Mathematical model identifies threshold for optimal containment of pandemics

Researchers at the have developed a mathematical framework that weighs infection control against its social and economic costs, revealing an abrupt threshold between minimal intervention and strong containment

A mathematical model developed by researchers at the Max Planck Institute for Dynamics and Self-Organisation (MPI-DS), Göttingen, Germany has identified a clear threshold that could help societies determine when strong measures to control an infectious disease become preferable to little or no intervention.

During a pandemic, governments face challenging problems in how to limit the spread of disease but without imposing too many restrictions on everyday life. Measures such as requirements to wear face coverings or limits on social contact can reduce transmission but also impose social, economic and psychological costs.

Researchers from the Theory of Complex Systems group at Max Planck have developed a mathematical model to optimise countermeasures while accounting for the costs of disease containment.

The framework uses numerical methods to calculate the optimal intensity of intervention according to factors that include the characteristics and severity of a disease. Rather than assess individual pathogens or recommend particular public-health measures, the researchers sought to identify general principles that could underpin pandemic control.

The framework can also be combined with different models of infectious disease transmission.

“Through this, we hope to contribute to preparations for possible future outbreaks,” said Laura Müller, first author of the study and a doctoral candidate at the institute.

To identify general patterns of optimal infection control, the researchers used a classical epidemiological model in which individuals are susceptible to infection, infected or temporarily immune after recovery.

Rather than identify a gradual progression from weak to increasingly stringent interventions as disease severity rose, the model revealed a threshold at which the optimal response changed abruptly.

“Surprisingly, it becomes very clear that optimal measures follow a threshold structure,” said Professor Viola Priesemann, group leader at MPI-DS.

“For mild diseases, the model shows that it is optimal to not impose any containment measures. However, once the disease reaches a certain severity, a high level of containment [becomes the] optimal [strategy].”

The result suggests that, within the assumptions of the idealised model, intermediate intervention can sometimes prove less efficient than either strong containment or no containment.

“It is very surprising that the transition in the idealised model occurs absolutely abruptly,” Priesemann said.

Compromise measures could therefore result in greater total costs. Restrictions might be more extensive than necessary for a relatively mild disease, while measures that are too weak could fail to suppress transmission if infection has sufficiently severe consequences.

The researchers stressed that the model does not determine which interventions authorities should adopt. Such choices remain societal and political questions and depend on factors including disease characteristics, the effectiveness of interventions and the costs that society attributes to them.

The team also examined how seasonal changes in infection rates could affect the optimal response. The model indicated that containment during winter should increase in parallel with the probability of infection. For seasonal diseases such as influenza, an optimised approach could be to substantially suppress the typical winter period of the epidemic, leaving – at most – a relatively small infection wave in spring.

Under the assumptions of the model, this residual wave would occur exactly three months after the seasonal peak in transmission.

The framework can also calculate how authorities could reduce containment as vaccination programmes increase population protection. As vaccination reduces susceptibility to severe infection or transmission, the balance between the benefits and costs of restrictions changes.

The researchers also assessed delays in the introduction of containment measures. Because infections can increase exponentially during the initial phase of an epidemic, even comparatively short delays can result in substantially more infections before controls take effect. For diseases with a high health burden, such delays produced considerable additional overall costs. Conversely these consequences were much smaller for milder diseases.

The study relies on a simplified representation of infectious disease transmission and cannot provide direct guidance for a particular pathogen or outbreak. Real epidemics involve factors such as differences in susceptibility, population structure, behaviour, immunity, healthcare capacity and the effectiveness and acceptability of interventions.

Nevertheless, the researchers have identified a general mathematical principle whereby under certain conditions, the optimal response can switch sharply from little or no intervention to strong containment once disease severity crosses a critical threshold.

Such principles could help scientists and policymakers to assess strategies during the early stages of future outbreaks and provide a basis for more detailed models that incorporate epidemiological, economic and social circumstances.

The researchers emphasised that mathematical optimisation cannot determine society’s priorities. Decisions about acceptable costs and how to balance public-health, economic and social objectives ultimately remain matters for society and policymakers.

For further reading please visit: 10.1073/pnas.2527395123