A spatial SIS point pattern model with movement

Eka Suci Pramana Sari; Nanang Susyanto; Fajar Adi-Kusumo; Atina Husnaqilati; Fugo Takasu; Eka Suci Pramana Sari; Nanang Susyanto; Fajar Adi-Kusumo; Atina Husnaqilati; Fugo Takasu

doi:10.3934/mbe.2026080

Mathematical Biosciences and Engineering

2026, Volume 23, Issue 7: 2208-2227. doi: 10.3934/mbe.2026080

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Research article

A spatial SIS point pattern model with movement

1.
Department of Mathematics, Universitas Gadjah Mada, Yogyakarta, Indonesia
2.
Department of Environmental Science, Nara Women's University, Nara, Japan

Received: 13 April 2026 Revised: 26 May 2026 Accepted: 06 July 2026 Published: 17 July 2026

Spatial epidemic dynamics are strongly affected by local interactions and by the movement of individuals. This study develops a susceptible–infected–susceptible (SIS) model based on point pattern dynamics with individual movement. The model extends the spatial SIS point pattern framework of Hamada and Takasu by allowing susceptible and infected points to move in continuous space. Disease transmission is represented by a distance-dependent infection kernel, and Gaussian, step-function, and exponential kernels are compared. Heuristic singlet and pair-density equations are derived to describe how infection, recovery, birth, death, and movement affect the spatial distribution of susceptible and infected individuals. Numerical simulations are used to illustrate the effects of kernel shape, infection range, and movement on singlet densities, pair correlations, and the temporal evolution of point patterns. The results show how spatial movement and distance-dependent transmission jointly shape infection spread in a recurrent-infection framework.
- spatial transmission,
- point pattern dynamics,
- SIS model,
- individual movement,
- mathematical modeling
Citation: Eka Suci Pramana Sari, Nanang Susyanto, Fajar Adi-Kusumo, Atina Husnaqilati, Fugo Takasu. A spatial SIS point pattern model with movement[J]. Mathematical Biosciences and Engineering, 2026, 23(7): 2208-2227. doi: 10.3934/mbe.2026080

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Abstract

Spatial epidemic dynamics are strongly affected by local interactions and by the movement of individuals. This study develops a susceptible–infected–susceptible (SIS) model based on point pattern dynamics with individual movement. The model extends the spatial SIS point pattern framework of Hamada and Takasu by allowing susceptible and infected points to move in continuous space. Disease transmission is represented by a distance-dependent infection kernel, and Gaussian, step-function, and exponential kernels are compared. Heuristic singlet and pair-density equations are derived to describe how infection, recovery, birth, death, and movement affect the spatial distribution of susceptible and infected individuals. Numerical simulations are used to illustrate the effects of kernel shape, infection range, and movement on singlet densities, pair correlations, and the temporal evolution of point patterns. The results show how spatial movement and distance-dependent transmission jointly shape infection spread in a recurrent-infection framework.

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