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Transmission of the agent of tuberculosis,

Cellular automata (if a deterministic model) and stochastic automata (if implemented with varying probabilities of transmission) are methods that allow us to study the dynamics of the population at large, based on local interactions among neighbors, as shown in models of physical systems [

Tuberculosis is both an ancient, familiar infectious disease (the White Plague, “consumption”) and a dangerous, resurgent threat to the lives and health of millions worldwide [

However, tuberculosis is also one of the slowest acting infectious disease agents known, making it possible to intervene in its transmission dynamics and dramatically change both the prevalence of latent infection and the incidence of active, infective cases in a population [

Once an active case is recognized, that person can be treated to push the infection back into latency, not a cure but effective in stopping transmission and greatly reducing both morbidity and mortality. Inadequate or incomplete treatment courses can backfire, producing drug-resistant TB strains that are much more difficult and resource intensive to treat, while also allowing a case to remain infective. The recommendation of the World Health Organization for treatment of tuberculosis disease has been with directly observed short course therapy (DOTS), where individuals are required to come into clinics to take chemotherapeutics targeted at

In the USA and worldwide the goal for the elimination of tuberculosis has been set, with the primary strategy for accomplishing this goal, the implementation and expansion of directly observed therapy (DOTS) [

The social-environmental structural factors that promote

In the dense metropolitan regions of the world’s rapidly urbanizing populations, housing markets are highly segmented and segregated based on wealth, income, and social status, including race and ethnicity in racialized societies such as the United States. A city’s residents are constrained in their choice of neighborhoods in which they can live and segregated into neighborhoods with specific clusters of characteristics.

TB is therefore highly dependent on social context and environment factors, meaning the social structure of one’s cluster of closest contacts and spatial distribution of those clusters in neighborhoods matter. Traditional transmission probability models fail in that they often assume homogeneity across populations or independence of individuals’ risks. For that reason, we propose to develop a stochastic automaton-based simulation of TB epidemic transmission dynamics. We believe it may be an alternative or extension to compartmental (differential equation based) transmission models which can capture probabilistic (stochastic) transmission rates in heterogeneous population groups (compartments) and compartment mixing [

Lastly, because this spatially based simulation is mapped to the structure of a prototypical urban area,

Models for epidemic spread, whether stochastic or deterministic, have been based on interactions between infected populations and susceptible ones, incorporating other epidemiological features such as latency and acquisition and subsequent loss of immunity [

Because of its spatial flexibility, cellular automaton makes possible the construction of neighborhoods differing in their characteristics such as densities and vulnerabilities, thereby studying the dynamics of epidemics throughout spatially diverse populations. It is specifically appropriate for studying diseases that are mainly concentrated in deprived neighborhoods and that absorb immigrants from countries with endemic disease [

The stochastic automata simulation of tuberculosis transmission dynamics was implemented using J software (version 6.01).

The “city” consists of 36 neighborhoods (a 6 by 6 grid), with each neighborhood defined as a grid of 3136 cells (56 cells by 56 cells) (Figure

Structure and distribution of neighborhood of different densities.

Figure

The stochastic automata models presented here consist of a lattice of cells populated by individuals in one of four states: (i) noninfected, (ii) infected-latent (at various levels of progression), (iii) infected-infectious (active), and (iv) dead (Figure

Example of high density (a, b) and low density (c, d) local spatial population matrixes at time 0 (a, c) and time

High density

High density

Low density cluster

Low density cluster

Each of the states is coded with numbers: 0 through 44 (Figure

Flow diagram of states and transition probabilities per month for each individual cell.

The total initial number of latent infected individuals was 14 in high density areas (2%), 3 in medium density areas (0.6%), and 2 in low density areas (0.8%).

At each level of distance from an infectious cell, other noninfected occupied cells in state 1 have the possibility of becoming infected and advancing to state 2 with probabilities shown in Figure

Individuals in higher density clusters will in general have a higher probability of becoming infected, due to more adjacent (and possibly active) individuals.

Figure

The different scenarios that we evaluate with the model are based on the percent of individuals infected who receive successful treatment in the middle and low density neighborhoods and the percent of individuals who receive successful treatment in the highest density neighborhoods which are considered to be vulnerable and are referred to as “targeted treatment.” We consider in each neighborhood a mix of two kinds of populations: the general infected and the immunocompromised. There is a 5% prevalence of immunocompromised individuals in the densest neighborhoods and a 1% prevalence in the medium and least dense neighborhoods. The model scenarios are presented in the table, where percent of successful treatment in the population is the product of percent of case ascertainment and percent of successful treatment.

Mean values (and standard deviations) of 50 runs for the number of deaths, the total number of infections during the duration of the simulation, and the number of people infected within high density, medium density, and low density neighborhoods (with results for treatment possible beginning 6 months after initial infection or beginning 3 months after initial infection are all presented in Table

Treatment scenarios and resulting number of deaths and number of people infected by type of area after 240 months.

Scenarios |
Deaths |
Total number of |
Number of infected |
Number of infected |
Number of infected people | |||||
---|---|---|---|---|---|---|---|---|---|---|

Delay in diagnosis | 6 mo | 3 mo | 6 mo | 3 mo | 6 mo | 3 mo | 6 mo | 3 mo | 6 mo | 3 mo |

(1) 0% general |
4 (3) | 4 (2) | 574 (170) | 560 (146) | 440 (147) | 428 (138) | 109 (55) | 110 (61) | 20 (15) | 19 (17) |

(2) 40% general |
0 (0) | 0 (1) | 56 (14) | 32 (8) | 43 (12) | 25 (7) | 11 (7) | 5 (3) | 2 (1) | 2 (1) |

(3) 40% general |
0 (0) | 0 (0) | 44 (12) | 27 (6) | 33 (9) | 20 (5) | 9 (6) | 5 (2) | 2 (1) | 2 (1) |

(4) 70% general |
0 (0) | 0 (0) | 46 (13) | 22 (6) | 36 (11) | 17 (4) | 8 (5) | 4 (2) | 2 (1) | 2 (1) |

(5) 70% general |
0 (0) | 0 (0) | 48 (13) | 22 (4) | 37 (12) | 17 (3) | 8 (5) | 3 (1) | 2 (1) | 2 (1) |

(6) 0% general |
1 (1) | 0 (1) | 140 (62) | 135 (73) | 35 (11) | 20 (10) | 85 (57) | 89 (67) | 18 (17) | 25 (27) |

The no treatment model is presented as baseline. The model that simulates 40% successful treatment across neighborhoods is based on the level of treatment that is standard practice in many areas. 70% treatment is a level that is reasonably achieved with a high level of surveillance and funding for treatment. 90% case ascertainment and treatment is what may be achieved through high levels of surveillance and funding for close to complete treatment.

All scenarios were run 50 times. Mean values for number of infected individuals after 240 months in low density, medium density, and high density neighborhoods and the total number of individuals who are infected during the course of the simulation, including the number of deaths, are given in the table.

As shown in Table

In order to examine the importance of duration of time between latent infection and the possibility of treatment we also ran all treatment scenarios with possible treatment beginning at 3 months rather than 6 months. As shown in the table, this resulted in a substantial decline in the number of infected individuals in all types of regions, most significantly.

In Figure

Spatial distribution of infected cases and deaths: initial (black), occupied (blue), latent (green), early stage (yellow), and late stage (red).

Mathematical models, in particular those with spatial components, may be useful in situations where actual data are not available, where there is interest in investigating disease processes over long periods of time, where it is difficult to separate individual and spatial attributes within data, and where it is difficult or unethical to conduct randomized experiments. They may also be useful for informing policy on different types of treatment and prevention options for controlling infection and disease.

In examining the impact of evenly distributed case ascertainment and treatment, our results indicate that this appears insufficient in limiting the spread of TB in densely populated areas, especially among those with a number of susceptible individuals (due to, e.g., age, HIV, or diabetes). Our results support additional treatment beyond 40% of successful case ascertainment and treatment, especially treatment and control programs in dense, high risk neighborhoods, similar to findings from studies on programs targeting homeless groups [

All the transition probabilities are approximations from the literature. As in all models, this simulation makes simplifications of reality and is based on numerous assumptions. These assumptions should and can be tested systematically over ranges of starting conditions and settings in a series of sensitivity analyses. Refinements (including diabetes as another class of susceptible) and extensions (age structures and competing mortality risk) of the basic stochastic automata model are both possible and necessary.

Other limitations are the simplified states over transmission models that have been demonstrated to produce incidence, prevalence, and mortality rates consistent with historical data [

The simulation results produced some answers to the set of questions we proposed regarding different treatment scenarios. Certainly, TB has an explosive spread in the dense neighborhoods and into contiguous dense neighborhoods, especially when active cases are initially seeded in those neighborhoods, and those neighborhoods also have higher proportions of more susceptible persons. Even if a dense neighborhood is somewhat isolated from areas with high levels of epidemic spread, there is still significant spread within that neighborhood. But there is also a high degree of patchiness in TB spread, with less active cases spreading in medium density neighborhoods, even if initially seeded with active cases. In isolated low density neighborhoods there is very limited spread of disease. Eventually, however, there are active cases in neighborhoods, even if not initially seeded with active cases. But all this is in the case of no effective treatment of active cases.

For some of our parameters there was not an extensive literature on which to base parameter estimates. Notably the proportion of infections with decreasing but still close proximity is not well documented, and other models of tuberculosis disease have demonstrated that number of new infections from an infectious case contribute importantly to the disease dynamics [

As we see cellular automata models may be useful in answering fundamental questions regarding the relationship between socioeconomic characteristics of populations and disease spread. Here it helped us identify the course of spread across segregated neighborhoods over a chosen period of time. In addition we demonstrated its value in assessing treatment strategies, such as identifying populations to be targeted by the sociodemographic makeup of their neighborhood. Important questions remain open, for example, how would housing policies affect the course of TB spread through changes in crowding and demographic characteristics? Immigration from a region where TB has been endemic and highly prevalent is a risk factor for latent infection in the USA. Immigrant populations are clustered in neighborhoods. How would this affect the spread across other neighborhoods? If individual level differences in populations have a large impact on the spread of TB, which has the bigger impact differential susceptibilities or treatment rates? Stochastic or cellular automata have the potential of becoming a powerful tool for answering questions to guide policy makers and support their decisions.

The authors declare that there is no conflict of interests regarding the publication of this paper.

David H. Rehkopf, Sc.D., conceived the study with AFD, implemented the model for various treatment scenarios, and interpreted the analytical results. Alice Furumoto-Dawson, Ph.D., conceived the study with DHR and prepared the background on which the study was designed as articulated in the Introduction. In addition she provided through her research estimate values of the parameters in the model. Anthony Kiszewski, Ph.D., developed the mathematical and computational aspects of the stochastic cellular automata used in this study and supervised its implementation. Tamara Awerbuch-Friedlander, Ph.D., supervised all aspects of the study implementation, contributed to the interpretation of the results, and edited the final versions of the paper. All authors helped to develop ideas and choose scenarios for investigation and wrote parts of the paper.

The authors wish to acknowledge the contribution of the course “Mathematical Models in Biology” offered in the spring of 2003 at the Harvard School of Public Health for the interdisciplinary approach that motivated the study. The authors thank Nathalie Marchand and Isabel Madzorera for their editorial contribution.