{"paper":{"title":"Crime hotspot dynamics in residential burglary models with police response","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Response delays in police deployment can destabilize stable crime equilibria through Hopf bifurcations, producing oscillating and moving hotspots.","cross_cats":["cs.NA","math.NA"],"primary_cat":"math.DS","authors_text":"Annalisa Quaini, Baoli Hao, Kamrun Mily, Ming Zhong","submitted_at":"2026-05-18T00:09:04Z","abstract_excerpt":"We develop and analyze mathematical models for residential burglary that incorporates police deployment through a delayed feedback mechanism. Motivated by empirical observations from publicly available crime and policing data, we extend a well-known agent-based model by introducing a dynamic police response driven by crime information that becomes available only after a finite delay. Taking the mean-field limit, we derive a coupled continuum system consisting of three partial differential equations and one ordinary differential equation describing the interactions among criminal density, envir"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Linear stability analysis of homogeneous steady states reveals that response delays can destabilize otherwise stable equilibria through Hopf bifurcations. As a result, the model predicts sustained temporal oscillations and dynamically evolving crime hotspots. Timely access to crime data plays a more important role than police density in stabilizing crime levels.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The mean-field limit of the agent-based model with the chosen form of delayed crime-signal feedback accurately represents real-world police deployment dynamics and neighborhood effects.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A mean-field continuum model with delayed police feedback, derived from an agent-based burglary model, shows via linear stability analysis that information delays trigger Hopf bifurcations and produce dynamic, moving crime hotspots.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Response delays in police deployment can destabilize stable crime equilibria through Hopf bifurcations, producing oscillating and moving hotspots.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"f4387d8e1f48af9d52a337160a0aa51cd6e4b0ad7cb421b65dc17f5137a46338"},"source":{"id":"2605.17709","kind":"arxiv","version":1},"verdict":{"id":"4605755f-8635-4142-94ba-40c6a1f85a28","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T22:54:57.966577Z","strongest_claim":"Linear stability analysis of homogeneous steady states reveals that response delays can destabilize otherwise stable equilibria through Hopf bifurcations. As a result, the model predicts sustained temporal oscillations and dynamically evolving crime hotspots. Timely access to crime data plays a more important role than police density in stabilizing crime levels.","one_line_summary":"A mean-field continuum model with delayed police feedback, derived from an agent-based burglary model, shows via linear stability analysis that information delays trigger Hopf bifurcations and produce dynamic, moving crime hotspots.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The mean-field limit of the agent-based model with the chosen form of delayed crime-signal feedback accurately represents real-world police deployment dynamics and neighborhood effects.","pith_extraction_headline":"Response delays in police deployment can destabilize stable crime equilibria through Hopf bifurcations, producing oscillating and moving hotspots."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17709/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T23:01:19.407556Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T23:00:54.107750Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"shingle_duplication","ran_at":"2026-05-19T21:49:43.476187Z","status":"skipped","version":"0.1.0","findings_count":0},{"name":"citation_quote_validity","ran_at":"2026-05-19T21:49:43.305789Z","status":"skipped","version":"0.1.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T21:33:23.509303Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"cited_work_retraction","ran_at":"2026-05-19T21:21:58.650612Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T21:21:57.415447Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"040d3099d68881f046df15c8b722b0f4241b792766e94032ca913d3de3367903"},"references":{"count":34,"sample":[{"doi":"","year":null,"title":"https://github.com/baolihao/UrbanCrime-Sim","work_id":"4563c523-ad65-4c34-8e3b-8ff3aac805fe","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Chicago Data Portal, Boundaries - Police Beats.https://data.cityofchicago.org /Public-Safety/Boundaries-Police-Beats-current-/aerh-rz74","work_id":"cffccbe5-1663-43e8-904b-4e0b26a24a48","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2001,"title":"Chicago Data Portal, Crimes - 2001 to Present.https://data.cityofchicago.org /Public-Safety/Crimes-2001-to-Present/ijzp-q8t2","work_id":"9ccd1269-3dc4-482c-917e-9b268f6d5b16","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Chicago Police Department Investigatory Stop Report data.https://www.chicagop olice.org/statistics-data/isr-data/. 30","work_id":"b5583ed2-96f4-4073-aca4-ba9bed437c40","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Sworn CPD Member Staffing Map.https://igchicago.org/information-portal/ data-dashboards/sworn-cpd-member-staffing-map-2/","work_id":"3a12c425-7e05-4606-a87a-6b3f7b3b9236","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":34,"snapshot_sha256":"9e19eb5f38864670724bc31a1a92d79e58ed1858827cdb8b569570699fe35592","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"f507e1d317b8fb8ba7e85d657f046da39813c3276527822653a68f6fe32b23ec"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}