Conceived and designed the experiments: LCdCM CARC. Performed the experiments: LCdCM. Wrote the paper: LCdCM CARC CB WVdS LR AMVM. Conceived and designed the model: LCdCM CARC. Discussed the model parameters: LCdCM CARC CB WVdS LR. Analyzed, re-analyzed and interpreted the results: LCdCM CARC CB WVdS LR. Critical revision: AMVM.
The authors have declared that no competing interests exist.
Dengue is a disease of great complexity, due to interactions between humans, mosquitoes and various virus serotypes as well as efficient vector survival strategies. Thus, understanding the factors influencing the persistence of the disease has been a challenge for scientists and policy makers. The aim of this study is to investigate the influence of various factors related to humans and vectors in the maintenance of viral transmission during extended periods.
We developed a stochastic cellular automata model to simulate the spread of dengue fever in a dense community. Each cell can correspond to a built area, and human and mosquito populations are individually monitored during the simulations. Human mobility and renewal, as well as vector infestation, are taken into consideration. To investigate the factors influencing the maintenance of viral circulation, two sets of simulations were performed: (1st) varying human renewal rates and human population sizes and (2nd) varying the house index (fraction of infested buildings) and vector per human ratio. We found that viral transmission is inhibited with the combination of small human populations with low renewal rates. It is also shown that maintenance of viral circulation for extended periods is possible at low values of house index. Based on the results of the model and on a study conducted in the city of Recife, Brazil, which associates vector infestation with
This study contributed to a better understanding of the dynamics of dengue subsistence. Using basic concepts of metapopulations, we concluded that low infestation rates in a few neighborhoods ensure the persistence of dengue in large cities and suggested that better strategies should be implemented to obtain measures of house index values, in order to improve the dengue monitoring and control system.
Dengue is the most rapidly spreading mosquito-borne viral disease in the world and approximately 2.5 billion people live in dengue endemic countries. In Brazil it is mainly transmitted by
Dengue is currently the most important arthropod-borne disease, affecting around 50 million people worldwide every year, mostly in urban and semi-urban areas
Dengue can be caused by four distinct but antigenically related serotypes which are mainly transmitted by
A wide variety of factors influence the spatial and temporal dynamics of mosquito populations and, therefore, dengue transmission patterns in human populations
While the development of dengue vaccines is still underway
For dengue control programs to be effective, information on the occurrence of infection and disease in the population are essential. However, as most dengue infections are asymptomatic or unapparent, presenting themselves as non-differential fevers of unknown etiology, surveillance systems based on the monitoring and notification of symptomatic cases have low sensitivity and are not capable of detecting low or sporadic transmission
Mathematical and statistical models have been developed in order to provide a better understanding of the nature and dynamics of the transmission of dengue infection, as well as predict outbreaks and simulate the impact of control strategies in disease transmission
Another class of models used to investigate the disease transmission process is that of cellular automata (CA)
We propose a stochastic cellular automata model that simulates dengue transmission in a hypothetical population, aiming to perform a qualitative analysis of factors that influence disease transmission. Unlike the mathematical models based on differential equations, the proposed CA-based model of diffusion of dengue fever uses heterogeneous rules for human mobility. The role of human mobility in the transmission of infectious diseases has been previously investigated
The proposed model takes into account existing knowledge about the biological cycle and disease transmission of dengue infection in humans and vectors. Although the model supports the assumption of co-circulation of all serotypes, in this initial approach the simplest scenario which considers the circulation of only one serotype was simulated. Some parameters of the model are constant while others follow a probability distribution (
Cell Parameters | |||
Parameter | Symbol | Description | Assumption |
Number of humans in cell |
Gaussian distribution with average 4 and deviation 2 | To ensure that 68% of occupied cells will have between 2 and 6 people | |
Vector-human ratio in cell |
Uniform probability density function within the interval [0, |
To be proportional to the number of humans in cell |
|
Number of vectors in cell |
Defined |
Individual Parameters | |||
Parameter | Symbol | Description | Assumption |
Intrinsic incubation period (days) | Gaussian distribution with average 5.5 and deviation 1.5 | To ensure that 68% of cases will be between 4 and 7 days | |
Infective period in humans (days) | Gaussian distribution with average 4.5 and deviation 1.5 | To ensure that 68% of cases will be between 3 and 6 days | |
Extrinsic incubation period (days) | Gaussian distribution with average 9 and deviation 0.25 | To ensure that 95% of cases will be between 8.5 and 9.5 days |
Individual Parameters by Unit of Time | ||
Parameter | Symbol | Description |
Number of bites of mosquito |
Defined by |
Infected human individuals are not contagious during the intrinsic incubation period that ranges between 4.5 and 7 days, with a small probability of exceeding 10 days
These vectors feed almost exclusively on humans
With relation to the spatial distribution of vectors, based on a survey conducted in the city of Recife using geo-referenced ovitraps
Research conducted from April 2004 to April 2005, counting 13 cycles of 28 days each, in four urban areas with the presence of
The CA-based model consists of two bidimensional square lattices,
Assuming that
In the
At the beginning of each simulation, the model generates an initial configuration for the H and M lattices, assuming that the entire population (humans and vectors) are susceptible, except for a single randomly chosen infected human. For this initial configuration, the following parameters in each cell are pre-determined: (1) the human population (
The dynamics of human-mosquito interactions is based on the following rule: every day each mosquito randomly selects one or a few humans to bite, according to a daily frequency of bites
During the process of interaction between humans and mosquitoes, each human can assume one of four states with respect to each serotype: susceptible (S), exposed (E), infectious (I) or recovered (R) and each vector can assume one of three states with respect to each serotype: susceptible (S), exposed (E) and infectious (I). The duration of the exposed state (infected but not infectious) corresponds to the incubation period. If there is a contact between a susceptible human and an infectious vector, the human may become exposed with probability
The human population was modeled considering a single annual renewal rate (
The boundary conditions are periodic, which means that opposite borders of the lattice are connected to each other to form a toric topology
The model also assumes that the probability that vectors bite humans decreases as the distance from its cell of origin increases. The random selection of the target-cell by the mosquito depends on its flight range
At first, without considering human movements in the model, for every mosquito a random human target is chosen in three steps: (1st) Draw of a neighborhood ring, according to the vector of predetermined probabilities
We consider random human mobility in which a daily percentage of the human population leaves its residence and randomly chooses other buildings to visit. Movements can be homogeneous (assuming that all households have the same characteristics) or concentrated in public locations. The daily rate of human mobility (
As these humans do not leave their homes during their infectious periods, only the fraction
In the simulations that there are no public locations in the cellular space (
With human mobility taken into account, the third stage of the choice of target by the mosquito changes to:
(3rd) with probability
For public locations, the maximum number of vectors per human was calculated assuming that there is a fixed amount of people in these places that spends the whole day on this site. This amount is based on the same rule for de
Considering the initial dengue-naïve open population in the sense that human renewal is taken into account, after a dengue epidemic, the small number of susceptible individuals in addition to the births and immigration of new healthy individuals allows the maintenance of viral transmission, despite low rates. Through time, the number of susceptible humans increases until it is sufficient to initiate a new outbreak. This is a classic framework that helps to understand the periodicity of epidemics
However, the SET model approaches the daily number of renewed humans at time
To investigate the values of the minimum parameters required for maintenance of viral transmission for extended periods, two sets of simulations were performed with the number of iterations corresponding to seven years. Our tests showed that this period is sufficient for steady-state establishment.
For both sets of simulations, the stochastic parameters used are those shown in
Constant Parameters | ||
Parameter | Symbol | Values |
Percentage of human occupation | 0.9 | |
Transmission probability from human to vector | 0.9 | |
Transmission probability from vector to human | 0.9 | |
Mosquito daily survival probability | 0.983 | |
Neighborhood selection probabilities | (0.7, 0.3) | |
Percentage of asymptomatic infected humans | 0.65 | |
Overall rate of human mobility | 0.5 | |
Mobility rate to public locations | 0.9 | |
Percentage of public locations | 0.05 |
Parameter | Symbol | Values |
Human population size | {2000, 4000, …, 12000} | |
Annual human renewal rate | {1, 2, …, 6} % | |
House index | 0.9 | |
Maximum ratio of vectors per human | 2 | |
Mosquito daily bite rate | {1, 1.5} |
Parameter | Symbol | Values |
Human population size | 8,000 | |
Annual human renewal rate | 5% | |
House index | {0.5, 2, 5, 10, 20, 30, 50, 70, 90} % | |
Maximum ratio of vectors per human | {0.5, 1, 2} | |
Mosquito daily bite rate | {1, 1.5} |
At first, we simulated the spread of dengue infection varying human parameters (population sizes and renewal rates), while the other model parameters remained fixed. Based on the observed data of the
With the results of the first set of simulations, we fixed the size of the human population and human renewal rate to values which ensure a high chance of maintaining viral transmission for extended periods. Then we performed the second set of simulations, varying the house index and vector per person ratio, in order to investigate the values that are able to eliminate viral transmission. For each combination of variable parameters in
To study the effects of human movement, we conducted simple experiments with different mobility configurations. The stochastic parameters used are those shown in
Spread of infection for humans (top) and for mosquitoes (bottom). Color legend in
Every day 50% of the population leaves its home. Among these, 90% of them go to public locations (which corresponds to 5% of the cells) while the remainder visits other domiciles. Spread of infection for humans (top) and for mosquitoes (bottom). Color legend in
No public locations were considered: 50% of people leave home every day and visit other domiciles. Spread of infection for humans (top) and for mosquitoes (bottom). Color legend in
Humans | Mosquitoes | |
Empty lot | Without vectors | |
There is at least a susceptible human and no infected in the cell | All mosquitoes are susceptible | |
Increasing number of infected humans | Increasing number of infected mosquitoes | |
Immunes only | - |
The different propagation speeds of the disease can be observed in
(
The periodicity of the epidemics is shown in
After the epidemic (shown at the left of A), few cases sustain viral transmission (magnified in B).
For the first set of simulations, using the range of parameters described in
Simulations performed using the range of parameters described in
The results showed that for both frequencies of bites and for all population sizes, the human renewal rate of 1% was not sufficient to maintain viral transmission for more than three years, while for 2% of human renewal, in very few cases, viral circulation was maintained for many years. The viral transmission was not sustained with the combination of small human population with low human renewal. In order to maintain viral transmission for a long period it was necessary that at least one of these parameters were not low. In the case of 8,000 inhabitants and 5% of annual human renewal rate, the chance of sustained viral circulation was higher than 50% (for both biting frequencies). Therefore, we chose these values for the second set of simulations.
200 replications were performed for each combination of parameters described in
Simulations performed using the range of parameters described in
Noting the limitations inherent to any mathematical modeling, we discuss the problem of viral transmission maintenance between successive epidemic periods. This question was motivated by the high incidence rates of dengue in densely populated areas of Recife
With respect to the investigation of the maintenance of viral transmission for extended periods, the question to be answered was: Since the number of susceptible individuals in a naive population is virtually exhausted after an epidemic outbreak, how can the virus remain active between outbreaks? This issue was exhaustively addressed in different scenarios, where we analyzed the influence of some human and vector factors in the maintenance of viral circulation during seven years, a sufficient period for equilibrium of viral transmission
The results of numerical experiments showed that with high house index values combined with high/moderate vector/human ratio, viral transmission was maintained for long periods, whereas it was not when considering the combination of small human population and low human renewal rates. The latter combination led to disease extinction in the model. Therefore, for the maintenance of viral transmission it was necessary that at least one of these parameters were not low. The extinction situation also happened when we considered house index values below 10%, for human populations with approximately 8,000 inhabitants in all cases of vector/human ratio. However, the SET model also showed that viral transmission is possible for several years (with low probability) considering low house index (between 20% and 30%), moderate ratio of vector per human (0–1 vector per person) and small human populations (approximately 4,000 people). For these cases, we believe that the random combination of factors in the initial configuration of the CA-based model allowed the virus to circulate for many years. The results of the SET model are consistent with findings from the model of Newton and Reiter
As the neighborhoods of large cities generally have populations of at least 8,000 inhabitants, the model suggests that it is possible that in these cities a small percentage of its neighborhoods have the potential to sustain the virus for extended periods. For example, considering a hypothetical metropolis of 6 million inhabitants with house index of 30% and 750 neighborhoods of approximately 8,000 inhabitants, the SET model showed that about 1.5% of the city's neighborhoods sustain viral circulation for 5 years (or roughly 11 neighborhoods). The persistence of viral circulation is in agreement with the classic notion of extinction risk and persistence in metapopulations
Considering three values of
However, in real situations, the vector population fluctuates according to a combination of meteorological factors
In practice, house index values should be zero or very close to zero in order to eliminate viral transmission
Special thanks go to André Freire Furtado and Maria Alice Varjal de Melo Santos, from Aggeu Magalhães research center (CPqAM/Fiocruz), for several helpful conversations and their contributions in model development. We are grateful to Cláudia Torres Codeço, Marília Sá Carvalho, from Fiocruz, and the members of SAUDAVEL project, for their collaboration and suggestions during the progress of work. Special thanks also to Etienne Tourigny for technical support and comments on the manuscript and to Newton Brigatti, Karine Reis Ferreira and Carolina Moutinho Duque de Pinho for technical support.