UnEncrypted

Alejandro Acosta, Francisco Almeida

Listing 4: Implementation of the runByRows function of the solution engine for GPU systems.

5dim3 dimBlock ( NTHREAD , 1 , 1 ) ;

6

7 f o r ( i n t i = 0 ; i < setup . getNumStages ( ) ; i++) {

8 kernelRunByRows<<<dimGrid , dimBlock >>>(i , ls , table ) ;

9}

10cudaDeviceSynchronize ( ) ;

11}

Listing 5: Implementation of the KernelrunByRows kernel.

6}

3Computational Results

In order to validate the skeleton developed, we tested them on several dynamic programming problems (Table 1). We must remember that the data dependencies are di erent in most of the formulas considered. This means that we have to use di erent parallel traversing in the DP table. The GPU skeleton used for RAP and KP access the table by rows, those for TCP access it diagonally, starting from the main diagonal upward, and those for GCP move diagonally downward until the secondary diagonal is reached.

The parallel platform used to execute the GPU skeleton is a Nvidia C2050 GPU installed on an Intel i7 host processor. We compare this new skeleton with skeletons developed for system consisting of four AMD Opteron 6128 processors with eight cores each. We used the sequential skeleton and the parallel skeletons that fit better to each problem. For the KP we used the shared-memory skeleton, for the RAP the heterogeneous skeleton and for the TCP and GCP we used the MPI version. The parallel executions on the AMD were developed using 4, 16 and 32 processes. To simplify the experiment, the tests were carried out using square matrices of order 1000, 2000 and 5000.

Table 2 shows the execution times of all the problems proposed using the GPU skeleton. This table provides an overview of each problem’s granularity. All of the times are expressed in seconds. We can see how the KP problem have the finest granularity, while the GCP problem exhibits the coarsest granularity.

Skeletal based programming for Dynamic Programming on GPUs

Table 2: Execution Time of the problems analysed

Figure 2 shows the speedup for each one of the problems when compared against the sequential and parallel skeletons. We first observe that the RAP performance is over the remaining problems. One of the reasons for this good behaviour resides in the granularity of the problem. In the case of the RAP, with non-fine grain granularity, the reduction in computational time overpass the cost associated to the data memory movements. In the KP, with a lower speedup, we find a fine grain problem where the total time is coerced by the memory accesses. In both former problems, each iteration launches a thread to compute a column of the table. In the case of the TCP the table is traversed by diagonals starting from the main diagonal. The number of threads is di erent on each iteration and when the number of iterations increases, the number of threads decreases. In the GCP we have a similar situation, in this case the number of threads increases and later decreases. This kind of traversing where the number of threads becomes very low, negatively influences the performance in GPU architectures.

In figure 2(a) we show the speedup of the GPU skeleton compared with the sequential skeleton. In this case all problems show a positive speedup. In the case of the KP, this speedup is constant with the size of the problem, while in the other problems, the speedp increases when the size of the problem increases. This behaviour remains when we increment

Alejandro Acosta, Francisco Almeida

Figure 2: Speedup of the GPU skeleton

the number of CPUs, however in figure 2(b) we compare against the parallel version using 4 CPUs, in this case, the GPU skeleton for the KP is slower than parallel one using 4 CPUs. Figures 2(c) and 2(d) shows the performance using 16 and 32 CPUs, where we can see as only the RAP has a positive speedup. As a consequence of a non-e cient use of the threads in the GPU for the TCP and GCP, they are overpassed by the parallel CPU skeletons. For this classes of recurrences the GPU skeleton should be launched as a hybrid skeleton that dynamically on each iteration choose the use of GPU or CPU when be more e cient.

Skeletal based programming for Dynamic Programming on GPUs

4Conclusion

We developed a Dynamic Programming Skeleton for GPUs. The skeleton developed shows the known advantages of skeletal programming of ease of use, programmability and e ciency while hiding the parallelism at the same time. The high productivity of the approach is tested by using four combinatorial optimization problems. The GPU skeleton is compared against parallel skeletons developed for CPUs and the performance is analysed. Although resource allocation problems show very good performances, we observe that some other problems could take advantage of the future development of skeletons that, on each iteration, dynamically select the use of the CPUs or the GPU.

Acknowledgements

This work has been supported by the EC (FEDER) and the Spanish MEC with the I+D+I contract number: TIN2008-06570-C04-03 and TIN2011-24598

References

[1]K. J. Brown, A. K. Sujeeth, H. J. Lee, T. Rompf, H. Chafi, M. Odersky, and K. Olukotun. A heterogeneous parallel framework for domain-specific languages. In Proceedings of the 2011 International Conference on Parallel Architectures and Compilation Techniques, PACT ’11, pages 89–100, Washington, DC, USA, 2011. IEEE Computer Society.

[2]M. Christen, O. Schenk, and H. Burkhart. Automatic code generation and tuning for stencil kernels on modern shared memory architectures. Comput. Sci., 26(3-4):205–210, June 2011.

[3]J. Enmyren and C. W. Kessler. Skepu: a multi-backend skeleton programming library for multi-gpu systems. In Proceedings of the fourth international workshop on High-level parallel programming and applications, HLPP ’10, pages 5–14, New York, NY, USA, 2010. ACM.

[4]I. Pel´aez, F. Almeida, and F. Su´arez. Dpskel: A skeleton based tool for parallel dynamic programming. In Seventh International Conference on Parallel Processing and Applied Mathematics, PPAM2007, 2007.

[5]M. Steuwer, P. Kegel, and S. Gorlatch. Skelcl - a portable skeleton library for highlevel gpu programming. In Proceedings of the 2011 IEEE International Symposium on Parallel and Distributed Processing Workshops and PhD Forum, IPDPSW ’11, pages 1176–1182, Washington, DC, USA, 2011. IEEE Computer Society.

Proceedings of the 12th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2012 July, 2-5, 2012.

Descriptive and Predictive models of dengue epidemiology: an overview

Ma´ıra Aguiar1, Rick Paul2, Aanavaj Sakuntabhai2, Nico Stollenwerk1 and

Sumonmal Uttayamakul3

1 Centro de Matem´atica e Aplica¸c˜oes Fundamentais, Universidade de Lisboa Avenida Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal

2 Institut Pasteur, Paris, France

3 Bamrasnaradura Infectious Diseases Institute, Department of Disease Control, Ministry of Public Health, Nonthaburi, Thailand Mahidol University, Bangkok, Thailand

emails: maira@ptmat.fc.ul.pt, relpaul@hotmail.com, anavaj.sakuntabhai@pasteur.fr, nico@ptmat.fc.ul.pt, sumonmal@health.moph.go.th

Abstract

In the DENFREE (Dengue Research Framework for Resisting Epidemics in Europe) project, with large emphasis on medical research under the heading of the Pasteur Institute, Paris, the Bio-mathemathics group at CMAF, will manage a work packed on mathematical modeling of spreading of dengue fever and its data analysis of potentially complex dynamics. The main objectives of DENFREE are (1) to identify key factors determining dengue transmission, outcome of infection and epidemics; (2) the development of less-invasive diagnostic tools to detect asymptomatic infections; (3) to evaluate the risk of dengue emergence into non-endemic areas by using Asian communities as a model. In this manuscript we present the o cial notification dengue hemorrhagic fever (DHF) data from the Ministry of Public Health in Bangkok, Thailand. The multi-strain dengue model, where the notion of at least two di erent strains is needed to describe di erences between primary infection, mainly leading to mild dengue fever (DF) and secondary dengue infections with high risk of DHF, is also described. We present the properties of the minimalistic multi-strain model with a summary of the analysis of the dynamics. The goal is to introduce notation, terminology, and results that will be used in future for formal parameter estimation procedures.

Key words: dengue fever, mathematical models, complex dynamics, data analysis

Descriptive and Predictive models of dengue epidemiology

1Introduction

Epidemiology of infectious diseases studies the incidence of disease cases and factors that influence disease emergence, transmission and spread in a well-defined population. Epidemiological models are a formal framework to convey ideas about the components of a host-parasite interaction and can act as a tool to predict, understand and develop strategies to control the spread of infectious diseases by helping to understand the behavior of the system under various conditions. They can also aid data collection, data interpretation and parameter estimation. The purpose of epidemiological models is to take di erent aspects of the disease as inputs and to make predictions about the numbers of infected and susceptible people over time as output.

Dengue fever is a viral mosquito-borne infection, a major international public health concern with more than 55% of the world population at risk of acquiring the infection. Two variants of the disease exist: dengue fever (DF), a non-fatal form of illness, and dengue hemorrhagic fever (DHF), which may evolve toward a severe form known as dengue shock syndrome (DSS). Epidemiological studies support the association of DHF with secondary dengue infection due to a process described as antibody-dependent enhancement (ADE), where the pre-existing antibodies to previous dengue infection cannot neutralize but rather enhance the new infection. Treatment of uncomplicated dengue cases is only supportive, and severe dengue cases require hospitalization and proactive treatment of hemorrhagic symptoms [1]. A vaccine against dengue is not yet available, although several candidates of vaccines are at various stages of development [2].

The dynamics of dengue fever epidemiology shows large fluctuations of disease incidences and mathematical models describing the transmission of dengue viruses appeared in the literature as early as 1970. Retrospective data and the possibility to estimate hidden states in dengue models from such data [3, 4] have been discussed, specially, primary versus secondary infection, and symptomatic versus asymptomatic cases that can be studied via the first already available models [3, 5, 6].

One of the main objectives of the DENFREE project will be the implementation of a new theoretical framework for modelling and preventing dengue. A work packed on descriptive and predictive models of dengue epidemiology will contribute to estimate the risk of spreading DENV to uninfected areas, especially in Southern Europe where susceptible vector exists. The major tools generated will be predictive models that enable specific interventions, whether concerning the environment, mosquito or human, to be made and that can prevent an epidemic. Field epidemiological data collection programs in DENFREE, as part of already established programs and those to be established within the framework of the project proposal, will provide the fine-scale relevant information needed to fine-tune the top-down epidemiological models to be built.

From the Ministry of Public Health MoPH in Bangkok, Thailand, extensions of the already existing time series data sets were obtained up to the present months of 2012. The

Aguiar et al.

time series can now be extended to 30 years.

In this manuscript we present the o cial notification DHF data from the Ministry of Public Health in Bangkok, Thailand. The minimalistic multi-strain model, motivated by dengue fever epidemiology, where the notion of at least two di erent strains is needed to describe di erences between primary (DF) and secondary dengue infections (DHF) is also described and the validation of this models with a formal parameter estimation procedure is discussed.

2Modeling dengue fever epidemiology

For e ective disease management, whether control, prediction or risk mapping, a better understanding of the transmission dynamics of the virus is a key. In dengue fever epidemiology there are four antigenically related but distinct serotypes (DEN-1 to DEN-4). The occurrence of the virus as four distinct serotypes raises many complications in the analysis and interpretation of serological data. Antibodies generated by exposure to any one strain are known to be cross-reactive for other strains, but they are believed only to provide strain-specific lifelong immunity to reinfection [11]. The immunological response on exposure to a second strain is complex and depends on factors such as patient age, strain type and the interval between exposure to one serotype and exposure to the second serotype. The high antibody titers attained after primary infection appear to generate a degree of cross-protection for a while, but if secondary exposure occurs after antibody levels begin to decline, cross-reactivity appears to act to enhance the growth rate of the new invading viral strain. This is called antibody-dependent enhancement (ADE) and its occurrence in dengue has been used to explain the etiology of serious disease (DHF and DSS) [1, 15, 14, 16, 17, 12, 13].

Epidemic models have been important in understanding the spread of infectious diseases and to evaluate the introduction of intervention strategies like vector control and vaccination. Infectious disease dynamics are by nature nonlinear. To understand such nonlinear epidemiological processes is vital for any modern society from the human as well as the economic perspective, but intrinsically mathematically di cult. Significant progress has been achieved in modelling childhood diseases, because they are so extraordinarily contagious, whereas other often more harmful diseases lag behind in terms of modelling. To make the urgently needed progress in improving our understanding of the dynamics of such diseases, concepts from other fields of mathematics are needed. Although the multi-strain interaction leading to deterministic chaos via ADE has been described previously, e.g. [18, 19, 20], the role of temporary cross-immunity has been neglected leading to unrealistic biological parameter estimation. More recently, despite incorporation of temporary cross immunity in rather complicated models, the possible dynamical structures were not deeply analyzed [21, 22, 23]. When including temporary cross immunity into the ADE models, a rich dy-

Descriptive and Predictive models of dengue epidemiology

namic structure including chaos in wider and more biologically realistic parameter regions was found [5, 6]. The model described in [5] is a basic two-strain SIR-type model for the host population motivated by modeling dengue fever epidemiology with its peculiar ADE phenomenology, being associated with secondary infection after primary infection with a di erent strain. The model consists of a set of classes representing parts of the population to be susceptible, infected and recovered individuals and two virus strains. The processes of infection are described in uniform mixing approximation and immunity to a particular strain is assumed to be lifelong. The demography of the host population is also included, assuming constant population size.

In the following sections we present the available DHF data and the model framework that has shown qualitatively very good results when comparing empirical DHF data and model simulation, o ering a promising perspective on inference of parameter values from dengue case notifications.

2.1Data

The first recorded epidemic of DHF in Thailand (see Fig.1a), population of approximately 66 million people [7]) was in 1958 [1]. The co-circulation of all four dengue serotypes and their capacity to produce severe dengue disease was demonstrated as early as 1960 in Bangkok, Thailand [8]. A system for reporting communicable diseases including DHF/DSS was considered fully installed in 1974 and the data bank of DHF and DSS is available at the Ministry of Public Health, Bangkok [10]. Reasonable data from all provinces exist since the beginning of the 1980s.

In figure 1b), a time series for the DHF cases in Thailand is shown. Note the yearly fluctuations, suggesting that the disease incidence is seasonal. This data come from the Ministry of Public Health of Thailand and consists in monthly incidences of DHF cases.

Large part of the data available to theoretical epidemiologists consists of time series tracking the evolution of a subset of states variables of an underlying dynamical system through a surveillance system. Partially-observed nonlinear stochastic dynamical systems (also known as hidden Markov models are a natural tool to simulate the processes potentially giving rise to such data sets. A hiden Markov model consists of a process model, governing the evolution of states variable along with an observation process. The formulation of various mechanistic process models through nonlinear stochastic dynamical system have been a major focus of theoretical epidemiology. However, di culties associated with inference from time-series data of the unknown parameters of these models have been a constraint on their applications.

Frequently, the time series of empirical data are used as a qualitative check on model output, however, fitting every detail of the chaotic model to that of the empirical data is not possible. Parameter estimation based on empirical data to estimate initial conditions and model parameters have received great attention and is notoriously di cult for chaotic time

Figure 1: In a) The provinces of Thailand. DHF case data are available for all provinces. [24]. In b) Time series of monthly DHF cases for the whole country of Thailand.

series. Temporally local approaches are possible using iterated filtering algorithms [25, 26, 27], and at the moment only minimalistic models would have a chance to be qualitatively understood well and eventually tested against existing data.

2.2Multi-strain model

The basic n-strain epidemiological model with primary and secondary infections can be written as follows

Descriptive and Predictive models of dengue epidemiology

and for i = 1, ..., n

i=1 j=1,j=i

dividing the population into 6 groups: susceptible to all existing strains (S), primarily infected with one of the strains (Ii), recovered from the first infection (Ri), susceptible with a previous infection (Si), secondarily infected with a di erent strain than the one acquired during the previous infection (Iij ) and recovered and life-long immune against all strains (R).

The four-strain epidemiological model can be written as an Eq. system of 26 ODEs (for the complete ODE system). It can be simplified to a three or two-strain model just by neglecting the existence of specific strains. A three-strain model can be obtained by putting I4 = 0 at t0 and initially no secondary infected for example, where the complete system of ODEs would be reduced to a system of 17 ODEs, and by putting I3 = 0, I4 = 0 at t0 and initially no secondary infected, we get back the original two-strain model without any loss of generality, a system of 10 ODEs, once the respective import terms are set to zero. It turns out that the distinction between primary infection and secondary infection is dynamically more important than the exact number of strains to be considered in the model [28], so for the modeling construction at least 5 ingredients are used, here given by the models parameters. Parameter β is the infection rate, describing the transmissibility of the disease. Individuals can become infected with two di erent infection rates, depending from whom they are getting the infection, β when the infection is coming form an individual in its first infection and φβ when the infection is coming from an individual in its secondary