TITLE:
On a Mathematical Model with Non-Pharmaceutical Interventions for Long-Term Dynamics of Viral Infections
AUTHORS:
James A. Njongwe, Miranda I. Teboh-Ewungkem, Gideon A. Ngwa
KEYWORDS:
Behavioural Dynamics, Environmental Contributors, Viral Disease Dynamics
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.9,
September
24,
2026
ABSTRACT: A non-linear ordinary differential equation mathematical model is developed and used to study long-term dynamics for viral disease transmission. The model explicitly incorporates human behaviour and environmental transmissions from viral particles shed into the environment by infected humans. The proposed compartmental model is sufficiently general to capture key epidemiological features common to a broad class of viral infections, while accounting for behavioural changes as individuals transition between disease states and adapt their contact patterns within each stage of infection. The model allows a contaminated environment to act as a reservoir of infection, thereby capturing environmentally mediated transmission arising from viral particles shed by infected humans. The population is stratified into quarantined and non-quarantined classes, allowing the model to assess the impact of non-pharmaceutical interventions on disease progression. Analytical results are used to establish the existence and stability of disease-free and endemic equilibria. The basic reproduction number
ℛ
0
is calculated taking into consideration different combinations of transition and new infection pathways, and its threshold-like nature is well established. Numerical simulations are performed using COVID-19 informed parameter values to illustrate the model dynamics under different scenarios. The theoretical and numerical results from the model’s output highlight the significant role of human behavioural dynamics and environmental transmission in shaping long-term disease outcomes.