Leptospirosis Relative Risk Estimates based on Continuous-Time, Discrete-Space Stochastic SIR-L-SI Transmission Model


  • Sufi Hafawati Ideris School of Mathematical Science, College of Computing, Informatics and Media Univerisiti Teknologi MARA, 40450 Shah Alam, Selangor, Malaysia
  • Norshahida Shaadan School of Mathematical Science, College of Computing, Informatics and Media Univerisiti Teknologi MARA, 40450 Shah Alam, Selangor, Malaysia
  • Syazreen Niza Shair School of Mathematical Science, College of Computing, Informatics and Media Univerisiti Teknologi MARA, 40450 Shah Alam, Selangor, Malaysia
  • Nor Azah Samat Department of Mathematics, Faculty of Science and Mathematics, Universiti Pendidikan Sultan Idris, 35900 Tanjong Malim, Perak, Malaysia




Leptospirosis, SIR-L-SI model, asymptotic, numerical, relative risk


Leptospirosis is a re-emerging global disease that has become endemic in Malaysia. The transmissions usually occur between animals especially rats to rats and rats to humans. Since, it is an easily contractable disease that can be transmitted directly through contact with infected rat’s urine or indirectly from the environment such as via water and soil, it is very challenging to curb this disease from infecting humans. Poor understanding of the disease and lack of epidemiological data also made leptospirosis is difficult to control. To cope with this problem, a leptospirosis disease transmission model is developed to study the mechanism of leptospirosis disease spread over continuous-time that may help to predict future caused of an outbreak. This study aims to construct a continuous-time and discrete-space stochastic SIR-L-SI (Susceptible, Infected, Recovered Humans-Leptospires in the Environment-Susceptible, Infectious Rats) of leptospirosis disease transmission to estimate the risk involved. A simple method of asymptotic and numerical analyses is applied as an alternative approach for solving simultaneous differential equations in the leptospirosis SIR-L-SI transmission model. The application of the proposed model is demonstrated using leptospirosis data for Malaysia. The results of asymptotic behaviour and numerical analysis provide useful information about susceptible and infective rat and human populations as well as offer relative risk estimates that can be used as one of the control measures in identifying hot-spot areas for this disease.


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