Global properties of a delayed model for the dynamics of lumpy skin disease with vaccination efficacy

Nada A. Almuallem; Nada A. Almuallem

doi:10.3934/math.2025922

AIMS Mathematics

2025, Volume 10, Issue 9: 20642-20669. doi: 10.3934/math.2025922

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Global properties of a delayed model for the dynamics of lumpy skin disease with vaccination efficacy

Nada A. Almuallem ^,

Department of Mathematics and Statistics, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi Arabia

Received: 14 July 2025 Revised: 25 August 2025 Accepted: 29 August 2025 Published: 09 September 2025
MSC : 34K20, 92D30, 37N25, 93C10

Lumpy skin disease (LSD) is a viral infection in cattle caused by the lumpy skin disease virus (LSDV). The purpose of this study was to examine the qualitative dynamics of the LSD model, including vaccine efficacy and different types of discrete time delays. The model considers LSD transmission in both susceptible and vaccinated populations. Local and global stability analyses have been conducted. A Lyapunov functional was developed, and LaSalle's invariance principle was utilized to demonstrate the global asymptotic stability of the model's equilibria. We have calculated the basic reproduction number $ \mathsf{R_0} $. The LSD-free equilibrium is globally asymptotically stable (GAS) when $ \mathsf{R_0}\leq1 $, whereas the LSD-endemic equilibrium is GAS when $ \mathsf{R_0} > 1 $. The theoretical results have been confirmed through numerical simulations. The results have indicated that utilizing a combination of time delays is more effective in eradicating the LSD virus outbreak. From a biological perspective, the delay functions similarly to antiviral vaccines and treatments in mitigating the LSD outbreak. Furthermore, even when the vaccine efficacy vanishes, an extended incubation period and increased delays before infection with LSD significantly restrict viral transmission and inhibit the spread of the virus. Moreover, the results indicate that even with a high level of vaccine efficacy, the elimination of the disease from populations is unlikely without the implementation of supplementary mitigation strategies.
- LSD mathematical model,
- global stability,
- time delays,
- vaccine efficacy,
- numerical simulations
Citation: Nada A. Almuallem. Global properties of a delayed model for the dynamics of lumpy skin disease with vaccination efficacy[J]. AIMS Mathematics, 2025, 10(9): 20642-20669. doi: 10.3934/math.2025922

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Abstract

Lumpy skin disease (LSD) is a viral infection in cattle caused by the lumpy skin disease virus (LSDV). The purpose of this study was to examine the qualitative dynamics of the LSD model, including vaccine efficacy and different types of discrete time delays. The model considers LSD transmission in both susceptible and vaccinated populations. Local and global stability analyses have been conducted. A Lyapunov functional was developed, and LaSalle's invariance principle was utilized to demonstrate the global asymptotic stability of the model's equilibria. We have calculated the basic reproduction number $ \mathsf{R_0} $. The LSD-free equilibrium is globally asymptotically stable (GAS) when $ \mathsf{R_0}\leq1 $, whereas the LSD-endemic equilibrium is GAS when $ \mathsf{R_0} > 1 $. The theoretical results have been confirmed through numerical simulations. The results have indicated that utilizing a combination of time delays is more effective in eradicating the LSD virus outbreak. From a biological perspective, the delay functions similarly to antiviral vaccines and treatments in mitigating the LSD outbreak. Furthermore, even when the vaccine efficacy vanishes, an extended incubation period and increased delays before infection with LSD significantly restrict viral transmission and inhibit the spread of the virus. Moreover, the results indicate that even with a high level of vaccine efficacy, the elimination of the disease from populations is unlikely without the implementation of supplementary mitigation strategies.

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