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Frank Fenner was a researcher of viruses whose work also helped establish that smallpox had been eradicated worldwide. He chaired the international commission responsible for certification and presented its conclusion to the World Health Assembly in May 1980. In his firsthand interview, he describes certification as an evidential problem distinct from conducting the campaign itself.^[ref]

The distinction is scientifically useful: recording no recent cases in one place is different from establishing worldwide interruption of transmission. Certification requires examining the coverage and credibility of the observations, not merely counting successful interventions. Fenner also helped assemble a detailed historical account of the campaign, linking epidemiology, vaccination and evidence from different countries. His contribution illustrates how public-health science depends both on changing outcomes and on documenting why a conclusion is justified.^[ref]

A fatal-infection model with vaccination proportional to infective-susceptible encounters adds $v_t=\eta qs$ and $m_t=q$. The total count including deaths is conserved. Integrating $dq/ds$ and $dv/ds$ gives $q=q_0+[s_0-s+\log (s/s_0)]/(1+\eta )$ and $v=\eta (s_0-s)/(1+\eta )$. The selected final root has $0<s_{\infty }<\min (s_0,1)$, and deaths are $q_0+(s_0-s_{\infty })/(1+\eta )$. The model is a specified deterministic reaction scheme, not a general vaccination policy rule.

These are row percentages, so the comparisons estimate the conditional probability of recorded vaccination within each NHS trust category. The percentages increase across the ordered trust categories asThe extreme-category difference is approximately 49.31 percentage points, rather than a 49.31% relative increase. The ordinal categorical variable records an order; the categories do not establish equal numerical trust increments.

Under the null hypothesis of independence, the expected count in a cell is its row total times its column total divided by 748. The Pearson chi-squared statistic compares observed and expected counts throughThere are $(5-1)(2-1)=4$ statistical degrees of freedom, giving $p\approx 6.49\times 10^{-21}$. The minimum expected count is approximately 5.015, so the usual expected-count concern is much less acute here than in the income strata. The table gives strong evidence of a statistical association. The likelihood-ratio test of independence in a contingency table agrees: its statistic is approximately 81.361 on four statistical degrees of freedom, with a very small p-value. The displayed 0.000 is rounding, not a zero probability.

For magnitude, Cramér's V isIt describes the size of the departure from independence without using the category order or giving a direction. Calling it “moderate” is only a rough contextual description; the actual percentage differences are easier to interpret.

The ordered coefficients describe a positive direction. Goodman and Kruskal's gamma uses only concordant pairs and discordant pairs:It excludes tied pairs, which are frequent when the response has only two categories. It is therefore not a claim that 54.7% of respondents were vaccinated because of their trust. The directional Somers' D for vaccination as response includes pairs tied on that response in its denominator:The smaller value reflects the retained ties; it does not contradict the positive statistical association.

The small column-dependent Goodman and Kruskal's lambda has a useful prediction interpretation. Without trust information, always predicting the more frequent vaccinated category gives 121 mistakes. Predicting the majority category separately in each trust row gives $13+15+26+34+28=116$ mistakes. HenceTrust reduces this particular modal-prediction error by only about 4.13%, despite the strong evidence against independence. Most rows retain the same majority category. Evidence of association, its ordinal direction and improvement in a chosen prediction rule are different questions. None establishes that changing trust would change an individual's vaccination status.

Infantile syphilis, varicella, modified smallpox and ordinary smallpox are distinguished by the whole illness: age and history, prodrome, distribution, lesion depth, successive crops and the speed of evolution. No isolated “spot” supplies an infallible diagnosis. Varioloid is the historical name for modified smallpox, particularly after vaccination; it is not a synonym for varicella.

In an infant, the relevant syphilis is chiefly congenital syphilis, transmitted during pregnancy. It can be unapparent at birth and declare itself later with persistent nasal discharge or “snuffles,” poor growth, fissuring near the mouth, a copper-coloured maculopapular or peeling eruption, and lesions on palms and soles. Bullae or pustular appearances can occur. Associated hepatosplenomegaly, jaundice and painful bone disease support a systemic process rather than a simple crop of transient vesicles. The eruption is not required to pass synchronously through the stereotyped smallpox sequence. Maternal infection and appropriate serology matter; absence of a rash at birth does not exclude infection.^[ref][ref]

Varicella, or chickenpox, typically produces delicate superficial itchy vesicles on a red base, especially on the anatomical trunk, with successive crops. Consequently fresh papules, vesicles and crusts coexist in one area. In children the preceding illness is often mild or absent. Individual lesions evolve quickly, and ordinary superficial lesions usually heal without the deep pitting associated with destructive smallpox lesions, although secondary infection can scar. The distribution is generally centripetal rather than most dense on the distal limbs; palms and soles are much less commonly affected.^[ref]

Ordinary variola, or smallpox, classically begins with a marked febrile prodrome, often including headache, backache and prostration, before the rash. Lesions favour the face and extremities, can affect palms and soles, and proceed through firm papules to deep-seated vesicles and pustules, often with central depression. Within a given region they are relatively alike in developmental stage. The lesions evolve over days rather than the rapid successive crops of varicella; crusting and separation may leave depressed scars. Severe confluent, flat or haemorrhagic forms depart from the uncomplicated textbook pattern, so the absence of perfect pustules is not a universal exclusion.^[ref][ref]

Varioloid retains the cause of smallpox but has a modified course. Prior vaccination can yield fewer and more superficial lesions, faster evolution and less secondary fever. The prodrome can still be substantial. Sparse lesions and shortened evolution can resemble chickenpox, so vaccination history and the entire time course help, but neither a vaccination scar nor mild disease proves the diagnosis. Varioloid and variola belong on a spectrum of the same infection, whereas varicella is caused by a different virus.^[ref]

Thus the characteristic contrast is persistent multisystem infantile infection for congenital syphilis, superficial lesions in mixed-stage crops for varicella, deeper regionally synchronous lesions with a severe preceding fever for ordinary smallpox, and attenuation of the latter pattern for varioloid. Current confirmation uses tests appropriate to the suspected organism; the historical clinical distinctions remain valuable descriptions, but they do not replace modern laboratory confirmation.

Use $t=aT$ and scale each count by the susceptible infection threshold $a/r$:For example $I_T=(a^2/r)q_t$, while $rIS=(a^2/r)qs$. Dividing all equations by $a^2/r$ therefore givesThus $\eta$ measures the vaccination response relative to transmission in this mass-action model. The total dimensionless count $q+s+v+m$ is conserved, and the instantaneous infection growth threshold is $s=1$.

The law of mass action assigns infection rate $rIS$, death rate $aI$, and vaccination rate $pIS$. Vaccination consumes a susceptible animal and produces a vaccinated one while leaving the triggering infected animal present. In dimensional time $T$,All compartment counts remain nonnegative. Adding the equations gives $(I+S+V+R{)}_T=0$; deaths are retained in the bookkeeping compartment rather than removed from that conserved count. There is no recovery class of live immune survivors in the infection reaction: in this particular model every animal that becomes infected ultimately dies.

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 $s_0-s_{\infty }$.

For the vaccinated epidemic-wave final size, write $s_{-}=s(-\infty )$. From the travelling-wave equations, $dv/ds=-\eta /(1+\eta )$ and $dm/ds=-1/[(1+\eta )s]$. Using the ahead-of-wave boundary data givesAdding the four wave equations yields $q^{\prime \prime }+c(q+s+v+m{)}^{\prime }=0$. Integrate once and use $q^{\prime }(\infty )=0$ to obtain $q^{\prime }+c(q+s+v+m)=c{s}^{\ast }$. At minus infinity the assumptions $q=q^{\prime }=0$ therefore give $s_{-}+v_{-}+m_{-}=s^{\ast }$.

Combining these identities shows that $s_{-}$ satisfiesThe nontrivial positive epidemic-wave root has $0<s_{-}<1<s^{\ast }$; the other root $s_{-}=s^{\ast }$ is the disease-free solution. The assumed positive wave consequently requires $s^{\ast }>1$; for $s^{\ast }\le 1$ the displayed relation has no distinct smaller root.

At the back of the pulse there are no infectives, so the live biological population consists of susceptible and vaccinated animals:The final-size root is independent of vaccination strength in this travelling-wave problem; vaccination instead changes the split between deaths and survivors for the total biological population lost from susceptibility.

A record of which vaccinations a person has received at a specified time. A binary categorical variable may collapse this into two recorded categories; the categories alone do not specify number of doses, timing, protective immunity or willingness to receive a vaccine.

Historical name for modified smallpox, especially following vaccination; distinct from chickenpox.