## 1 Introduction  
One promising strategy for managing the exponential growth of mobile data traffic within cellular networks involves the adoption of device-to-device (D2D) opportunistic communications to offload traffic from congested cellular base stations [1]. Specifically, D2D communications leverage short-range out-of-band communication media, such as IEEE 802.11, or are used as an underlay to the cellular network [2], to enable direct data exchange between proximate mobile devices without relying on cellular network infrastructure. In cases where content reception may tolerate a certain degree of delay (e.g., podcasts), the D2D-based information dissemination process can employ a delay-tolerant networking (DTN) approach [3]. Delay-tolerant networks embrace the so-called _store-carry-forward_ communication paradigm, where a node receiving a message holds it while it moves, and subsequently passes it to an encountered node.  
Various D2D-based offloading schemes have been proposed, differing in the methods used to select the subset of nodes that assist in the content distribution [4], the DTN routing algorithm [5], or the caching policy for buffering the content in node caches [6]. In this study, we consider a _\"frugal\"_ offloading scheme inspired by [6] and initially described in [7], which aims to minimise memory and network resource usage during the offloading process. Specifically, only nodes interested in a given content can store it in their local cache, and content copies are not disseminated epidemically but are forwarded to a newly encountered node only if it has already requested the content. Additionally, content copies are stored in local device caches for a limited time, called _sharing timeout_. Hereafter, we refer to this offloading scheme as Opportunistic Offloading with Resource Saving, or OORS for short.  
Previous research has proposed ordinary differential equation (ODE)-based mathematical frameworks, inspired by the susceptible-infectious-recovered (SIR) epidemic model, to assess the performance of DTN-based routing and information diffusion mechanisms [8; 9; 10; 11]. The SIR model draws an analogy between a content item and an infectious disease, where nodes become susceptible to 'infection' after requesting the content, become 'infected' upon receiving and sharing it, and transition to a'recovered' state once they stop content sharing. The foundational work [9] developed ODE models for several variants of epidemic routing under the assumption that the inter-meeting time of any two nodes is exponentially distributed. Fluid models based on ODEs have also been proposed to include multiple classes of nodes [10], community-based mobility models [11], or different packet-discarding strategies aimed at controlling the spread of packet copies in the network [8; 9; 12]. Previous ODE-based models of epidemic-like routing have included time-limited caching to save buffer space and communication resources under rather restrictive assumptions. For instance, the model in [9] assumes that either an exponentially distributed timeout is associated with each content copy, or a global fixed timeout exists, which deletes all copies of the content at all nodes at the same time. To the best of our knowledge, how to model delayed D2D offloading with constant and asynchronous timeouts on both content sharing and content delivery using a fluid model is still an unsolved problem.  
Before presenting the main contribution of this paper, we provide an overview of the relevant related work, highlighting its advantages and limitations.