article in connection with ronald ross's work

8/13/2019 Article in connection with Ronald Ross's work

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Indian Science Cruiser, Vol. 27, No. 5, Special issue on Sir Ronald Ross.

Mathematical modeling of infectious disease

N.C. Ghosh*

A B S T R A C T

Human suffers from infectious disease since prehistoric time. Some times epidemic infectious

disease causes mass death toll. So attempts were been taen to sa!e human ind from suchinfectious diseases. "ith the ad!ent of science branches of it are ha!e been associate with this

endea!our. In recent #earsmathematical modelin$ has become a !aluable tool in the anal#sis of

infectious disease d#namics and to support the de!elopment of control strate$ies.%athematical

modelscan pro&ect howinfectious diseasespro$ress to show the liel# outcome ofan epidemicand help informpublic healthinter!entions. %odels use some basic assumptions

and mathematics to findparametersfor !arious infectious diseasesand use those parameters to

calculate the effects of possible inter!entions, lie mass !accinationpro$rammes. Here author

attempt to discuss basic problems from mathematical !iew.

Introduction :

It is not known when infectious diseases first started to attack human race; but like otheranimals man kind suffers from it from earl !eriod. There are some natural measures to !rotectanimal" but men isolated from animal race are #ictim infectious diseases; some times in wideran$. Attem!ts were been taken to sa#e mankind from infectious diseases and is continuin$ inmodern era. %irstl mathematical conce!ts were been used to #isualise the !roblem in scientificmanner andbut from last centur mathematicalideas are been used for man other analticalas!ects of it. &athematical modelin$now are im!ortant tools to stud these !roblems.In recentears understandin$ of infectious'disease e!idemiolo$ and control has been $reatl increased

throu$h mathematical modelin$. Insi$hts from this increasin$l'im!ortant" e(citin$ field are nowinformin$ !olic'makin$ at the hi$hest le#els" and !lain$ a $rowin$ role in research. Thetransmissible nature of infectious diseases makes them fundamentall different from non'infectious diseases" so techni)ues from classical e!idemiolo$ are often in#alid and hence leadto incorrect conclusions ' not least in health'economic analsis. So mathematical analsis are notonl im!ortant to measure $ra#it of infectious diseases" its causal as!ects" !hsiolo$ical as wellclinical as!ects are also been taken care of. &athematical modelin$ now !las a ke role in!olic makin$" includin$ health'economic as!ects; emer$enc !lannin$ and risk assessment;control'!ro$ramme e#aluation; and monitorin$ of sur#eillance data. In research" it is essential instud desi$n" analsis +includin$ !arameter estimation, and inter!retation.

-arl !ioneers in infectious disease modelin$ were illiam /amer and Ronald Ross"

who in the earl twentieth centur a!!lied the law of mass actionto e(!lain e!idemicbeha#iour. 0owell Reedand ade /am!ton %rostde#elo!ed the Reed1%rost e!idemic modeltodescribe the relationshi! between susce!tible" infected andimmuneindi#iduals in a!o!ulation.2#er the last two decades" mathematical models ha#e seen a hu$e de#elo!ment in allas!ects of infectious diseases" from microbiolo$ to e!idemiolo$ and e#olution. In recent earsunderstandin$ of infectious'disease e!idemiolo$ and control has been $reatl increased throu$h

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http://en.wikipedia.org/wiki/Mathematical_modelshttp://en.wikipedia.org/wiki/Mathematical_modelshttp://en.wikipedia.org/wiki/Infectious_diseaseshttp://en.wikipedia.org/wiki/Infectious_diseaseshttp://en.wikipedia.org/wiki/Epidemichttp://en.wikipedia.org/wiki/Epidemichttp://en.wikipedia.org/wiki/Public_healthhttp://en.wikipedia.org/wiki/Public_healthhttp://en.wikipedia.org/wiki/Parameterhttp://en.wikipedia.org/wiki/Infectious_diseasehttp://en.wikipedia.org/wiki/Vaccinationhttp://en.wikipedia.org/wiki/Vaccinationhttp://en.wikipedia.org/wiki/Ronald_Rosshttp://en.wikipedia.org/wiki/Law_of_mass_actionhttp://en.wikipedia.org/wiki/Lowell_Reedhttp://en.wikipedia.org/wiki/Wade_Hampton_Frosthttp://en.wikipedia.org/wiki/Reed%E2%80%93Frost_modelhttp://en.wikipedia.org/wiki/Susceptiblehttp://en.wikipedia.org/wiki/Immunity_(medical)http://en.wikipedia.org/wiki/Immunity_(medical)http://en.wikipedia.org/wiki/Mathematical_modelshttp://en.wikipedia.org/wiki/Mathematical_modelshttp://en.wikipedia.org/wiki/Infectious_diseaseshttp://en.wikipedia.org/wiki/Epidemichttp://en.wikipedia.org/wiki/Public_healthhttp://en.wikipedia.org/wiki/Parameterhttp://en.wikipedia.org/wiki/Infectious_diseasehttp://en.wikipedia.org/wiki/Vaccinationhttp://en.wikipedia.org/wiki/Ronald_Rosshttp://en.wikipedia.org/wiki/Law_of_mass_actionhttp://en.wikipedia.org/wiki/Lowell_Reedhttp://en.wikipedia.org/wiki/Wade_Hampton_Frosthttp://en.wikipedia.org/wiki/Reed%E2%80%93Frost_modelhttp://en.wikipedia.org/wiki/Susceptiblehttp://en.wikipedia.org/wiki/Immunity_(medical)


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Dr. Narayan ChGhosh, Professor in Mathematics, having thirty seven years teaching and

research experience in India and abroad. Email $hosnara#an'$mail.com, Mobile !"

!#$""%%$&', !" !($$#)%)*%, !" !#%(&&$!(!, !" %$$ &*#%$'*$, !" %$$ &*#%&$*'

mathematicalmodelin$. &artini" and 0otka has done a $ood 3ob on such modelin$. Startin$ in4567 8ermack and &c8endrick !ublished !a!ers on e!idemic models and obtained the e!idemic

threshold result that the densit of susce!tible must e(ceed a critical #alue in order for ane!idemic outbreak to occur. &athematical e!idemiolo$ seems to ha#e $rown e(!onentiallstartin$ in the middle of the 69th centur +the :rst edition in 45< of Baile=s book is anim!ortant landmark," so that a tremendous #ariet of models ha#e now been formulated"mathematicall anal>ed" and a!!lied to infectious diseases. Re#iews of the literature show thera!id $rowth of e!idemiolo$ modelin$.

As the transmissible nature of infectious diseases makes them fundamentall differentfrom non'infectious diseases and conse)uence of which techni)ues from classical e!idemiolo$are often in#alid as it leads to incorrect conclusions for health'economic analsis and e#en tocombat effect of diseases !rofessionals in these fields are now e(!osed to a wide ran$e ofmodels.

The recent models ha#e in#ol#ed as!ects such as !assi#e immunit" $radual loss of#accine and disease'ac)uired immunit" sta$es of infection" #ertical transmission" disease#ectors" macro!arasitic loads" a$e structure" social and se(ual mi(in$ $rou!s" s!atial s!read"#accination" )uarantine" and chemothera!.Mathematical Formulation :

ith infectious diseases fre)uentl dominatin$ news headlines" !ublic health and!harmaceutical industr !rofessionals" !olic makers" and infectious disease researchers"increasin$l need to understand the transmission !atterns of infectious diseases" to be able tointer!ret and criticall'e#aluate both e!idemiolo$ical data" and the findin$s of mathematicalmodelin$ studies. Recentl there has been ra!id !ro$ress in de#elo!in$ models and newtechni)ues for measurement and analsis" which ha#e been a!!lied to outbreaks and emer$in$e!idemics.

%or mathematical models it is essential to inte$rate the increasin$ #olume ofdatabein$$enerated on host'!atho$eninteractions. &an theoretical studies of the!o!ulation dnamics"structure and e#olution ofinfectious diseasesof!lantsand animals" includin$ humans" areconcerned with this !roblem. In such in#esti$ation Research to!ics include? a. Transmission"s!read and control of infection" b. -!idemiolo$icalnetworks" c. S!atial e!idemiolo$" d.@ersistence of !atho$ens within hosts" e. Intra'host dnamics" f. Immuno'e!idemiolo$" $.irulence"h. Strain +biolo$,structure and interactions" i. Anti$enic shift" 3. @hlodnamics" k.@atho$en!o!ulation $enetics" l.-#olutionand s!read ofresistance" m. Role of host $eneticfactors" n. Statistical and mathematical tools and inno#ations" o. Role and identificationof infection reser#oirs.

Basic ideas for mathematical formulation de!ends on the conce!ts of i. @o!ulationdnamics" ii. eterministic and stochastic models" iii. Network analsis" i#. ithin'hostdnamics of #iral and bacterial infections" #. &athematical re#iew +calculus" !robabilities...," #i.A!!lied !ro$rammin$ with R" #ii. Statistical modelin$" #iii. Com!uter'based simulations

&odels are onl as $ood as the assum!tions on which the are based. If a modelmakes !redictions which are out of line with obser#ed results and the mathematics iscorrect" the initial assum!tions must chan$e to make the model useful. Rectan$ular a$edistribution is often well'3ustified for de#elo!ed countries where there is a low infant

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mailto:[email protected]://en.wikipedia.org/wiki/Numerical_datahttp://en.wikipedia.org/wiki/Numerical_datahttp://en.wikipedia.org/wiki/Host_(biology)http://en.wikipedia.org/wiki/Pathogenhttp://en.wikipedia.org/wiki/Population_dynamicshttp://en.wikipedia.org/wiki/Population_dynamicshttp://en.wikipedia.org/wiki/Infectious_diseasehttp://en.wikipedia.org/wiki/Infectious_diseasehttp://en.wikipedia.org/wiki/Plantshttp://en.wikipedia.org/wiki/Plantshttp://en.wikipedia.org/wiki/Transmission_(medicine)http://en.wikipedia.org/wiki/Epidemiologicalhttp://en.wikipedia.org/wiki/Spatial_epidemiologyhttp://en.wikipedia.org/wiki/Immune_systemhttp://en.wikipedia.org/wiki/Virulencehttp://en.wikipedia.org/wiki/Virulencehttp://en.wikipedia.org/wiki/Strain_(biology)http://en.wikipedia.org/wiki/Antigenic_shifthttp://en.wikipedia.org/wiki/Phylodynamicshttp://en.wikipedia.org/wiki/Population_geneticshttp://en.wikipedia.org/wiki/Evolutionhttp://en.wikipedia.org/wiki/Evolutionhttp://en.wikipedia.org/wiki/Drug_resistancehttp://en.wikipedia.org/wiki/Drug_resistancehttp://en.wikipedia.org/wiki/Infection_reservoirhttp://en.wikipedia.org/wiki/Population_pyramidhttp://en.wikipedia.org/wiki/Population_pyramidmailto:[email protected]://en.wikipedia.org/wiki/Numerical_datahttp://en.wikipedia.org/wiki/Host_(biology)http://en.wikipedia.org/wiki/Pathogenhttp://en.wikipedia.org/wiki/Population_dynamicshttp://en.wikipedia.org/wiki/Infectious_diseasehttp://en.wikipedia.org/wiki/Plantshttp://en.wikipedia.org/wiki/Transmission_(medicine)http://en.wikipedia.org/wiki/Epidemiologicalhttp://en.wikipedia.org/wiki/Spatial_epidemiologyhttp://en.wikipedia.org/wiki/Immune_systemhttp://en.wikipedia.org/wiki/Virulencehttp://en.wikipedia.org/wiki/Strain_(biology)http://en.wikipedia.org/wiki/Antigenic_shifthttp://en.wikipedia.org/wiki/Phylodynamicshttp://en.wikipedia.org/wiki/Population_geneticshttp://en.wikipedia.org/wiki/Evolutionhttp://en.wikipedia.org/wiki/Drug_resistancehttp://en.wikipedia.org/wiki/Infection_reservoirhttp://en.wikipedia.org/wiki/Population_pyramidhttp://en.wikipedia.org/wiki/Population_pyramid


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number" the lower the !ro!ortion of the !o!ulation susce!tible must be" and #ice #ersa.%orabo#e cited first assum!tion +abo#e, lets us sa that e#erone in the !o!ulation li#es toa$e*and then dies. If the a#era$e a$e of infection is+" then on a#era$eindi#iduals oun$er then+are susce!tible and those older than+are immune +or infectious,.Thus the !ro!ortion of the !o!ulation that is susce!tible is $i#en b

But the mathematical definition of the endemic stead state can be rearran$ed to $i#e?

Therefore"

This !ro#ides a sim!le wa to estimate the !arameterR9usin$ easil a#ailable data.%or a !o!ulation with an e(!onential a$e distribution"

This allows for the basic re!roduction number of a disease $i#en+and*in either t!e of!o!ulation distribution.Mathematics of mass vaccination

If the !ro!ortion of the !o!ulation that is immune e(ceeds the herd immunit le#el forthe disease" then the disease can no lon$er !ersist in the !o!ulation. The infectious disease can beeliminated if herd immunit le#el can be e(ceeded b #accination. An e(am!le of this bein$successfull achie#ed worldwide is the $lobal eradication of small!o(" with the last wild case in45


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This chan$e occurs sim!l because there are now fewer susce!tible in the !o!ulationwho can be infected.Ris sim!lR)minus those that would normall be infected but thatcannot be now since the are immune. As a conse)uence of this lowerbasic re!roductionnumber" the a#era$e a$e of infection + will also chan$e to some new #alue +)inthose who

ha#e been left un#accinated./ere one can recall the relation that linked R9"+and*. Assumin$ that life e(!ectanchas not chan$ed" now?

ButR9F*E+so?

The #accination !ro$ramme will raise the a#era$e a$e of infection" another mathematical

3ustification for a result that mi$ht ha#e been intuiti#el ob#ious. Pn#accinated indi#iduals nowe(!erience a reduced force of infectiondue to the !resence of the #accinated $rou!.When mass vaccination exceeds the herd immunity

If a #accination !ro$ramme causes the !ro!ortion of immune indi#iduals in a !o!ulationto e(ceed the critical threshold for a si$nificant len$th of time" transmission of the infectiousdisease in that !o!ulation will sto!. This is known as elimination of the infection and is differentfromeradication.Elimination

Interru!tion of endemic transmission of an infectious disease" which occurs if eachinfected indi#idual infects less than one other" is achie#ed b maintainin$ #accination co#era$eto kee! the !ro!ortion of immune indi#iduals abo#e the critical immunisation threshold.

EradicationReduction of infecti#e or$anisms in the wild worldwide to >ero. So far" this has onl been

achie#ed for small!o(and rinder!est. To $et to eradication" elimination in all world re$ions mustbe achie#ed.Essential to be careful for Studying epidemic infectious disease :-!idemiolo$ists deal withman uncontrollable" inde!endent #ariables usin$ ad#anced statistical techni)ues and carefulldesi$ned e(!eriments. %orStudin$ such esi$nsone need be careful on followin$ toolsA. ariables

4. e!endent #ariable 1 the outcome #ariable' -(am!les +in e!idemiolo$,? Risk of de#elo!in$ a disease" se#erit of disease"inde( associated with disease such as e3ection fraction +fraction of blood enterin$

the heart that is e3ected out in one beat,6. Inde!endent #ariables 1 the #ariables ou mani!ulate" measure" andEor record' -(am!les? A$e" $ender" Q bod fat" amount of !hsical acti#it

. -(traneous #ariables 1 #ariables not measured or controlled" assumed to ha#e norelation to the outcome #ariable or to be accounted for b randomi>ation

' -(am!le? -e color" number of siblin$sB. Correlation #ersus Causalit

4. A correlation is a $reater than nothin$ relationshi! between two #ariables

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6. Correlation does N2T necessitate causalit' Cows lie down before it rains' In the 69thCentur" the murder rate rose and the shee! farmin$ rate fell' Smokin$ causes lun$ cancer" or is it factor K

. Correlation is measured usin$ r or R6

' r ran$es from 14 to 4' R6ran$es from 9 to 4C. -(!erimental esi$n

4. -(!erimental studies in#ol#e the mani!ulation of one or more inde!endent #ariables' -(am!le? A clinical trial for a treatment 1 dru$" er$o$enic aid" or e(ercise e.$.' -(am!le? ietar mani!ulation such as low fat or low C/2" hot or cold

en#ironment" Q o($en in air6. &ore than two cate$ories of an inde!endent #ariable are called le#els ' e.$." hi$hintensit" moderate intensit" low intensit" or no e(ercise. -(!erimental studies are intended to establish mechanisms ' causalit

. -(!erimental and Control Grou!s

4. -(!erimental studies randoml assi$n sub3ects to either an e(!erimental or a control$rou!6. The e(!erimental $rou!+s, recei#e+s, the inter#ention 1 the I of interest 1 a s!ecialdiet" a dru$" or an e(ercise !ro$ram. The control $rou! is as identical as !ossible to the e(!erimental $rou! in e#er facet-KC-@T the inde!endent #ariable of interest 1 no s!ecial diet" no dru$" no e(ercise!ro$ramU. @otential confounders +#ariables that could affect the , are ho!efull balancedbetween the two +or more, $rou!s

-. Bias control4. -(!erimental studies are susce!tible to both sub3ect and e(!erimental bias

' A sub3ect who thinks that creatine +e.$., will make him +or her, stron$er maunconsciousl tr harder durin$ testin$ after su!!lementation than before

' @lacebo effect' An e(!erimenter who e(!ects a dru$ to im!ro#e sm!toms ma unconsciousl

rate sm!toms as more se#ere without the dru$ than with the dru$6. To control for these effects" e(!eriments can bea. Blind 1 the sub3ect does not know which $rou! he or she is in 1 the e(!erimental or

control $rou!b. ouble'blind 1 neither the sub3ect nor the e(!erimenter knows which $rou! the

sub3ect is in%. 2bser#ational Stud esi$ns4. 2bser#ational studies do not in#ol#e direct mani!ulation of a #ariable" althou$h sub3ects mabe $rou!ed accordin$ to the le#el of a #ariable" e.$. has disease or does not" a$e $rou!"hi$hEmediumElow !hsical acti#it le#el,

V Cross'sectionalV Retros!ecti#e +case'control,V @ros!ecti#e +lon$itudinal,

6. Cross'Sectional' Indi#iduals are selected at random from a !o!ulation and both the I=s and the

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Case control studies select sub3ects based on their disease status. A $rou! of indi#idualsthat are disease !ositi#e +the XcaseX $rou!, is com!ared with a $rou! of disease ne$ati#eindi#iduals +the XcontrolX $rou!,. The control $rou! should ideall come from the same

!o!ulation that $a#e rise to the cases. The case control stud looks back throu$h time at !otentiale(!osures that both $rou!s +cases and controls, ma ha#e encountered. A table of 6Y6 !attern isbeen constructed" dis!lain$ e(!osed cases +A," e(!osed controls +B," une(!osed cases +C, andune(!osed controls +,. The statistic $enerated to measure association is the odds ratio+2R,"which is the ratio of the odds of e(!osure in the cases +AEC, to the odds of e(!osure in thecontrols +BE," i.e. 2R F +AEBC,

If the 2R is clearl $reater than 4" then the conclusion is Xthose with the disease are morelikel to ha#e been e(!osed"X whereas if it is close to 4 then the e(!osure and disease are notlikel associated. But if the 2R is far less than one" then this su$$ests that the e(!osure is a!rotecti#e factor in the causation of the disease. Case control studies are usuall faster and morecost effecti#e than cohort studies" but are sensiti#e to bias +such asrecall biasand selection bias,.The main challen$e is to identif the a!!ro!riate control $rou!; the distribution of e(!osureamon$ the control $rou! should be re!resentati#e of the distribution in the !o!ulation that $a#erise to the cases. This can be achie#ed b drawin$ a random sam!le from the ori$inal !o!ulationat risk. This has as a conse)uence that the control $rou! can contain !eo!le with the diseaseunder stud when the disease has a hi$h attack rate in a !o!ulation.

A ma3or drawback for such case control studies is that" in order to be considered to bestatisticall si$nificant" the minimum number of cases re)uired at the 5Q confidence inter#al isrelated to the odds ratio b the e)uation?Total cases F +aMc,F +4.57,Z6Y+4MN,Y+4[ln+2R,,Z6Y++2RM6\2RM4,[\2R,]4.Y+4MN,Y+4[ln+2R,,Z6where N F The ratio of cases to controls. As the odds ratio a!!roached 4" a!!roaches 9;renderin$ case control studies all but useless for low odds ratios. %or instance" for an odds ratioof 4. and cases F controls" the table shown abo#e would look like this?

!!!!!as

esontrols

-(!osed 49 HU

8

!!!!!as

esontrols

-(!osed A B

Pne(!osed C
http://en.wikipedia.org/wiki/Case-control_studyhttp://en.wikipedia.org/wiki/Odds_ratiohttp://en.wikipedia.org/wiki/Cohort_studieshttp://en.wikipedia.org/wiki/Recall_biashttp://en.wikipedia.org/wiki/Selection_biashttp://en.wikipedia.org/wiki/Case-control_studyhttp://en.wikipedia.org/wiki/Odds_ratiohttp://en.wikipedia.org/wiki/Cohort_studieshttp://en.wikipedia.org/wiki/Recall_biashttp://en.wikipedia.org/wiki/Selection_bias


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Pne(!osed HU 49%or an odds ratio of 4.4?

!!!!! ases ontrols

-(!osed 4


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codin$. Random error affects measurement in a transient" inconsistent manner and it isim!ossible to correct for random error.Sampling error

There is random error in all sam!lin$ !rocedures which is calledsamplingerror.@recision in e!idemiolo$ical #ariables is a measure of random error. @recision is also

in#ersel related to random error" so that to reduce random error is to increase !recision.Confidence inter#als are com!uted to demonstrate the !recision of relati#e risk estimates. Thenarrower the confidence inter#al" the more !recise the relati#e risk estimate.

To reduce random error in anepidemiological studythere are two basic was. The first isto increase the sam!le si>e of the stud. In other words" add more sub3ects to our stud. Thesecond is to reduce the #ariabilit in measurement in the stud. This mi$ht be accom!lished busin$ a more !recise measurin$ de#ice or b increasin$ the number of measurements.

Systematic error

A sstematic error or bias occurs when there is a difference between the true #alue +in the!o!ulation, and the obser#ed #alue +in the stud, from an cause other than sam!lin$ #ariabilit.

An e(am!le of sstematic error is if" unknown to in#esti$ator" thepulse oximeterhe is usin$ isset incorrectl and adds two !oints to the true #alue each time a measurement is taken. Themeasurin$ de#ice could beprecise but not accurate. Because the error ha!!ens in e#erinstance" it is sstematic. Then conclusions drawn based on that data will still be incorrect. Butthe error can be re!roduced in the future +e.$." b usin$ the same mis'set instrument,.

A mistake in codin$ that affectsallres!onses for that !articular )uestion is anothere(am!le of a sstematic error.

The #alidit of a stud is de!endent on the de$ree of sstematic error. alidit is usuallse!arated into two com!onents? Internal validity is de!endent on the amount of error in measurements" includin$ e(!osure"

disease" and the associations between these #ariables. Good internal #alidit im!lies a lack of error in measurement and su$$ests that inferences

ma be drawn at least as the !ertain to the sub3ects under stud. External validity!ertains to the !rocess of $enerali>in$ the findin$s of the stud to the

!o!ulation from which the sam!le was drawn +or e#en beond that !o!ulation to a moreuni#ersal statement,. This re)uires an understandin$ of which conditions are rele#ant +orirrele#ant, to the $enerali>ation. Internal #alidit is clearl a !rere)uisite for e(ternal #alidit.(t the end of discussion it is necessary to note that biasness can threaten the validity of a

study! So no discussion on &athematical modelin$ of infectious disease is com!lete with outsain$ an thin$ on Bias.$hree are three types of bias:i! Selection bias) ii! Information bias) and iii! onfounding

Selection bias :Selection bias occurs when stud sub3ects are selected or become !art of the

stud as a result of a third" unmeasured #ariable which is associated with both the e(!osureand outcome of interest.

Information bias :It is bias arisin$ from sstematic error in the assessment of a #ariable.Ane(am!le of this is recall bias.

onfounding :onfoundinghas traditionall been defined as bias arisin$ from the co'occurrence or mi(in$ of effects of e(traneous factors" referred to as confounders" with themain effect+s, of interest.A more recent definition of confoundin$ in#okes the notionofcounterfactualeffects.Accordin$ to this #iew" when one obser#es an outcome of

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field a priori!athometr" or constructi#e e!idemiolo$ " but it is now more widel knownas mathematical e!idemiolo$.

Se#eral )uantities are commonl defined as !art of the Ross'&acdonald model; the!o!ulation densit of humans"H; the !o!ulation densit of mos)uitoes"%; the number ofinfected humans"0; the number of infected" but not et infectious mos)uitoes" ; the number of

infectious mos)uitoes"6; the human blood feedin$ rate" the !ro!ortion of mos)uitoes that feedon humans each da" a; mos)uito sur#i#al as either the !robabilit of sur#i#in$ one da"p" or theinstantaneous death rate"$+p e8$or$F Llnp,; the !atho$ens #ertebrate latent !eriod" oftencalled the intrinsic incubation !eriod" the number of das from infection to infectiousness inthe human" u; the !atho$ens mos)uito latent !eriod" often called the e(trinsic incubation!eriod" the number of das from infection to infectiousness in the mos)uito" !; the dail rateeach human reco#ers from infection" r; the !ro!ortion of infected humans that are infectious" oralternati#el" the !robabilit a mos)uito becomes infected after bitin$ an infected human" c; andthe !ro!ortion of bites b infectious mos)uitoes that infect a human" b. It is also sometimesuseful to consider the human blood feedin$ rate as the !roduct of a blood feedin$ rate"f" and thefraction of blood meals on humans" or more $enerall" the !atho$ens host" 9 3a f94.

So these Im!ortant and measurable )uantities need be reco$ni>ed in models includin$?the !re#alence of malaria" malaria rate" or !arasite rate +( 0:H,; the fraction of infected but notinfectious +# :%, or infectious mos)uitoes +; 6:%,; the ratio of mos)uitoes to humans +m %:H,; the number of bites b #ectors !er human !er da" called the human bitin$ rate+/BR" ma," the number of infectious bites !er human !er da" called the entomolo$icalinoculation rate +-IR" ma;or,. %ormulas are $i#en in the main te(t for the#ectorial ca!acit +V, or dail re!roducti#e number and basic re!roducti#e number +R), and thecritical densit of mos)uitoes re)uired for sustainin$ transmission +m?,.

-ach #ersion of the Ross'&acdonald model has used a subset of these !arameters" buteach one has also utili>ed a different notation.

$he Ross#Waite#*ot&a Model

Rosss first dnamic model of malariawas further de#elo!ed b aite and0otka. 0otkawrote the model more ele$antl as a sim!le difference e)uation?

Ross formulated a )uantit" " is #er similar to #ectorial ca!acit. The deri#ation is #er similar"

but there are some differences. Rosss time ste! was one month" and his formula considered atmost two bites !er mos)uito each month" one that infected it and one that transmitted the!arasites. Thus" in the ali$nment of notation" the inter!retation of Rosssf+or e)ui#alentlbp, isnot identical to the human blood feedin$ rate"a. aites time ste! was the inter#al between bites"but he retained the inter!retation off.

$he Ross#*ot&a Model

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The second dnamic model of malaria was !ublished b Ross twice in 4544" first as anaddendum to the second edition of@he -re!ention of %alaria" and then inNature. 2ne earlater" 0otka !ro!osed a closed'form solution" and in the ear 456 0otka thorou$hl anal>ed it.The model formulation was more focused on mathematical details" and not on the entomolo$icalones. The !arameters here ha#e been su!!lied from ali$nment ?

It must be noted that Ross also considered births and deaths in both the human and #ector!o!ulations" but he set these e)ual to each other so the !o!ulations would be in their stead statefor analsis.

he /harpe01ot2a Model

Shar!e and 0otka e(tended Rosss model to consider the latent !eriod in both humans andmos)uitoes?

The analsis is focused on mathematical details" not biolo$ical ones" and so the model ne$lectsmos)uito mortalit durin$ the latent !eriod and therefore the conclude that the dela has noeffect on the e)uilibrium.

Macdonald) Ir+in) Diet,) and Superinfection

&acdonalds com!lete model was !resented in a series of !a!ers and e(ce!t for the

ori$inal !a!er on su!erinfection" usuall rele$ated to brief summaries in the a!!endices of his!a!ers. The model he uses is essentiall the followin$

The ha!!enin$s rate is defined b the formula?

&acdonald and Irwin first defined a function describin$ the reco#er rate under su!erinfection. Themathematical model is !erfectl #alid" but it was not consistent with the !rocess the described ofindi#idual infections bein$ ac)uired and clearin$ inde!endentl. iet> later described this !rocesscorrectl in the Garki model. /ere it is !aired with the sim!ler formulation to become the &cdonald'iet> model.

&acdonald simulated e!idemics. In so doin$" he used e)uations similar to Rosss first model?

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Ross#Macdonald Style Models

Se#eral models ha#e been !ublished as a Ross'&acdonald model. Aron and &a firstwrote it in the followin$ wa in the ear 45H6?

This model considers infected but not infectious mos)uitoes" so it i$nores the dela for !atho$en latencin mos)uitoes. There are se#eral was to consider the dela or its effects. Smith and &c8en>ie wrotedown a sim!le model with two e)uations that does incor!orate mos)uito mortalit durin$ the latent!eriod but that i$nores the dela?

@erha!s" the best sim!le im!lementation of the Ross'&acdonald model which Aron and &a formed asecond model" a dela differential e)uation.

0ater" Anderson and &a wrote down the followin$ #ersion of the Ross'&acdonald model?

Integrated ontrol

It is more useful to writeR)in a sli$htl different" but e)ui#alent wa for the !ur!ose ofdescribin$ control effect si>es of different inter#entions alone or in combination. It is moreuseful to writeR)in a sli$htl different" but e)ui#alent wa. 0et denote the number of adult

mos)uitoes that are born each da" di#ided b the !o!ulation densit of humans. Pnder theconsensus assum!tions of the Ross'&acdonald model"

so at e)uilibrium?

An e)ui#alent e(!ression for the basic re!roducti#e number is then?

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-ach set of terms in the models corres!onds to a different !art of the !rocess that is sub3ect tocontrol? lar#al ecolo$ and lar#al control + ," adult blood feedin$ and sur#i#al and adult #ectorcontrol the

duration of infection and control b treatin$ infections with dru$s +=:r," usin$ #accines or dru$chemo!ro!hla(is to block infection +b," and usin$ dru$s or #accines that block transmissionfrom humans +c,.$he -irth of a $heory: ./001.020

%or Ross" )uantitati#e thinkin$ came naturall. &athematical models were a wa tocodif" refine" and communicate the )uantitati#e lo$ic of biolo$ical !henomena" es!eciallmos)uito'borne !atho$en transmission" in a form that was ri$orous and testable. In hiscorres!ondence with &anson in the ear 4H5one was lar$e enou$h" an area in the middle would be mos)uito'free.Ross concluded that lar#al control could work if it could de!lete lar#al mos)uitoes in a lar$eenou$h area; but no conclusions about the #alidit of lar#al control"per se" could be reached if ithad not been done intensi#el enou$h" for lon$ enou$h" at a lar$e enou$h scale.

After one ear of Ross #isited &auritius to ad#ice on the control of malaria" heformulated and described a model of mos)uito'borne disease transmission in 459H in hisReport

on the -re!ention of %alaria in %auritius" and he e(!anded on these ideas in the first editionof his @he -re!ention of %alaria. The model was an a prioridescri!tion of the number ofinfections in humans based on his )uantitati#e reasonin$ about the number of mos)uitoes andtheir infection dnamics. It can be formulated as a difference e)uation as stated abo#e. At Rosssin#itation" aite anal>ed the model and wrote a clear descri!tion of the model assum!tions andlimitations. The model was concisel !resented and anal>ed a$ain b 0otka . Rosss mainconclusions from the models were that there is a causal relationshi! between the ratio ofmos)uitoes to humans and the number of infected humans and that it was not necessar to kill

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e#er mos)uito to end transmission. The models demonstrated that there was a critical mos)uitodensit" " such that $reater densities would sustain transmission while lesser ones would not.Rosss formula +makin$ some liberal allowances in the inter!retation of !arameters, is e)ui#alentto the followin$?

Ross was unsatisfied with some minor numerical discre!ancies between his results and aites.These discre!ancies arose because the had !icked different time ste!s for simulation. Ross thendecided to reformulate a $eneral model that would not de!end on an !articular time ste!. /eformulated the model usin$ a sstem of cou!led differential e)uations in continuous time;thou$h mathematicall different" the second model was the limitin$ case of Rosss first modelwith an infinitesimall small time ste!. At the same time" he wanted to de#elo! a more e(!ansi#etheor. Rosss second malaria transmission model was !ublished as an addendum to the secondedition of @he -re!ention of %alariain ear 4544and inNature.

Mathematical models

Because of this !arasites im!ortance and com!le( life ccle models ha#e been de#elo!edto hel! to understand its dnamics.

*ife cycle

The !attern of alternation of se(ual and ase(ual re!roduction which ma seem confusin$at first is a #er common !attern in !arasitic s!ecies. The e#olutionar ad#anta$es of this t!e oflife ccle were reco$nised bGre$or &endel.

Pnder fa#ourable conditions ase(ual re!roduction is su!erior to se(ual as the !arent iswell ada!ted to its en#ironment and its descendants share these $enes. Transferrin$ to a new hostor in times of stress" se(ual re!roduction is $enerall su!erior as this !roduces a shufflin$of $enes which on a#era$e at a !o!ulation le#el will !roduce indi#iduals better ada!ted to the

new en#ironment.The ad#anta$es to ase(ual re!roduction within a host can be seen from this sim!le modeltaken from Cook. The !ro!ortion of hosts that are !arasitised is assumed to be small. This bein$the case the@oisson distributionis a reasonable model. If the !arasite is self'fertili>in$ then thechance of successful re!roduction is 4 ' e'mwhere mis the !ro!ortion of the !o!ulation!arasitised. If the !arasite is a facultati#e bise(ual one ' one that re)uires the !resence of another!arasite on the same host the likelihood of success is 4 ' +4 M m,e'm. If the !arasite has twodistinct se(es and re)uires both for re!roduction" then the chance of success is +4 ' 6 4'n,+mnE ne'm, where the sum is taken between nF 6 and infinit. If mF 9.4 then the chance of success ofthe self'fertili>in$ !arasite is U9 times that of one with distinct se(es. The chance of success ofthe bise(ual !arasite is twice that of the !arasite with distinct se(es. %or smaller #alues of m" the

ad#anta$es of self'fertili>ation are e#en $reater.Gi#en that this !arasite s!ends !art of its life ccle in two different hosts it must use a!ro!ortion of its a#ailable resources within each host. The !ro!ortion utili>ed is currentlunknown. -m!irical estimates of this !arameter are desirable for modelin$ of its life ccle.$ransmission

The basic model of transmission is known as the Ross'&aconald modelafter itsauthors.It is a set of four cou!led non linearordinar differential e)uations. This model is alsoused for other mos)uito borne infections includin$ filariaand den$ue. hile modifications to the

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http://en.wikipedia.org/wiki/Gregor_Mendelhttp://en.wikipedia.org/wiki/Gregor_Mendelhttp://en.wikipedia.org/wiki/Gregor_Mendelhttp://en.wikipedia.org/wiki/Geneshttp://en.wikipedia.org/wiki/Poisson_distributionhttp://en.wikipedia.org/w/index.php?title=Ross-MacDonald_model&action=edit&redlink=1http://en.wikipedia.org/wiki/Ordinary_differential_equationhttp://en.wikipedia.org/wiki/Filariahttp://en.wikipedia.org/wiki/Denguehttp://en.wikipedia.org/wiki/Gregor_Mendelhttp://en.wikipedia.org/wiki/Geneshttp://en.wikipedia.org/wiki/Poisson_distributionhttp://en.wikipedia.org/w/index.php?title=Ross-MacDonald_model&action=edit&redlink=1http://en.wikipedia.org/wiki/Ordinary_differential_equationhttp://en.wikipedia.org/wiki/Filariahttp://en.wikipedia.org/wiki/Dengue


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model ha#e been !ro!osed to encom!ass !articular conditions" the basic model is still re$ardedas a reasonable first a!!ro(imation to most scenarios.

The model is

In this model0h is the number of susce!tible humans" h is the number of infectedhumans"0mis the number of susce!tible mos)uitoes" mis the number of infectedmos)uitoes" bis the bitin$ rate" Ais the reco#er rate in humans"Bis the mortalit rate of themos)uitoes" @hm is the transmission rate from humans to mos)uitoes" @mhis the transmissionrate from mos)uitoes to humans andNF0hM h. The reco#er rate is the reci!rocal of the meanof the duration of a human infection.

There are a number of sim!lifin$ assum!tions in this model?

mos)uito and human !o!ulations are of constant si>e

susce!tibilit of both humans and mos)uitoes to infection is constant

no incubation !eriod

neither humans nor mos)uitoes are or can become immune to infection

both !o!ulations are well mi(ed

the bitin$ rate is constant

%urthermore the e)uations are deterministic ' that is the i$nore the !ossibilit of randomfluctuations. This model is not sol#able in closed form and instead re)uires numericalinte$ration.

-asic reproductive number

An im!ortant !arameter ' the basic re!roducti#e number +R9, ' can be deri#ed from thismodel. This is the a#era$e number of new infectious hosts that a t!ical infectious human will!roduce durin$ his or her infectious !eriod. An infectious a$ent can onl !ersist if the basicre!roducti#e number is $reater than one. In the Ross'&aconald model

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where mis the number of mos)uitoes !er human. The term on the ri$ht of the e)uation is the!roduct of the a#era$e number of humans infected b a mos)uito and the a#era$e number ofmos)uitoes infected b a human. It is worth notin$ that the bitin$ rate a!!ears here as a )uadraticterm.

To eliminate an infectious a$ent it is necessar to reduce the R9to below 4. This can be

achie#ed b reducin$ the bitin$ rate" reducin$ the number of mos)uitoes" shortenin$ themos)uito life s!an or shortenin$ the duration of infection in the human. If a #accine is a#ailableit is onl necessar to #accinate 4 ' 4 ER9!ro!ortion of the !o!ulation to eradicate the infectiousa$ent as this will brin$R9below 4.

This method of estimatin$R9makes a number of sim!lifin$ assum!tions

transmission rates are constant bitin$ rates are constant

duration of human infection is constant

mos)uito infection rates are constant

I$norin$ the #ariabilit that ma be !resent in these !arameters can lead to underestimationofR9.

In the case of malaria in AfricaR9has been estimated to be 9'H4.

onclusion :

This is a short note of mathematical a!!roach for e!idemic diseases. In this short noteon&athematical modelin$ of infectious disease author has tried to focus on recentl stressedconce!ts. As there was no sco!e to discuss details of those de#elo!ed conce!ts or man othermathematical #iews author has confined himself with a torch li$ht #iew onl on main features.

References

4. Alfredo &orabia +699U, ? A histor of e!idemiolo$ic methods and conce!ts. Birkhuser.!. 5.ISBN'" 8. +45HH, ? ensit de!endence in !arasite transmission dnamics" @arasit. Toda" U"

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49. iet>" 8. and Schen>le" . +45H, ? &athematical models for infectious disease statistics" inA Celebration of Statistics" A. C. Atkinson and S. -. %einber$" eds." S!rin$er'erla$" Nework. 47 R" Sanche> Gome> 0&" Carmona 0" Ro)u i %i$uls &" BonfillCos! K.+699, ? /ormone re!lacement thera! for !re#entin$ cardio#ascular disease in !ost'meno!ausal women. Cochrane atabase of Sstematic Re#iews" Issue 6. Art. No.?C996665. doi?49.4996E4U74HH.C996665.!ub6.

4. Grassl" N. C." %raser" C. +699H, ? &athematical models of infectious diseasetransmissionX.Nat. Re!. %icrobiol.3+7,? U" S.; Robins" `. &. +699U, ? A structural a!!roach to selectionbias.


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6H. &i)uel@orta +699H, ? A ictionar of -!idemiolo$. 2(ford Pni#ersit @ress. !!. 49144. ISBN5klo && W Nieto %` +6996,. ? -!idemiolo$? beond the basics" As!en @ublishers" Inc.U4.Statistical methods in e!idemiolo$? 8arl @earson" Ronald Ross" &a3or Greenwood and

Austin Bradford /ill" 4599 1 45U. Trust Centre for the /istor of &edicine at PC0" 0ondonU6.Rile" S. +699


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UH.ickwire" 8. +45

article in connection with ronald ross's work

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