Propagation dynamics of an epidemic model with infective media connecting two separated networks of populations

Highlights

• An epidemic model based on two contact-networks is proposed, where cross network infection occurs via vectors.

• The basic reproduction number for this system is calculated and global stability is proven.

• The effect of animal–animal interactions and contact with vectors can induce endemic states for even low infection rates.

• Increased heterogeneity in either of the networks can exacerbate epidemic spreading.

Abstract

Based on the fact that most human pathogens originate from animals, this paper attempts to illustrate the propagation dynamics of some zoonotic infections, which spread in two separated networks of populations (human network I and animal network II) and cross-species (vectors, or infective media). An epidemic time-evolution model is proposed via mean-field approximation and its global dynamics are investigated. It is found that the basic reproduction number in terms of epidemiological parameters and the network structure is the threshold condition determining the propagation dynamics. Further, the influences of various infection rates and contact patterns are verified. Numerical results show that the heterogeneity in connection patterns and inner infection in network I can easily trigger endemic dynamics, but when a pathogen, such as H7N9, has weak infectivity in humans, the effects of animal–animal interactions and the contacts with vectors tend to induce endemic states and enhance the prevalence in all the populations.

Introduction

Infectious diseases are always serious threats to mankind’s health and life, and the recent outbreaks of endemic and pandemic events (such as AIDS, SARS, H1N1 and currently H7N9) signify that the danger does not go away [1]. As an analytical approach in epidemiology, mathematical modeling, which might be traced back to as early as 1927 [2], can help predict epidemic dynamics (including the propagation process, the outbreak time and place, the final state, and so on) and thus suggests effective measures to mitigate and prevent the spread of the diseases.

After nearly a century of development in epidemic modeling, mathematical epidemiologists now pay much more attention to meticulous and practical details of different diseases, and focus more on individual-based analysis. In the last decade, spurred by the availability of real data and the maturation of the complex networks theory [3], [4], a bridge between epidemiology and network science has emerged. The connections among individuals that allow an epidemic to propagate naturally form a network, which can provide many new insights into the epidemiological dynamics [5], [7], [6]. In a contact network, for example, each node represents an individual and each link describes the direct contact between two of them. Network modeling usually rejects the over-simplified homogeneous-mixing assumption and explicitly captures the diverse patterns of interactions that underly disease transmission [8], [9]. Many network-based models, such as the networked SIS [5], [10], SIR [11], [12] and SI [13] models, have been built to study the effects of network structures on infection mechanisms [14], [15], [16]. For a structurally homogenous network, the epidemic threshold is the reciprocal of the average degree of the network. For a scale-free network however, it was formulated that the epidemic threshold is near zero and thus the infection could always persist [5], [11], [12], [13]. Further, it was verified that the leading role of hub and the innermost dense core is the main activation mechanism [17]. Generally speaking, the topological fluctuation of the contact network exacerbates the outbreak of a disease [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16].

On the other hand, a recent survey reveals that over half of human pathogens are zoonotic [18], and almost all of them are either zoonotic or originated as zoonoses before affecting humans [19]. For example, the three most devastating pandemic diseases in human history, namely, Black Death, Spanish Influenza and AIDS, were caused by zoonoses [20]. Moreover, many diseases like black plague, dengue, malaria and yellow fever, are transmitted through vectors (i.e., infective media) such as mice and mosquitoes [1]. These underlying patterns of complex multi-host ecology of zoonotic infections are widespread challenges to public health, which also accelerate environmental and anthropogenic changes today. They as well alter the rate and nature of contact between human and animal populations [21]. Recently, several epidemic models have been built with human-animal interface based on complex contact networks. Particularly, a new networked SIS model with infective media was proposed [22], in which the disease spreads by contacts among individuals (considered as an scale-free network) and also between individuals and vectors (considered to be homogeneous). Lately, the model was further improved and its global behaviors were investigated [23], [24].

Noticeably, the dynamics of most zoonoses involve multiple phases: propagation in the animal reservoir, spillover propagation into humans, and sustained propagation among humans [21], [20]. A typical example is the Black Death, which came from mice and then reached people through fleas. Here, the fleas work as a cross-specie connecting humans and mice, but humans and mice do not have direct contact.

In this paper, we propose a new epidemic model, where humans form a complex network through contact and the animals form another one, while the vectors can interact with animals and humans respectively thus transmitting diseases from one network to another. The whole contact patterns can be seen as a layered network. In Section 2, we describe the model in detail. In Section 3, we investigate the global dynamics of this new model. In Section 4, we report some numerical simulations and sensitivity analysis. Finally, we present a brief conclusion in the last section.