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Rearrange the final-size equation as $s_{\infty }=s_0\exp [s_{\infty }-s_0-(1+\eta )q_0]$. With fixed $q_0>0,s_0>0$ and $\eta \to \infty$, the selected root is exponentially small:Substitution into the death count yieldsThus strong vaccination suppresses most secondary fatal infections but cannot save the $q_0$ animals already infected, as required by the stated reaction scheme. The limit holds for the fixed positive initial infection assumed here; it need not be uniform if $q_0$ is allowed to shrink with increasing $\eta$.

While $q>0$ and $s>0$, division of the equations gives $dq/ds=(1-s)/[(1+\eta )s]$ and $dv/ds=-\eta /(1+\eta )$. Integrating from the initial data yieldsLet $s_{\infty }$ be the final susceptible level. Using $q_{\infty }=0$ givesThis is the smaller positive root selected by decreasing susceptibility; the function $s-\log s$ is decreasing on $(0,1)$, so that selected root is unique. Its upper bound below $s_0$ also follows directly because the added positive term moves the decreasing-branch root left.

Conservation of the total count, or integration of $m_t=q$, then gives the number killed, including those initially infected:The remaining susceptible loss is split between infection and vaccination in fixed ratio $1:\eta$; this explains the factor multiplying $s_0-s_{\infty }$.

The proposed asymmetry is not generally defensible: animal behaviour and immunity form a reciprocal system, and “more” has no common meaning without an outcome, scale and intervention. Avoiding infection, controlling pathogen growth, tolerating tissue damage and changing social contact are different outcomes. Their causal effect sizes cannot be ranked by counting examples in either direction.

The definition of immunity matters first. In its narrower physiological sense, it concerns cellular and molecular defences. A behavioural immune system describes actions that reduce exposure before those defences are challenged, while social immunity includes protection generated by interactions among group members. If avoidance is defined as part of immunity, a statement that animal behaviour influences immunity becomes partly true by definition. A useful analysis instead specifies whether an action changes exposure, immune activity, pathogen burden or the damage associated with that burden. disease resistance reduces burden; disease tolerance reduces damage at a given burden. Lower illness alone does not distinguish these routes.

One strong route from animal behaviour to defence is the management of exposure. Choice of food, mates and social partners determines contact with infectious material. grooming can remove external parasites, and hygienic behaviour can remove contaminated material from a nest. Conversely, aggregation and some forms of close social contact can increase opportunities for transmission. These effects depend on the pathogen's transmission route: removing one kind of contact need not reduce infection spread by an environmental reservoir. An observed association between sociability and infection can also arise because infection changes sociability, making temporal or experimental evidence essential.

honeybee colonies selected for hygienic behaviour provide a causal evolutionary and experimental example. Selection based on brood-removal assays was combined with exposure to American foulbrood disease, allowing disease outcomes in selected and comparison colonies to be assessed. Removal of infected brood can reduce a colony's infectious reservoir and protect other individuals. ^[ref] The result concerns colony protection, not a demonstration that each worker's cellular immunity is stronger. Selection can change correlated traits as well as brood removal, so a mechanistic follow-up should match colony size, brood age, exposure and background, and manipulate the opportunity for removal. Colonies are independent experimental units; scoring many cells within one colony does not create many independently selected colonies.

Choice of an offspring’s food plant can also change the chemical environment encountered by parasites. Infected monarch butterflies preferentially laid eggs on plants that reduced parasite growth and disease in their offspring. This supported transgenerational medication: neither a simple cure of the mother nor merely a demonstration that a toxic plant can inhibit a parasite. ^[ref] The inference needs both the infection-dependent change in oviposition and the offspring benefit. Tests should include infected and uninfected mothers offered the same plant alternatives, standardized parasite exposure of offspring, and measurements of larval growth and survival as well as parasite burden. Nutritional quality and general plant preference are competing explanations. A cost of the medicinal plant in parasite-free conditions would additionally help explain why every individual does not always choose it.

animal behaviour can influence physiological immunity as well as exposure. Activity, feeding and sleep affect resource acquisition and allocation, while social interactions can alter endocrine and neural signals. These are plausible routes, but a study must measure the relevant immune process and not infer it from survival alone. An immune assay can increase because the animal is more infected, rather than because it has greater protective capacity. randomization, controlled nutrition and repeated observations can help separate these interpretations. Any claim of a trade-off should identify the limiting resource and the competing biological fitness demands; a change in one circulating marker is insufficient.

The reverse direction is equally substantive. Immune challenges can change activity, appetite, social investigation and priorities, producing sickness behaviour. Such changes may conserve resources, reduce exposure to further hazards or alter transmission, but some can instead reflect damaging inflammation or a pathogen's manipulation. Reduced activity after infection is therefore not automatically adaptive host defence. Comparing host biological fitness with and without the response, and separating immune activation from direct pathogen action, is necessary for an evolutionary interpretation.

lipopolysaccharide provides one experimental way to induce an inflammatory response without a replicating pathogen. Work on lactating house mice showed that the resulting effects depended on the activity and environmental context: pup retrieval could be maintained, and cold conditions elicited nest building despite the challenge. ^[ref] This supports a reorganization of competing motivations rather than simple global paralysis. It does not show that all illness-related inactivity is strategic. Controls should match handling and injection, assess motor capacity and temperature, and distinguish loss of appetite from inability to acquire food. Dose and timing also matter: an artificial inflammatory challenge need not reproduce a natural infection's trajectory. Blocking a candidate immune mediator, followed by a rescue intervention, would provide stronger evidence for a specific immune-to-neural pathway than a challenge alone.

Immune influences are not confined to overt sickness. In house mice, work on interferon-gamma linked immune deficiency and altered social preference to changes in neural connectivity and inhibitory physiology. Immune restoration and targeted signalling interventions strengthened a mechanistic interpretation beyond a correlation between illness and social withdrawal. ^[ref] The experiments support a role for immune signalling in the tested neural and behavioural context; they do not establish a universal explanation of sociability or justify extrapolating a house mouse social-preference assay directly to human psychiatric conditions. Sensory ability, general activity, anxiety-related responses, housing and genetic background remain important controls when interpreting such assays.

The two directions can form the same feedback loop. An immune response changes social contact; changed contact alters subsequent exposure and infection; the new infection state changes the immune response again. The medication example similarly links infection-dependent action to offspring defence. Neither direction is an independent input fixed while the other acts. A useful factorial design would manipulate a specific action or opportunity for action and a specific immune challenge, then measure both behavioural and physiological outcomes over time. Baseline health, age, sex and reproductive state should be controlled, and the manipulation of action should avoid introducing injury or stress that independently changes immunity.

A claim that one direction is larger would require comparable interventions and a shared endpoint, such as lifetime reproductive success or pathogen transmission, with uncertainty over an explicitly chosen time horizon. Different answers may follow for individual survival, colony persistence and epidemic spread. The evidence supports bidirectional, context-dependent regulation: actions govern exposure and defence, while immune signals govern behavioural priorities and sometimes normal neural function. A universal ranking conceals precisely the causal distinctions needed to understand this system.

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.

Historical category used in explanations of bodily disturbance. In the Celsus cupping passage, wind and corrupted material should not be silently translated into a proven modern infection or a measured gas compartment. ^[ref]

A historical claim that typhoid fever could originate from decomposing organic material, as opposed to transmission from an existing source of the specific infection.

The removed compartment consists of people permanently excluded from both acquiring and passing on infection in the SIR model. Its count is $R(t)$ and its fraction is $r(t)=R(t)/N$. Removal can represent regained health with lasting immunity, or another permanent exclusion from acquiring or passing on infection; it need not mean being healthy again. Here there are no deaths. In a variant that counts deaths within $R$, conservation of population would concern the original cohort, rather than the number currently alive.^[ref]

For the closed SIR model with $\gamma >0$, the infectious compartment's fraction approaches $i_{\infty }=0$. One proof outline uses the bounded, increasing removed compartment's fraction: since $r(t)-r_0=\gamma {\int }_0^{t}i(u)du$, the total time-integrated $i$ is finite. The bounded derivative $di/dt$ from the SIR model equations prevents indefinitely recurring spikes, so $i(t)$ tends to zero.

For $s_0>0$, the final size relation for an epidemic follows by dividing the SIR model equations for $s$ and $r$ while $i>0$, giving $ds/dr=-(\beta /\gamma )s$. Integration then yieldsThis also follows from the SIR first integral and conservation of population. For the stipulated initial value problem, solveThe bisection method givesAbout $5.88\%$ of the original population remains in the susceptible compartment. The $94.12\%$ is the final removed compartment's fraction, including the initially $1\%$ in the infectious compartment. The newly acquired infection during the modelled period accounts for $s_0-s_{\infty }\approx 0.93120263520$ of the total population, not $1-s_{\infty }$. Dividing by $s_0$ instead gives the proportion of initially susceptible compartment members who acquire infection; its denominator must be stated.

At the SIR infection peak, the flow into the infectious compartment only balances its outflow; it does not stop. Further infection explains why the final susceptible compartment's fraction is well below $1/3$. Also $s_{\infty }=s_0{e}^{-R_0(r_{\infty }-r_0)}>0$ for $s_0>0$ and finite $R_0$: the closed SIR model does not require everybody to acquire infection before its infectious compartment empties in the long-time limit.

The SIR model is a mathematical model of an epidemic that follows three groups: a susceptible compartment whose members can acquire infection, an infectious compartment whose members can pass it on, and a removed compartment whose members neither acquire nor pass on infection. The letters stand for susceptible, infectious and recovered or removed. Here we study the simplest closed, constant-parameter version, with homogeneous mixing and no births, deaths, migration or waning immunity. A person who leaves the infectious compartment never returns to the susceptible compartment in this version.^[ref]

The appeal of the SIR model is that one can explain a rise and fall in the infectious compartment without changing anyone's behavior: fewer opportunities for new infection eventually reverse the growth. The SIR first integral even determines the SIR infection peak without computing its time. These are useful deductions from explicit assumptions. The stipulated numbers below illustrate the mathematical model; they are neither estimates for an actual disease nor a forecast or medical advice.

Kermack and McKendrick's 1927 paper, "A contribution to the mathematical theory of epidemics", in Proceedings of the Royal Society A, is an early foundation for the SIR model. Its DOI is 10.1098/rspa.1927.0118, also recorded in the Royal Society's archive of contemporary referee reports.^[ref][ref]

Collective protection against pathogens arising through interactions among individuals, including hygienic behaviour. A colony-level reduction in infection need not be increased cellular immunity in every worker.

The susceptible compartment consists of people who can acquire infection. In the closed SIR model, $S(t)$ is their count and $s(t)=S(t)/N$ their fraction of the fixed total $N$. Membership expresses the assumptions of the mathematical model, rather than measuring the strength of an individual's immunity.