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At the retained order define $K=1+\tau r(I-S)$. Neglecting terms of order $a\tau$ in the incidence correction givesDividing and expanding consistently,Thus the susceptible count $S_{\ast }$ at the first threshold crossing obeys the same retained-order equation as before:It is important to distinguish completed case transfers at quarantine from all infections already initiated. Let the quarantine time be $T$. Isolation prevents exposures after $T$, but animals exposed during $(T-\tau ,T)$ still finish their incubation period during $(T,T+\tau )$. Their number, in the printed bookkeeping, isThe infection cases transferred by time $T$ among the initial $S$ class number $S_0-S_{\ast }$. Adding those pending infection cases gives the eventual number who catch the infectious disease:This adds only infections caused by exposures before isolation; it does not allow secondary transmission after quarantine. If one asks instead only for the transfers completed by the instant of quarantine, the answer is $S_0-S_{\ast }$. The distinction is necessary because the model keeps incubating animals in its $S$ compartment.

The Holocene history combines natural climatic and geographic filtering with increasingly strong human restructuring through hunting, habitat change, domestication, introductions and conservation. The balance varies through time and between Britain and Ireland. The Holocene begins approximately 11.7 thousand calibrated years before present; for radiocarbon dating, “present” conventionally means 1950. Uncalibrated radiocarbon ages must not be silently treated as calendar ages.

At the beginning, warming following the last glaciation changed vegetation and the distribution of food and cover. Open, cold-adapted assemblages gave way to a changing mixture of woodland, wetland and open-habitat mammals. Rising seas progressively restricted natural colonization; Britain retained connections to continental Europe later than Ireland's faunal assembly can simply be assumed to have done. The modern Irish fauna is therefore not a complete British fauna subsequently pruned by identical extinctions. A missing Irish species may never have colonized, rather than having been exterminated there. pollen analysis, stratigraphy and directly dated bones help distinguish habitat change from collecting gaps.

Chronology prevents an equally misleading division into wholly “Pleistocene” losses and wholly modern survivors. Some conspicuous cold-stage mammals, such as the woolly rhinoceros, disappeared from Britain before the Holocene and should not automatically be included in a Holocene hunting-extinction list. Conversely, directly dated giant deer from the northern Irish Sea basin demonstrate early Holocene survival of a lineage once thought to have disappeared regionally earlier. ^[ref] A date from one part of this region does not establish persistence everywhere in Ireland or Britain. Local disappearance, regional disappearance and global extinction are different endpoints; environmental deterioration and human activity need to be evaluated at the appropriate geographic and chronological scale.

Later prehistory saw a profound change in mammalian abundance as farming spread. Domestic cattle, sheep, goats and pigs became major components of archaeological assemblages and living biomass. Grazing, clearance and woodland management altered habitat for wild herbivores and their predators; settlement and stored food also favoured commensal rodents. This was more than adding several species to an unchanged ecosystem. Distinguishing domestic from wild forms requires skeletal comparisons, context and, where available, Ancient DNA and stable isotope analysis. A large number of livestock bones in a settlement midden is evidence of human use, not a direct census of every mammal in the surrounding landscape.

Losses of large native species were staggered. British aurochs, wild boar, lynx, brown bears, beavers and wolves disappeared at different times through combinations of hunting, persecution, habitat transformation and habitat fragmentation. The list cannot be transferred intact to Ireland: a secure native record must be established for each taxon, and a late imported specimen does not demonstrate an earlier indigenous population. Small and elusive species are also less visible in many archaeological deposits, so the absence of their remains should not be equated with a known extinction date.

The dated British lynx record is especially instructive. Survival into the early medieval period makes an explanation based solely on warming at the start of the Holocene inadequate. Human habitat change and direct or indirect persecution become more plausible proximate causes, although the last dated specimen cannot identify which pressure killed the final population. ^[ref] brown bear history is less tidy: review of the British evidence supports either late Neolithic/early Bronze Age or early medieval extinction, and later archaeological bear remains can represent imported captive animals rather than surviving wild populations. ^[ref] Traditional stories of the “last wolf” likewise require caution; historically documented persecution is much stronger evidence of human pressure than an unverified exact last-kill year.

Human transport makes apparent continuity equally uncertain. Combined skeletal, Ancient DNA and phylogeography evidence indicates a complex Irish red deer history involving prehistoric and recent introductions, rather than simple uninterrupted survival of every modern population from the last glaciation. Neolithic-era records and the contrast between older Kerry-associated ancestry and many recent stocks must be interpreted with sampling gaps in mind. ^[ref] Modern distribution alone cannot distinguish an ancient native population, an early species translocation, and a recent estate introduction. Each may require a different conservation interpretation even if all animals bear the same species name.

fallow deer provide a clearer example of deliberate species translocation. Integrated archaeological and biomolecular research traces Roman movements and later medieval redistribution, including the British history of introduction and reintroduction. ^[ref] Managed rabbits and later introductions of sika deer, muntjac, grey squirrels and American mink further altered assemblages. Some escaped from breeding or management; others were deliberately released. Historical records establish routes and timing that cannot be deduced from climatic suitability alone. Modern invasions are thus a continuation of human-mediated assembly, with potentially stronger effects because transport crosses barriers that once limited colonization.

The replacement of red squirrels by grey squirrels illustrates multiple interacting mechanisms. interspecific competition for resources can matter, but experimental infection demonstrated severe infectious disease in red squirrels with little detectable effect on grey squirrels, supporting a role for squirrelpox in some declines. ^[ref] That experiment establishes differential susceptibility; it does not show that every field decline is caused by the virus. A field test must establish infection exposure as well as squirrel abundance and account for habitat and competitor density. Likewise, recovery of pine martens in Ireland has been associated with declines of grey squirrels and recovery opportunities for red squirrels; the evidence is informative but a spatial association alone is not a controlled demonstration that predation explains every local change. ^[ref]

Recent protection, woodland recovery and reintroductions also allow people to reverse some earlier losses. Such recoveries do not recreate a unique pristine baseline: a “native” reference chosen at the beginning of the Holocene, before farming, or before modern introductions gives different assemblages. The same applies to interpreting recovering coastal populations; marine and terrestrial records need their own detection and exploitation histories. Reliable attribution combines directly dated remains, habitat evidence, historical documentation and demographic or experimental tests, with explicit allowance for preservation and survey effort.

Climate and access strongly shaped early Holocene assembly; later changes increasingly reflect farming, persecution and transport, with modern infectious disease and conservation adding further layers. Human influence is therefore pervasive, but neither every absence nor every change in abundance can be assigned to humanity without establishing when and where the change occurred.

A per-capita mortality rate $a$ means that the death flux from a biological population of size $I$ is $aI$. With constant rate and no other transfers, $\dot{I}=-aI$ gives exponential survival $I(t)=I(0)e^{-at}$. In an epidemic model this can specify mortality from the infective compartment rather than mortality of the entire population.

In a compartmental epidemic model, a susceptible individual is one who can acquire the modeled infection. The susceptible compartment counts these individuals. An exposed but not yet infectious individual belongs to a separate latent class in an SEIR model; models that retain such individuals in a not-yet-infective compartment must state that bookkeeping convention explicitly.

An infectious disease caused by Treponema pallidum.

Willy Burgdorfer investigated tick-borne infectious diseases. In 1982 he and collaborators reported a spirochaete isolated from ticks associated with Lyme disease. Infected ticks produced persistent skin lesions in rabbits, and patients' blood serum contained antibodies reacting with the organism in indirect immunofluorescence tests.^[ref]

These were complementary observations: finding an organism in a suspected vector, observing an effect after feeding, and detecting a human immune response each supplied a different link in the causal argument. The original report cautiously proposed involvement in Lyme disease; it did not rest its conclusion solely on resemblance under a microscope.