance is assumed to stay with infectious individuals and, thus, within-AB simulation is defined as a method to obtain specific contacts (specific to the target infectious disease) from the space–time dynamics of individuals constrained by both the individuals’ social activity and the physical characteristics of the space.

One example of within-AB simulation was provided by Epstein et al. (2002). In this model, the spaces of all ABs are structured as raster grids and the whole day is discretized into a specific number of rounds. In each round and each social unit, one infectious agent interacts with one of their Moore neighbour agents. The locations of individuals within an AB change every day. Fewer contacts are assumed to occur at the workplace or school than at the home or hospital. The likelihood of an interaction resulting in a contact at home is 1.0 and at work is 0.3. These values were based on intuition instead of estimated from data. In fact, the cell occupied by an individual does not reflect the individuals’ spatial location within an AB since there is no “spatial distance” meaning between cells. It can be taken as a graph to describe possible contacts between individuals in different types of AB (with different weights such as the above values of 1.0 and 0.3).

Another example is provided by the EpiSims model (Eubank 2002) which maintains a list of individuals who are present and a disease load for every AB. Load here means the viral concentration in an individual or AB. The load grows or decays with time depending on the specific infectious disease. Contamination by shedding from infectious individuals may be restricted to a small region near the infected person, or may spread to the entire location. Due to the lack of data for the proximity of people, Eubank et al. (2004) split a large location into small sublocations small enough to assume that all individuals within the same sublocation are within the distance for infection. One problem is that this split assumes that no infection can occur between individuals in adjacent sublocations even when they are close enough for infection. Another problem is that the allocation of individuals to these sublocations cannot be avoided.

The concept of AB was applied previously within an individual space–time activity-based model (ISTAM) to simulate airborne infectious disease transmission (Yang and Atkinson 2008a). Taking advantage of the concept of an AB, ISTAM considers the activities of human individuals at two levels: between-AB level and within-AB level. At the within-AB level, raster space AB, vector space AB and role-based AB simulation were proposed and discussed (Yang and Atkinson 2006). In this paper, the parameters of raster space AB simulation were explored to further understanding of the transmission process at the within-AB level, that is, the relation between the transmission of airborne infectious diseases and humans’ space–time dynamics at fine space–time scales. This research may be considered as a supplement to the ISTAM model, providing advice on the various within-AB simulation parameter choices for use within raster space AB simulation models.

In Sect. 2, the individual space–time activity model and raster space AB simulation are introduced. The parameters of raster space AB simulation are explored in Sect. 3. Section 4 provides an example application of raster space AB simulation. Sections 4 and 5 present a discussion and conclusion, respectively.

2 Methods of simulation at fine scales

2.1 Individual space–time activity-based model

ISTAM is an individual-based model that integrates the infectious disease evolution process at the individual level, individual activity patterns and a stochastic infection model. The ISTAM model was developed by the integration of Repast (2004) and the Java Topology Suite (JTS 2005). ISTAM is a bottom-up model in which the transmission network is built on the physical contacts between individuals at a fine space–time scale. At this scale, human social behavior, the environment’s physical condition and specific infectious disease transmission modes are considered. ISTAM was applied to simulate hypothetical influenza outbreaks in the campus of the University of Southampton (Yang and Atkinson 2008a) and the city of Eemnes (the Netherlands), respectively (Yang et al. 2008b). Different control measures were tested. The results show that the model behavior is approximately consistent with expectations. Using ISTAM, data can be simulated at the micro-level and aggregated to several different higher levels. These data can be used for model calibration and validation purposes. Also further analyses can be based on these simulated data.

Three properties of humans’ space–time dynamics within ABs which have been observed in reality are considered during modeling the space–time dynamics of individuals within ABs: (1) individuals’ static spatial distribution pattern; (2) individuals’ movement pattern; (3) minimum distances between individuals. The spatial distribution pattern of individuals can be observed in most types of AB. One example is provided by a restaurant: people sit in clusters which reflect the existence of different groups. Another example is provided by individuals visiting a library: they try to find an empty table and sit as far as possible from each other (Given and Leckie 2003). At a fine spatial scale, individuals’ movement patterns are strongly confined by the physical condition of the current AB and the status of other individuals. In some ABs, individuals are assumed to remain static during the simulation time unit, such as in a lecture room. In other ABs, movements must be considered.

In ISTAM, two parameters are used to describe individuals’ movements inside an AB during one time unit: movement frequency and movement proportion. Movement frequency (M f ) refers to how often individuals move while movement proportion (M p ) refers to the proportion of individuals who choose to move instead of staying where they are. The importance of spatial distance between individuals to infection has been discussed above. It needs to be pointed out that there are social rules about how close humans can approach each other. Hall (1966) identified four distances that are normally used by people in North America in relation to others: (i) intimate, (ii) personal, (iii) social and (iv) public distance. Although these rules are general and differences exist between different populations, they provide a basis for the study of infectious disease transmission at fine spatial scales (i.e., focus on the locations where the distances between individuals are less than the necessary distance for infection).

2.2 Raster space for within-AB simulation and parameters

The space of an AB can be taken as a raster space or vector space depending on need and the availability of data. This paper focuses on raster space AB simulation in ISTAM in which the space inside each AB is represented as rectangular, and consequently, the position of one individual is represented as an explicit discrete grid coordinate (integer values for x and y) at one snapshot in time. Raster space AB simulation has several advantages including the ease with which certain spaces (e.g., lecture theatres, rectangular rooms) can be represented computationally, and efficiency in terms of computer time.

For raster space within-AB simulation in ISTAM, the parameters required to describe individuals’ space–time dynamics are: (1) size of AB (i.e., width and length); (2) the minimum possible distance between individuals in the x and y directions (i.e., MinX and MinY): these reflect such differences in reality (e.g., in a lecture room or computer room, the distance between individuals side-by-side may be less than that between adjacent rows); (3) spatial distribution types: four types of individuals’ spatial distribution within an AB were implemented in ISTAM (Table 1), which are assumed to be the most common types within the campus of the University; (4) movement frequency and movement proportion (i.e., M f and M p ).

Table 1 Four types of spatial distribution of individuals within an AB
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3 Parameter exploration of raster space AB simulation

Influenza, one common airborne infectious disease, was used for this exploration. To simplify the question, it was assumed that if the contact distance between infectious and susceptible individuals was less than 1.5 m, then infection will occur. For example, if the MinX and MinY are set to be 1 m, one infectious individual can have effective contacts with his or her eight neighbours within the Moore neighbourhood. Here, the contact duration was assumed to be less than the duration of one step in the simulation and, therefore, did not need to be considered further. To simplify, if two individuals come into contact more than once during a time unit (such as within a dynamic AB), only one effective contact was recorded.

Here, the output variable was set to be the proportion infected (P i ) which is the ratio of (i) the number of susceptible individuals who make effective contacts with infectious individuals (infection may occur) during the simulation duration to (ii) the total number of susceptible individuals before the simulation.

The input variables (see Table 2) included the parameters which are required to describe individuals’ space–time dynamics as mentioned in Sect. 2.2 and two additional variables; the number of index cases (N) and proportion occupied (P 0). Proportion occupied is the ratio of (i) the number of current individuals to (ii) the maximum number of individuals that can be accommodated (which is determined by the size of AB and size of cell). Proportion occupied is used to describe the saturation of the AB. It was assumed that no immune individuals existed within the target AB. To make the result of the analysis more generally applicable, the parameters of Width (W), Length (L), MinX and MinY were transformed to be functions of the ratio of AB \( {\left( {\frac{W} {L}} \right)}, \) size of AB (WL), ratio of cell \( {\left( \frac{ {\rm MinX} } { {\rm MinY} } \right)} \) and size of cell (MinX × MinY). Every simulation was repeated 1,000 times to obtain the average value.

Table 2 Input variables, their default values and value ranges
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3.1 Proportion occupied

This section explores the relation between proportion occupied and proportion infected for all types of spatial distribution. The value of proportion occupied was varied from 0.05 to 0.95 with a step value of 0.05. All other variables were fixed to the default values in Table 2. Figure 1 shows the relation of proportion infected with the proportion occupied given default values for all other parameters.

Fig. 1
figure 1

Proportion infected plotted against proportion occupied with default values for other variables

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3.1.1 Type 1

It is clear that type 1 AB (e.g., computer room) displays a positive asymmetric and non-linear relation which is quite straightforward to interpret: individuals in type 1 try to be as far as possible from each other and when the proportion occupied is large enough there is no space for individuals to avoid each other so the proportion infected approaches a constant asymmetrically. When P 0 < 0.15, individuals within the AB can always find a location where no effective contacts can occur, so P i remains as 0. When P 0 > 0.6, most individuals locate themselves randomly. When P 0 ∈ (0.15, 0.6), with an increase in P 0, the percentage of individuals who can keep expected distances with other individuals decrease. Regression analysis confirmed an inverse relationship between P i and P 0 (with R 2 = 0.992):

$$ P_{i} = 0.094 - \frac{{0.016}} {{P_{0} }} $$
(1)

3.1.2 Type 2

Type 2 AB (e.g., lecture room) displays a periodicity which ranges between 0.10 and 0.20 of the proportion occupied. The reason is that the width of this AB is 10 m and MinX is 1 m such that 10% proportion occupied implies ten individuals which take up just one row in a type 2 AB. In fact, according to the mathematical definition of type 2, the plot can be computed without real simulation as following:

If the total size of individuals within the AB is n, C n is the total number of effective contacts.

  1. 1.

    When n ≤ width, all individuals are in a line one by one. Individuals who are at the front and the end of this line have one effective contact and all others have two effective contacts. So C n  = 2 + 2(n − 2) = 2n − 2.

  2. 2.

    When n > width, individuals are in at least two lines. Each time the total size of individuals increases from (n − 1) to n, the number of newly added effective contacts (i.e., C n  − C n − 1) can be considered according to the location of the nth individual within the line (Fig. 2):

    1. (a)

      At the front, C n  = C n − 1 + 4;

    2. (b)

      In the middle, C n  = C n − 1 + 8;

    3. (c)

      At the end, C n  = C n − 1 + 6.

  3. 3.

    For any value of n, the \( P_{i} (n) = \frac{{C_{n} }} {n}.\)

As this can be certified from theory, there is no need to calculate R 2for the simulated result.

Fig. 2
figure 2

The relation of number of newly added effective contacts with the location of the nth individual (N is the nth individual and A, B, C and D are individuals whose effective contact numbers increase by 1 due to N)

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3.1.3 Type 3

Type 3 AB (e.g., a bar) produces a negative relation between proportion infected and proportion occupied. This can be explained by the fact that it is easy for infectious individuals to infect susceptible individuals within his or her own cluster, but hard to infect individuals from other clusters. Regression analysis showed an inverse relationship between P i and P 0 (with R 2 = 0.992):

$$ P_{i} = 0.049 + \frac{{0.016}} {{P_{0} }} $$
(2)

For type 3 ABs, firstly, the cluster number was computed as the total individual number divided by g (the expected cluster size, see Table 1). Each cluster was allocated one individual (selected randomly) as a key member. The location of the key member was selected randomly from the whole AB. Secondly, all other individuals were allocated randomly to a cluster, with a location within the Moore neighbourhood of the key member as shown in the sequence of Fig. 3. These two steps ensure that the average size of clusters equals g, and every cluster has at least one individual.

Fig. 3
figure 3

The shape of clusters with different g values (2–9) and the sharing sequence of adding individuals around the key member

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If P 0 is small enough that there are no effective contacts between different clusters, then all effective contacts occur between individuals within the same cluster. Based on the definition above, the shape of the clusters was determined by g (Fig. 3). The average number of effective contacts \( (\ifmmode\expandafter\bar\else\expandafter\=\fi{C}_{n} ) \) can be computed based on the value of g. Table 3 lists the average number of effective contacts for individuals within clusters with different g values from 2 to 9. P i can be computed as

$$ P_{i} = \frac{{\ifmmode\expandafter\bar\else\expandafter\=\fi{C}_{n} \times 0.01}} {{P_{0} }} $$
(3)

It is clear from Fig. 4 that with an increase in g, the simulated data were closer to the theoretical curve provided by Eq. 3. The reason is that with an increase in g, the ratio of effective contacts between individuals from the same cluster to effective contacts between individuals from different clusters increases.

Table 3 Average number of effective contacts for individuals within clusters with different g values (2–9)
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Fig. 4
figure 4

Proportion infected plotted against proportion occupied for a type 3 AB with values of (a) g = 2, (b) g = 3, (c) g = 4, (d) g = 5, (e) g = 6, (f) g = 7, (g) g = 8 and (h) g = 9 (solid dots are the simulated results and the solid lines are based on Eq. 3)

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3.1.4 Type 4

For type 4 AB (e.g., a gym), the proportion infected is approximately independent of proportion occupied. In this situation, the likelihood of one susceptible individual being within the infection distance of an infectious individual is determined only by the ratio of the infection area to the whole area of the AB (in this case the area of the AB is 100 m2). The size of infection area (the area where if individuals are present, effective contacts will occur with infectious individuals) varies according to the relative location of infectious individuals. Infection area is 8 m2 when an infectious individual is not at the border or corner of the AB with a probability of 64/100; Infection area is 5 m2 when an infectious individual is at the border of the AB with a probability of 32/100; Infection area is 3 m2 when an infectious individual is at the corner of the AB with a probability of 4/100. If the number of index cases is fixed to be 1, the probability of one susceptible individual having an effective contact will be \( \frac{8} {{100}}0.64 + \frac{5} {{100}}0.32 + \frac{3} {{100}}0.04 = {\text{ }}0.0684. \) According to the definition of the proportion infected and proportion occupied, the proportion infected is also 0.0684 which is very close to the average value of Fig. 1. This value was defined to be a constant k as it was used frequently in the following sections. More generally, for a type 4 rectangular AB with 1 m for MinX and MinY, the proportion infected will be computed by \( ({\frac{4}{WL}}3 + {\frac{{2((W - 2) + (L - 2))}}{{WL}}}5 +{\frac{{(W - 2)(L-2)}}{{WL}}}8){\frac{1}{{WL}}}.\) The formula is as follows:

$$ P_{i} = \frac{{8WL - 6W - 6L + 4}} {{W^{2} L^{2} }} $$
(4)

3.2 Number of index cases

This section explores the relation between the number of index cases and proportion infected for all types of spatial distribution. The value of number of index cases was varied from 1 to 10 with a step value of 1 for the type 4 AB and varied from 1 to 5 with a step value of 1 for all other types (Fig. 5). All other variables were fixed to the default values in Table 2.

Fig. 5
figure 5

Proportion infected plotted against proportion occupied with different numbers of index cases for (a) types 1, 2, 3 ABs and (b) type 4 AB

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It is not surprising that for every type of AB, the proportion infected increases with an increase in the number of index cases. Also, Fig. 5b shows the independent relation between proportion infected and proportion occupied for type 4 AB. The interesting point here is the increase in magnitude of the proportion infected over the increase in the number of index cases. Omitting the irregularity of proportion infected when the proportion occupied is close to 0, it is clear that the magnitude of increase in the proportion infected decreases for each increase in the number of index cases.

The type 4 AB was selected to be an example for the quantification of the relation between the proportion infected and the number of index cases. The value of proportion infected was averaged over the proportion occupied value (from 0.25 until 0.75) for each of the number of index cases from 1 to 10. A mathematical relation was obtained for type 4 ABs. The effect of an additional index case will make the P i (N + 1) = P i (N) + (1 − P i (N)) P i (N = 1), and P i (N = 1) = k (see Sect. 3.1.4). So the formula is deduced from theory as follows:

$$ P_{i} = 1 - (1 - k)^{N} $$
(5)

where N is the number of index cases, k is a constant with the condition that all other variables take the default values. The mathematical curve plotted based on Eq. 5 fits the values obtained by simulation very well (R 2 = 0.9958).

3.3 Size of AB

This section explores the relation between the size of AB and proportion infected for all types of spatial distribution. The R AB was set to be 1, and value of Width (W) and Length (L) varied from 3 to 15 m with a step value of 1 m (Fig. 6). All other variables were fixed to the default values in Table 2.

Fig. 6
figure 6

Proportion infected plotted against size of AB with different numbers of index cases and proportion occupied for types 1, 2, 3 and 4 ABs

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Three properties are apparent from Fig. 6: (1) for almost all types of AB, P i decreased with an increase in the size of AB with 25 combinations of number of index cases and P 0. One exception is when P 0 = 0.1 for the type 1 ABs: individuals have enough space to keep away from each other, and P i  = 0; (2) in almost all circumstances, P i (type 2) ≥ P i (type 3) ≥ P i (type 4) ≥ P i (type 1); (3) with an increase in P 0, the differences between the different types decrease. Especially when P 0 = 0.9, all types have a similar P i . It is clear that for the type 4 AB, the plots of P i against the size of AB are almost exactly the same for different P 0 values with the same number of index cases. This demonstrates the independence of P i on P 0 for type 4 AB again.

Equation 4 from the above section can be applied here for analysis of the type 4 AB. When W = L, since S AB = WL, Eq. 4 is transformed to be

$$ P_{i} = 4(2S_{\rm AB} - 3{\sqrt {S_{\rm AB} } } + 1)/S_{\rm AB} ^{2} $$
(6)

A theoretical curve was computed and plotted in Fig. 7 to compare with simulated data. It is clear that it fits very well, demonstrating a strong inverse relationship between P i and the size of AB for type 4 ABs

Fig. 7
figure 7

Proportion infected plotted against size of AB for type 4 AB (solid dots are the simulated results while the solid line is based on Eq. 6)

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For P 0 = 0.5 and number of index cases = 1, the regression analysis for types 1, 2 and 3 are as follows (Fig. 8, Table 4). The R 2 value for the power curve model is larger than from the inverse model. However, the power values for types 1, 2 and 3 AB are −0.846, −0.817 and −0.905, which are close to −1 indicating that the inverse model fits the relation between P i and size of AB for all types of AB well.

Fig. 8
figure 8

Proportion infected plotted against size of AB for (a) type 1, (b) type 2 and (c) type 3 ABs (solid dots are the simulated results while the solid lines are based on regression formulae)

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Table 4 Regression analysis results between proportion infected and ratio of AB for types 1, 2 and 3 ABs with P 0 equal to 0.5 and number of index cases equal to 1
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3.4 Ratio of AB

This section explores the relation between R AB and proportion infected for all types of spatial distribution. The size of AB was set to be 100. Since for types 1, 3 and 4 (see Fig. 9a), each AB’s two dimensions are interchangeable, the value of W was varied from 1 to 10 m with a step value of 1 m. Thus, Widths of AB are no larger than Lengths of AB, and the ratio of AB was constrained within the range 0 and 1. For type 2 ABs, since the AB’s dimensions cannot be interchanged, the value of W was varied from 1 to 10 m with a step value of 1, 11, 13, 14, 17, 20, 25, 33, 50 and 100 m. Thus, when W < L, R AB is between 0 and 1; when W > L, R AB is between 1 and 100. To display the relation, a transformation was made: when W > L, R AB was replaced by \( R^{*}_{{{\text{AB}}}} = {\left( {2 - \frac{1} {{R_{\rm AB} }}} \right)}. \) The range of R * AB is from 1 to 2 and also R AB and \( \frac{1} {{R_{\rm AB} }} \) are symmetrical to the point of R AB = 1. All other variables were fixed to the default values in Table 2.

Fig. 9
figure 9

Proportion infected plotted against ratio of AB for (a) types 1, 3, 4 ABs and (b) type 2 with different numbers of index cases and proportion occupied

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From Fig. 9a, the first property of interest is that the relation between the proportion infected and ratio of AB is positive generally. The reason is that for rectangular ABs with a fixed size, the closer the ratio of AB to 1, the smaller the proportion of the border and corner area in the whole AB. As discussed before, index cases at the border or corner positions of an AB can infect fewer individuals than at other positions. Take the type 4 AB as an example. Equation 4 can be transformed to

$$ P_{i} = \frac{{8S_{\rm AB} - 6W - 6L + 4}} {{S^{2} _{\rm AB} }} $$
(7)

It is clear that when S AB is fixed, the value of (W + L) reaches the minimum value when W = L. Secondly, almost in all circumstances, P i (type 3) ≥ P i (type 4) ≥ P i (type 1). Thirdly, with an increase in P 0, the differences between the different types of AB decrease. Especially when P 0 = 0.75, all AB types take a similar P i value.

For type 2 ABs, with an increase in P 0, the curve becomes more symmetrical to the point of R AB = 1 (Fig. 9b). The reason is that when the AB is full of individuals, the two dimensions can be interchanged. Secondly, when P 0 is small (5 or 10% which means 5 or 10 individuals in total in this simulation) and the width is large enough, all individuals are arranged in a line and there is no difference, even when W is larger (the ratio of AB is larger). This accounts for the flat tail which curves display when P 0 is smaller than 0.5. The size of the tail decreases within an increase in P 0.

3.5 Size of cell and ratio of cell

This section explores the relation between P i and the size and ratio of cell for all types of spatial distribution. In the current raster-based AB simulation, possible values for minX or minY are 0.5, 1.0 and 1.5 m. Further, minX and minY are interchangeable. Thus, all possible shapes of cell (considering both the ratio and size of the cell) are 0.25 m2 (0.5 × 0.5), 0.5 m2 (0.5 × 1), 0.75 m2 (0.5 × 1.5), 1 m2 (1 × 1), 1.5 m2 (1 × 1.5) and 2.25 m2 (1.5 × 1.5).

From Fig. 10, the most obvious characteristic to note is that all AB types for different proportions occupied show clearly two “peaks” (i.e., smaller values when the size of cell is 0.25, 0.75 or 1.5 m2 and larger values when the size of cell is 0.5, 1 or 2.25 m2). This can be accounted for by the relation between the circular shape of the infection area with the rectangular shape of the cells. The value of the circular infection area is determined only by the infection distance, which is assumed to be 1.5 m for influenza. When applying this circular area to ABs with different shapes of cells, the sum area of all cells which overlays with the circular area is different since it is assumed that if only the centre point of the cell is within the infection area then the whole cell is included (Table 5). Also almost in all circumstances, P i (type 2) ≥ P i (type 3) ≥ P i (type 4) ≥ P i (type 1). With an increase in P 0, the differences between the different types decrease.

Fig. 10
figure 10

Proportion infected plotted against size of cell for types 1, 2, 3 and 4 ABs with different numbers of index cases and proportion occupied

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Table 5 Different sum of covered areas for different shapes of cell
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3.6 Movement proportion and movement frequency

This section analyses the parameters of dynamic ABs. The spatial distribution type is assumed to be type 4, that is, randomly distributed for a dynamic AB both for the first allocation and subsequent movements. W, L, MinX and MinY take the default values from Table 2. Simulations were repeated based on the combination of four variables from their value ranges: P 0(0.1, 0.2, …, 0.9), N (1, 2, …, 10), M f (1, 2, …, 9) and M p (0.1, 0.2, …, 0.9). The simulation results showed again that P i has no relation with P 0. Figure 11 shows the relations of P i with the other three variables.

Fig. 11
figure 11

Proportion infected plotted against movement proportion with different numbers of index cases and movement frequency

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Initial regression analysis between P i and these three variables showed similar exponential relations. It is straightforward to suspect that for dynamic ABs, the number of index cases affects P i in the same way as it does for a static AB, as summarized in Eq. 5. M f and M p are suspected to affect P i jointly by their product form, that is, H(M f ) × F(M p ). Here, the whole relation was suspected to be as following:

$$ P_{i} = 1 - (1 - G(n))^{{1 + H(M_{f} )F(M_{p} )}} ,\quad {\text{where}}\;G(n) = 1 - (1 - k)^{n} $$
(8)

Firstly, it was assumed that H (M f ) = M f , then F (M p ) was generated from regression analysis. Simulated data with M f  = 1 and N = 1 were used for regression. The analysis generated Eq. 9 with R 2 = 0.996:

$$ F(M_{p} ) = 0.007 + 2.075M_{p} - 1.061M_{p} ^{2} $$
(9)

Figure 12a shows the regression curve and simulated data.

Fig. 12
figure 12

Proportion infected plotted against (a) movement percentage and (b) movement frequency for dynamic ABs (solid dots are the simulated results while the solid line is based on Eqs. 8, 9)

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Secondly, simulated data with N = 1 and M p  = 0.5 were used for comparison with the mathematical curves based on Eqs. 8 and 9 (Fig. 12b). The value of R 2was computed to be 0.998.

3.7 Rank importance of parameters

For static ABs, the correlation coefficients between the above six parameters and P i were computed. The results show that the rank importance of the parameters for any type of static AB is very similar. That is, the number of index cases is the most important parameter and the size of AB and the proportion occupied are the second and third most important (one exception is type 1 AB, which the size of AB is the third and the proportion occupied is the second). Size of cell, shape of cell and ratio of AB are less important. For dynamic ABs, the correlation coefficient between movement frequency and P i is larger than between movement proportion and P i .

4 An example application of raster space AB simulation

Raster space AB simulation was applied within ISTAM to simulate a hypothetical influenza outbreak amongst the first year undergraduate student population in the University of Southampton (see Yang and Atkinson 2008a for details). Twelve types of ABs were designed to describe the spatial environment of the part of the campus relevant to these students. Students’ daily activities were simulated at between-AB and within-AB levels. At the between-AB level, during the simulation, the changing numbers of individuals within a certain AB during the whole day can be collected. Figure 13 shows the number of individuals changing within five types of AB during a whole day (based on a simulation over 1,000 days). The ABs of computer room and library room have the same changing pattern because in ISTAM, students were assume to have the same probability of going to the computer room or library room.

Fig. 13
figure 13

Changing number of individuals within a few ABs during a day within an example application of ISTAM: (a) bar, (b) student union, (c) gym and (d) computer room or library room

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The changing number of individuals within a certain AB can be converted to the proportion occupied since the size of AB and cell of AB are known. Figure 14 shows the average proportion infected (based on 1,000 simulations) for the above five ABs during the whole day (the proportion occupied is assumed to be the average value from Fig. 14, and it is assumed that one individual is infectious and all others are susceptible within the AB). It is interesting to see that the computer room and library room have the same changing patterns of number of individuals and same physical size, but the proportion infected during the whole day are quite different.

Fig. 14
figure 14

Changing proportion infected within a few ABs during a day within an example application of ISTAM: (a) bar, (b) student union, (c) gym, (d) computer room and (e) library room

Full size image

5 Discussion

Sections 3 and 4 provided a detailed analysis of the parameters of raster space AB simulation. These results and plots are potentially useful to future investigators using AB simulation based on the raster data model. For example, the plots and equations clarify some of the relations that will arise during such modeling. It is not difficult to see that these quantified relations can be used in many cases to replace the need for within-AB simulation. This potentially could lead to great savings in computational time, with the relations being used as look-up tables.

Some of the observed relations occur for reasons of geometry, and it would be very interesting to compare the relations obtained for a vector space with the results obtained here for the raster space to illustrate which of the effects are caused by the discretization inherent in the raster data model representation, and to quantify the magnitude and influence of these unwanted, spurious effects.

Despite the potential use of the results of this paper, the parameters of raster space AB simulation are still within the domain of theoretical discussion and far from application for practical use. A few problems exist as follows:

  1. 1.

    Humans’ space–time dynamics are too complex to be expressed by a few parameters. In this research, it was assumed that the space–time dynamic patterns for individuals who are present within the same AB at the same time are the same at within-AB level. In fact, this assumes that the pattern is AB-based. In reality, humans’ space–time dynamics depend strongly on the different roles that they play in their joint activity.

  2. 2.

    The simulations are always based on patterns that we have already discovered or accepted. For example, the four spatial distribution types presented in this paper may be not suitable or there may be other distribution types which the modeller does not know. Computational complexity is a related problem. For example, complex dynamic types of AB were not implement within ISTAM.

  3. 3.

    The scale of space and time of the simulation may be not sufficiently fine to capture the real space–time dynamics of humans. For the spatial scale, raster space AB simulation uses MinX and MinY to express the minimum distances in two dimensions, while in reality the space within an AB where humans can move freely is not always a rectangle (it is also necessary to consider the layout of facilities such as furniture). For the temporal scale, real-time patterns of the dynamics of humans are ideal for simulation of infectious disease transmission, but such patterns are seldom available.

  4. 4.

    The process of infection is not clearly understood. Take influenza as an example: although 1.5 m is taken as a normal distance for the infection to occur, the precise attack duration and attack distance is not known.

Considering the problems above, vector space AB simulation and role-based AB simulation will be applied within the framework of ISTAM in future research. Vector space AB simulation takes the space within an AB as continuous. Vector space AB simulation allows representation of individual movements at a finer spatial scale than raster space and the distance between any two individuals within an AB can be represented as a real value (rather than in terms of pixels). Importantly, Hall’s (1966) distance rules can be applied directly. An example of vector space AB simulation involving the simulation of influenza transmission amongst customers and sales people inside one shopping mall is provided by Yang and Atkinson (2006). This model, although simple, shows the two merits of vector space simulation: (i) the spatial layout can be expressed explicitly by the use of vector data (e.g., within a geographical information system); (ii) the individuals’ movement patterns inside the AB can be as detailed as required such as to reflect their activities and roles. However, a fundamental question is: how much detail is needed for this simulation such as to generate a valid conclusion which can be applied to more general situations? Both the raster space and vector space simulations can be used to generate individuals’ contacts based on their space–time relations at a fine spatial scale which requires an individual space–time model to be defined at fine space–time resolutions. Unfortunately, such models are seldom available. Role-based AB simulation assumes that individuals’ physical distances (also their contact frequency and contact duration) between each other are determined mainly by the roles that they play in joint activities. Role-based AB simulation was applied to simulate a hypothetical influenza transmission within a city (Yang et al. 2008b). Yang and Atkinson (2006) also discussed the classification of ABs into different types such as public, social, personal and special AB, such that different methods of AB simulation can be applied separately or integrated together for one AB.

For a certain number of people in a certain AB, it is hard (if not impossible) to quantify the probability of infection. However, contact probability is used here as a substitute, with the assumption of a linear relation between contact and infection. Unfortunately, contact probabilities in different types of ABs are difficult to estimate as well. For random contacts between individuals in different ABs, Meyers et al. (2005) used different probabilities: 1 for households; 0.03 for workplaces and 0.003 for other public places. The authors admitted that these parameters were based on intuition instead of estimated from data. Another example is Brouwers (2005)‘ MicroPox model for the simulation of smallpox transmission. In MicroPox, ABs were assigned different transmission probabilities. It is possible to compare the probability of infection at two ABs if the primary factors which control the infection can be recognized and compared. In practice, the identification of key ABs for infection is important as this may assist control measures. If enough information about individuals’ activity patterns and an ABs’ physical condition is available, the percentage of infections at different types of AB for one epidemic outbreak can be generated by AB simulation. One example is provided by Longini et al. (2005) whose simulation results showed that 28% of infections occurred within the family, 20% in household clusters and 21% in schools. Ferguson et al. (2005) showed through simulation results that infection risk comes from three sources in roughly equal proportions: (1) household, (2) other places, and (3) random contacts in the community. The percentage of infection at ABs was generated by ISTAM for a hypothetical influenza outbreak in the campus of the University of Southampton. Empirical data are needed for comparation with the above simulation results for model calibration and validation purposes.

6 Conclusion

ISTAM is a novel model for simulating the transmission of infectious disease. The two-level structure (separating the between-AB and within-AB activities) makes ISTAM flexible such that it can be applied to novel circumstances. The concept of AB plays a key role: both the building of individual activity patterns and simulation within ABs depends on how well the ABs are defined and classified.

This paper explored and discussed the parameters of raster space AB simulation within ISTAM, thus, providing a useful reference for these wishing to apply such methods in the future. The parameter exploration showed that:

  1. 1.

    For different types of AB, P 0 has a different effect on P i . For example, a negative inverse relationship for type 1, a positive inverse relationship for type 3, a negative relationship for type 2 and P i has no relation with P 0 for type 4. With an increase in P 0, the difference between different AB types decreases.

  2. 2.

    For all types of AB, P i increases with an increase in the number of index cases while the magnitude of increase decreases as the number of index cases increases.

  3. 3.

    For all types of AB, the relationship between P i and size of AB is inverse.

  4. 4.

    The relation between the ratio of AB and P i is positive generally for all types.

  5. 5.

    For static ABs, the number of index cases is most important and size of AB and proportion occupied are the second and the third parameters in terms of the effect on the proportion infected.

  6. 6.

    For dynamic ABs, the relations between movement frequency, movement proportion and number of index cases with P i are exponential and a regression model fitted the simulated data very well.

Although the above results are valid for raster space AB simulation and the ISTAM model, they can be applied to the general situation. As shown by the application of ISTAM to a hypothetical influenza outbreak amongst a simulated first year undergraduate student population in the University of Southampton, raster space AB simulation could be an effective method to simulate infection at fine space–time scales and for a certain AB, the proportion infected can be quantified. This makes the estimation and analysis of the infection risk for a certain AB possible. Given sufficient data (about both the physical structure of ABs and humans’ social activities), within-AB simulation and ISTAM can be applied for practical use, for example, control measure testing for epidemics.