MARKOV CHAIN MODELING OF ENDEMIC CATTLE DISEASES IN THE TROPICS OF MEXICO

USO DA CADEIA DE MARKOV COMO MODELO PARA DOENÇAS ENDÊMICAS DE BOVINOS NOS TRÓPICOS DO MÉXICO

MARKOV CHAIN MODELING OF ENDEMIC CATTLE DISEASES IN THE TROPICS OF MEXICO.

USO DA CADEIA DE MARKOV COMO MODELO PARA DOENÇAS ENDÊMICAS DE BOVINOS NOS TRÓPICOS DO MÉXICO.

José Alfonso Barajas-Rojas¹ Hans Riemann² Charles Franti³

SUMMARY

Endemic diseases measured by prevalence of IgG antibodies by ELISA all showed remarkably stable overall prevalences over a study period of 3 years. This however does not mean that incidence is low or absent. Observed transition from the state of being positive for a disease to the state of being negative and vice versa suggested fairly high incidence rates (and rates of waning antibodies) for all 19 disease agents tested for. In spite of this dynamic situation Markov Chain simulations indicate that a steady prevalence is reached in a fairly short time. Markov Chain simulations may also be useful in elucidating seasonal changes in prevalence.

Key words: prevalence, Markov Chain, ELISA, México.

RESUMO

Doenças endêmicas medidas pela prevalência de anticorpos IgG pelo ELISA mostraram prevalências totais notavelmente estáveis num período de estudo de 3 anos. Isto entretanto, não significa que a incidência é baixa ou ausente. A transição observada entre o estado de positivo para uma doença para o estado negativo e vice-versa sugeriu taxas de incidência razoavelmente altas (e perda de anticorpos) para os 19 agentes infecciosos testados. Apesar desta situação dinâmica, simulações da Cadeia de Markov indicam que uma prevalência fixa é alcançada num período de tempo razoavelmente curto. A Cadeia de Markov como modelo pode também ser útil para a elucidação das alterações sazonais na prevalência.

Palavras-chave: prevalência, Cadeia de Markov, ELISA, México.

INTRODUCTION

Studies of IgG antibody levels to a number of infectious disease agents of cattie in the tropics of México showed remarkable fluctuations (Barajas-Rojas et al, 1993a, 1993b, 1993c). Some of the fluctuations could be related to state of pregnancy and seasonal variation. In spite of the fluctuations the overall prevalence of positive tests remained remarkably stable over a 3 year period.

In this paper we examine the possibility that fluctuations in antibody levels are caused by waning of antibodies followed by reinfections.

MATERIAL AND METHODS

Based on recorded ELISA reversals, from negative to positive and vice versa and total prevalence in 1988-1989 overall transition probabilities on a bimonthly basis were calculated.

Selected diseases were modelled using the Markov Chain approach to illustrate how quickly endemic disease with different incidence levels may reach an overall stable prevalence. A simulation of seasonal variation of prevalence of bluetongue antibodies was also carried out. The choice of the Markov Chain seems logical since the system we deal with can be described by two states (positive and negative) and the animals move between these two states with given probabilities. Furthermore the Markov Chain approach assumes that the present state of an individual does not depend on earlier states; this agrees with the assumption that infection of individual animals is a random process. Finally Markov Chain models with fixed transition probabilities tend to stabilize after relatively few iterations which seem to agree with the observed overall stable prevalences for the entire study period (1986, 1988, 1989).

The study area, its animal population and the method of ELISA has been described in an earlier publication (BARAJAS-ROJAS et al, 1993a).

RESULTS

The observed transition rates from positive to negative and from negative to positive during bimonthly ELISA tests of 181 producing cows are shown in table I.

From this data Markov Chain transition matrices were constructed for three disease agents: Bovine Viral Diarrhea virus, low overall prevalence (10%); Leptospira interrogans serovar hardjo, medium overall prevalence (41%) and Bluetongue virus, high overall prevalence (87. 7%). Markov Chain simulations were carried out and the results are shown in Tables II, ^III and ^IV. The herd size is fixed at 100 and it is assumed that at the beginning one cow is positive. The same results would be obtained if it was assumed that all cows were negative at the beginning, the source of infection being the environment or other animal populations. When the simulated prevalences reach steady state they correspond closely to the observed prevalences: 9.88% versus 10.00, 41.66 versus 41.00 and 86.95 versus 87.7. This is not surprising since observed and simulated prevalences are based on the same data set.

The simulated steady states are reached in a period of 1 - 1.5 years; this is a rather short period and in agreement with the fact that overall observed prevalences were almost the same over the total 3 years study period.

For diseases with pronounced seasonal variation simulation using overall transition probabilities misses interesting aspects of the disease. Antibodies to Bluetongue virus show strong seasonal variation (Figure 1). The decrease in prevalence is assumed to result from decreased vector activity that will reduce the number of animals remaining positive because re-infection occurs less frequently and also reduce the transition probability from negative to positive. Increasing prevalence is caused by the opposite situation. Table V and Figure 1 show the result of a simulation with transition probabilities varying with season. The same simulation can be obtained with other, different combinations of transition probabilities; here it was chosen to keep the probabilities of positive animals remaining positive fairly high because of longevity of antibodies and because these animals may be the most attractive for the vector. The transition negative to positive was allowed to fluctuate tenfold over seasons.

DISCUSSION

Although endemic diseases may show a very stable prevalence over years this does not mean that active disease transmission is absent. In this study of a cattie population of the tropics of México there are indications that a disease as Bovine viral Diarrhea with a long term prevalence of 10% actually has a 8% bimonthly incidence rate; for Leptospira irrterrogans serovar hardjo with a steady prevalence of 41% the bimonthly incidence rate may actually be 25% and for Bluetongue virus with a steady prevalence of 87.7% the overall bimonthly incidence rate may be 60%. The latter show strong seasonal variations that may be illustrated by means of Markov Chain modeling.

ACKNOWLEDGEMENTS

We appreciate the assistance in sample and data collections of Dr. Eusebio Garcia Neria and Dr. Rebeca Acosta and personnel of the Centro de Investigación Ensenanza y Extension en Ganadería Tropical-ClEEGT of the Faculty of Veterinary Medicine. National Autonomous University of México. México City.

²Doctor in Veterinary Medicine, PhD, Professor Dept of Epidemiology and Preventive Medicine, University of California, Davis, CA 95616. USA.

³Statistician PhD, Professor Dept of Epidemiology and Preventive Medicine, University of California, Davis, CA 95616, USA.

Recebido para publicação em 21.10.92. Aprovado para publicação em 07.04.93.

• BARAJAS-ROJAS, J.A, RIEMANN, H.P. and FRANTI, C.E. Application of enzyme linked immunosorbent assay (ELISA) for epidemiological studies of diseases of livestock in the tropics of México. Rev Sd Tech Off Int Epiz, v. 12, n.1, 1993a. In Press.
• BARAJAS-ROJAS, J.A., RIEMANN, H.P. and FRANTI, C.E. Serological screening for infectious cattie diseases I. Influence of reproductive status. Ciência Rural, v. 23, n. 1, p. 69-72, 1993b.
• BARAJAS-ROJAS, J.A., RIEMANN, H.P. and FRANTI, C.E. Serological screening for infectious cattie diseases II. Association between prevalence and level of ELISA responde. Ciência Rural, v. 23, n. 2, p. 193-196, 1993c.

¹

Doctor in Veterinary Medicine, MSc, PhD, Professor Titular, Dept of Virology and Immunology, Faculty of Veterinary Medicine, National Autonomous University of México, México City 04510, México.

Publication Dates

• Publication in this collection
  03 Sept 2014
• Date of issue
  Dec 1993

History

• Accepted
  07 Apr 1993
• Received
  21 Oct 1992