Effects of seasonal variation patterns on recurrent outbreaks in epidemic models

Impact of periodic vaccination in SEIRS seasonal model

We study three different strategies of vaccination in an SEIRS (Susceptible-Exposed-Infected-Recovered-Susceptible) seasonal forced model, which are (i) continuous vaccination; (ii) periodic short-time localized vaccination, and (iii) periodic pulsed width campaign. Considering the first strategy, we obtain an expression for the basic reproduction number and infer a minimum vaccination rate necessary to ensure the stability of the disease-free equilibrium (DFE) solution. In the second strategy, short duration pulses are added to a constant baseline vaccination rate. The pulse is applied according to the seasonal forcing phases. The best outcome is obtained by locating intensive immunization at inflection of the transmissivity curve. Therefore, a vaccination rate of 44.4% of susceptible individuals is enough to ensure DFE. For the third vaccination proposal, additionally to the amplitude, the pulses have a prolonged time width. We obtain a non-linear relationship between vaccination rates and the duration of the campaign. Our simulations show that the baseline rates, as well as the pulse duration, can substantially improve the vaccination campaign effectiveness. These findings are in agreement with our analytical expression. We show a relationship between the vaccination parameters and the accumulated number of infected individuals, over the years, and show the relevance of the immunization campaign annual reaching for controlling the infection spreading. Regarding the dynamical behavior of the model, our simulations show that chaotic and periodic solutions as well as bi-stable regions depend on the vaccination parameters range.

Dynamics of an SIS network model with a periodic infection rate

Seasonal forcing and contact patterns are two key features of many disease dynamics that generate periodic patterns. Both features have not been ascertained deeply in the previous works. In this work, we develop and analyze a non-autonomous degree-based mean field network model within a Susceptible-Infected-Susceptible (SIS) framework. We assume that the disease transmission rate being periodic to study synergistic impacts of the periodic transmission and the heterogeneity of the contact network on the infection threshold and dynamics for seasonal diseases. We demonstrate both analytically and numerically that (1) the disease free equilibrium point is globally asymptotically stable if the basic reproduction number is less than one; and (2) there exists a unique global periodic solution that both susceptible and infected individuals coexist if the basic reproduction number is larger than one. We apply our framework to Scale-free contact networks for the simulation. Our results show that heterogeneity in the contact networks plays an important role in accelerating disease spreading and increasing the amplitude of the periodic steady state solution. These results confirm the need to address factors that create periodic patterns and contact patterns in seasonal disease when making policies to control an outbreak.

Intervention threshold for epidemic control in susceptible-infected-recovered metapopulation models

Metapopulation epidemic models describe epidemic dynamics in networks of spatially distant patches connected via pathways for migration of individuals. In the present study, we deal with a susceptible-infected-recovered (SIR) metapopulation model where the epidemic process in each patch is represented by an SIR model and the mobility of individuals is assumed to be a homogeneous diffusion. We consider two types of patches including high-risk and low-risk ones under the assumption that a local patch is changed from a high-risk one to a low-risk one by an intervention. We theoretically analyze the intervention threshold which indicates the critical fraction of low-risk patches for preventing a global epidemic outbreak. We show that an intervention targeted to high-degree patches is more effective for epidemic control than a random intervention. The theoretical results are validated by Monte Carlo simulations for synthetic and realistic scale-free patch networks. The theoretical results also reveal that the intervention threshold depends on the human mobility network and the mobility rate. Our approach is useful for exploring better local interventions aimed at containment of epidemics.

The dynamical consequences of seasonal forcing, immune boosting and demographic change in a model of disease transmission

The impact of seasonal effects on the time course of an infectious disease can be dramatic. Seasonal fluctuations in the transmission rate for an infectious disease are known mathematically to induce cyclical behaviour and drive the onset of multistable and chaotic dynamics. These properties of forced dynamical systems have previously been used to explain observed changes in the period of outbreaks of infections such as measles, varicella (chickenpox), rubella and pertussis (whooping cough). Here, we examine in detail the dynamical properties of a seasonally forced extension of a model of infection previously used to study pertussis. The model is novel in that it includes a non-linear feedback term capturing the interaction between exposure and the duration of protection against re-infection. We show that the presence of limit cycles and multistability in the unforced system give rise to complex and intricate behaviour as seasonal forcing is introduced. Through a mixture of numerical simulation and bifurcation analysis, we identify and explain the origins of chaotic regions of parameter space. Furthermore, we identify regions where saddle node lines and period-doubling cascades of different orbital periods overlap, suggesting that the system is particularly sensitive to small perturbations in its parameters and prone to multistable behaviour. From a public health point of view - framed through the 'demographic transition' whereby a population׳s birth rate drops over time (and life-expectancy commensurately increases) - we argue that even weak levels of seasonal-forcing and immune boosting may contribute to the myriad of complex and unexpected epidemiological behaviours observed for diseases such as pertussis. Our approach helps to contextualise these epidemiological observations and provides guidance on how to consider the potential impact of vaccination programs.