D Shailend a Vashis ha e al. OPTIMAL CONTROL, VACCINATION AND AWARENESS IN TUBERCULOSIS AND HEPATITIS
B VIRUS INFECTION: A QUALITATIVE CLINICAL MODELLING STUDY. In . J Med. Pha m. Res., 6 (6): 561‐566, 2025
561
In e na ional Jou nal o Medical
and Pha maceu ical Resea ch
Online ISSN-2958-3683 | P in ISSN-2958-3675
F equency: Bi-Mon hly
A ailable online on: h ps://ijmp .in/
O iginal A icle
OPTIMAL CONTROL, VACCINATION AND AWARENESS IN TUBERCULOSIS AND
HEPATITIS B VIRUS INFECTION: A QUALITATIVE CLINICAL MODELLING STUDY
D Shailend a Vashis ha1, D Sandeep Rajo ia2, D B ijmohan Meena3, D K ishna Kuma Sha ma4, D Ama deep5
1 Assis an P o esso , HLA Lab, Depa men o Immuno-Haema ology & T ans usion Medicine, GMC, Ko a, Rajas han, India – 324010.
2 Senio Specialis , Depa men o Respi a o y Medicine, GSH, Tonk
3 Assis an P o esso , Depa men o Respi a o y Medicine, GMC, Ko a
4 Associa e P o esso , Depa men o Compu e Science, Uni e si y o Ko a
5 Assis an P o esso , Depa men o Ma hema ics, IIMT Enginee ing College, Mee u
A B S T R A C T
Co esponding Au ho :
D Shailend a Vashis ha
Assis an P o esso , HLA Lab,
Depa men o Immuno-
Haema ology & T ans usion
Medicine, GMC, Ko a, Rajas han,
India – 324010.
Recei ed: 10-10-2025
Accep ed: 14-11-2025
A ailable online: 20-11-2025
Backg ound: Tube culosis (TB) and ch onic hepa i is B i us (HBV) in ec ion
emain impo an causes o p e en able mo bidi y and mo ali y wo ldwide.
Ma hema ical models can help explo e how bes o combine accina ion and public
awa eness o educe disease bu den.
Objec i e: To use a ansmission model o s udy how accina ion and public-
awa eness campaigns can op imally educe he bu den o TB o HBV in ec ion,
while accoun ing o limi ed esou ces.
Me hods: In his mul i-cen ic collabo a i e quali a i e s udy, we ex ended a
s anda d compa men al in ec ious-disease model, di iding he popula ion in o
suscep ible (un accina ed), accina ed, acu ely in ec ed, asymp oma ic ch onic
ca ie s, symp oma ic ch onic ca ie s, indi iduals wi h complica ions and
eco e ed. Two ime-dependen con ol a iables we e in oduced: in ensi y o
accina ion and in ensi y o awa eness ac i i ies ha educe e ec i e con ac a es.
Using op imal con ol heo y (Pon yagin’s Maximum P inciple), we iden i ied he
combina ion o in e en ions o e a one-yea ho izon ha minimises bo h he o al
numbe o in ec ed people and he cos s o accina ion and awa eness p og ammes.
Model pa ame e s we e aken om published TB/HBV li e a u e. Resul s we e
summa ised using he basic ep oduc ion numbe (R0) and nume ical simula ions.
Resul s: Wi hou addi ional in e en ion, acu e in ec ion and ch onic ca ie
popula ions inc eased, wi h a co esponding ise in se e e complica ions. Wi h
op imal con ol, ea ly agg essi e use o accina ion and awa eness in he i s h ee
mon hs p oduced a ma ked inc ease in eco e ed pa ien s and a educ ion in acu e
and ch onic in ec ion. The model p edic ed ha accina ion has a s onge e ec
han awa eness campaigns on d i ing R0 below 1, bu he combina ion o bo h
s a egies is mo e e ec i e han ei he alone, especially when ansmission a es a e
high. Fo plausible pa ame e alues, R0 dec eased om app oxima ely 2.4 wi h no
con ol o <1 wi h easible le els o accina ion and awa eness.
Conclusions: In his model, p io i ising high-co e age accina ion, suppo ed by
in ensi e sho - e m awa eness campaigns, is an e icien s a egy o con ol TB o
HBV ansmission. These indings suppo exis ing clinical and public-heal h
ecommenda ions o scale up accina ion, pa icula ly o newbo ns and high- isk
g oups, while sus aining communi y educa ion on in ec ion p e en ion and linkage
o ca e.
Copy igh © In e na ional Jou nal o
Medical and Pha maceu ical Resea ch
Keywo ds: Tube culosis; hepa i is B; ma hema ical model; op imal con ol;
accina ion; heal h educa ion; basic ep oduc ion numbe .
D Shailend a Vashis ha e al. OPTIMAL CONTROL, VACCINATION AND AWARENESS IN TUBERCULOSIS AND HEPATITIS
B VIRUS INFECTION: A QUALITATIVE CLINICAL MODELLING STUDY. In . J Med. Pha m. Res., 6 (6): 561‐566, 2025
562
INTRODUCTION
Tube culosis (TB) and ch onic hepa i is B i us (HBV) in ec ion emain majo causes o p e en able mo bidi y and
mo ali y wo ldwide. TB, caused by Mycobac e ium ube culosis, can a ec he lungs as well as ex apulmona y si es
such as bones, kidneys and he cen al ne ous sys em. HBV is a blood-bo ne and sexually ansmi ed in ec ion ha
a ge s he li e and may p og ess om an acu e, o en asymp oma ic episode o ch onic in ec ion, ci hosis and
hepa ocellula ca cinoma.1
Despi e he a ailabili y o e ec i e accines ( o HBV) and p e en i e and cu a i e ea men ( o bo h TB and HBV),
many coun ies con inue o expe ience pe sis en o esu ging epidemics. Reasons include incomple e accina ion
co e age, delayed diagnosis, gaps in ea men comple ion and low awa eness abou modes o ansmission and
p e en ion. P og ammes mus he e o e decide how o alloca e limi ed esou ces be ween in e en ions such as
accina ion, case- inding, ea men and heal h-educa ion campaigns.
Ma hema ical models a e inc easingly used o suppo such decisions by allowing us o explo e "wha -i " scena ios ha
would be di icul o cos ly o es in eal li e.2 T ansmission models can es ima e he impac o di e en s a egies on
in ec ion incidence and p e alence and can inco po a e he cos s o in e en ions. Op imal con ol heo y is a
ma hema ical amewo k ha seeks he ime-dependen combina ion o in e en ions ha yields he bes ou come
acco ding o a speci ied objec i e unc ion.3-6
This s udy was conduc ed wi h he collabo a i e e o s o he expe s om he depa men s o Respi a o y Medicine,
T ans usion Medicine, Compu e Science and Ma hema ics om GMC, Ko a, GSH, Tonk and UoK. A de ailed
ma hema ical analysis o a TB/HBV model, including p oo s abou exis ence, uniqueness and s abili y o solu ions was
done. In p esen a icle, we ocus on he clinically o ien ed key ideas and indings ha a e mos ele an o medical
p ac i ione s. The model s uc u e has been ied o desc ibe in in ui i e e ms, o summa ise he op imal con ol app oach
and o highligh he main clinical and public-heal h implica ions.
METHODS
Model s uc u e
We used a de e minis ic compa men al model ha di ides he o al popula ion in o he ollowing mu ually exclusi e
g oups:
• S( ): Un accina ed suscep ible indi iduals – people who do no ye ha e TB o HBV and ha e no ecei ed a
p o ec i e accine dose.
• V( ): Vaccina ed indi iduals – people who ha e ecei ed he accine; hey ha e pa ial p o ec ion bu can s ill be
in ec ed i exposed.
• A( ): Acu ely in ec ed indi iduals – pa ien s in he ea ly phase o in ec ion.
• Cn( ): Asymp oma ic ch onic ca ie s – indi iduals wi h ch onic in ec ion bu no symp oms; hey can ansmi he
pa hogen.
• Cs( ): Symp oma ic ch onic ca ie s – indi iduals wi h ch onic in ec ion who ha e clinical symp oms.
• Dc( ): Indi iduals wi h disease complica ions – pa ien s wi h ad anced disease (e.g. ci hosis o li e cance o
HBV).
• R( ): Reco e ed indi iduals – people who ha e clea ed he in ec ion and a e assumed o ha e long-las ing immuni y.
A any ime , he o al popula ion N( ) is he sum o all compa men s:
N( ) = S( ) + V( ) + A( ) + Cn( ) + Cs( ) + Dc( ) + R( ).
The model assumes ha new indi iduals en e he popula ion h ough bi h a a e Π. A p opo ion p o newbo ns a e
accina ed a bi h, and he emaining (1−p) a e un accina ed. All indi iduals may die om backg ound causes a a
cons an na u al mo ali y a e μ, and hose wi h complica ions o ad anced ch onic disease may ha e an addi ional
disease- ela ed mo ali y a e δ.
In ec ion p ocess
Suscep ible and accina ed indi iduals can be in ec ed when hey ha e e ec i e con ac wi h acu ely in ec ed pa ien s o
ch onic ca ie s. The o ce o in ec ion ( a e a which suscep ibles become in ec ed) is w i en as:
λ = β [ A + ηn Cn + ηs Cs + ηc Dc ],
whe e β is he baseline ansmission a e and ηn, ηs and ηc a e modi ica ion ac o s cap u ing ha asymp oma ic ca ie s,
symp oma ic ca ie s and pa ien s wi h complica ions may di e in in ec iousness.
We assume ha symp oma ic ch onic ca ie s a e mo e in ec ious han asymp oma ic ca ie s, and ha pa ien s wi h
complica ions ha e a leas as high in ec iousness as symp oma ic ca ie s.
D Shailend a Vashis ha e al. OPTIMAL CONTROL, VACCINATION AND AWARENESS IN TUBERCULOSIS AND HEPATITIS
B VIRUS INFECTION: A QUALITATIVE CLINICAL MODELLING STUDY. In . J Med. Pha m. Res., 6 (6): 561‐566, 2025
563
Vaccina ed indi iduals ha e pa ial p o ec ion: hei isk o in ec ion is mul iplied by a ac o (0 < < 1), which e lec s
accine e icacy. Vaccine-induced p o ec ion may wane o e ime a a e ω, mo ing people om accina ed back o
suscep ible.
Disease p og ession
Among newly in ec ed indi iduals, a ac ion en e an asymp oma ic ou e and la e become ch onic ca ie s wi hou
ini ial symp oms. The emaining ac ion (1− ) de elop symp oma ic acu e in ec ion.
F om he acu e compa men , indi iduals may p og ess o asymp oma ic ch onic ca iage, p og ess o symp oma ic
ch onic ca iage, o eco e o die acco ding o model pa ame e s.
Asymp oma ic ch onic ca ie s can la e de elop symp oms a a e ξ o eco e a a e γ. Symp oma ic ch onic ca ie s
may eco e (e ec i e ea men a a e θ γ, whe e θ > 1 e lec s in ensi ied ea men ), p og ess o complica ions a a e
ν, die om he disease a a e δ, o expe ience na u al mo ali y μ. Pa ien s wi h complica ions can die ei he om na u al
causes μ o om disease- ela ed causes δ.
Reco e ed indi iduals a e assumed o ha e long-las ing in ec ion-acqui ed immuni y and do no become suscep ible
again wi hin he model ime ame.
Con ol a iables: accina ion in ensi y and awa eness
We conside ed wo ime-dependen con ol a iables:
• u1( ): in ensi y o accina ion e o s, ep esen ing how s ongly accina ion is scaled up beyond he baseline a e ψ ( o
example ou each, ca ch-up campaigns, o imp o ing co e age among high- isk g oups).
• u2( ): in ensi y o awa eness and heal h-educa ion e o s, ep esen ing communi y ac i i ies ha educe e ec i e
con ac a es ( o example knowledge abou cough e ique e, sa e injec ion p ac ices, sc eening and linkage o ca e).
Bo h u1 and u2 a e cons ained be ween 0 and 1, whe e 0 means no addi ional e o beyond ou ine p ac ice and 1 means
maximum easible e o .
In he model, u1( ) inc eases he low o people om S o V (mo e accina ion o suscep ibles). The con ol u2( ) educes
he e ec i e con ac a e (by lowe ing β in p ac ice) and inc eases he a e a which suscep ibles mo e o he eco e ed
compa men R h ough beha iou change and ea lie heal hca e seeking.
Objec i e unc ion and op imal con ol
The goal o he op imal con ol p oblem is o ind ime p o iles u1*( ) and u2*( ) on a ini e ime ho izon [0, T] ha
minimise a combined measu e o disease bu den and in e en ion cos .
Fo mally, he objec i e unc ion is:
J(u1, u2) = ∫0^T[a0 A( )+a1Cn( )+a2 Cs( )+a3Dc( )+½ (b1u1( )^2+b2u2( )^2 )]d .
He e, a0–a3 a e posi i e weigh s assigned o each in ec ed o diseased g oup, e lec ing hei clinical impo ance ( o
example complica ions may be weigh ed mo e hea ily han acu e in ec ion). The coe icien s b1 and b2 a e posi i e cos
pa ame e s o accina ion and awa eness, espec i ely. The quad a ic e ms penalise e y agg essi e con ol s a egies,
ep esen ing inc easing ma ginal cos o logis ical di icul y.
Ma hema ically, we applied Pon yagin’s Maximum P inciple o de i e necessa y condi ions o op imali y.4 In in ui i e
e ms, his p inciple in oduces shadow p ices o each compa men and leads o a se o di e en ial equa ions ha , when
sol ed oge he wi h he o iginal model, iden i y he op imal in e en ions o e ime.
We also calcula ed he basic ep oduc ion numbe (R0), which summa ises he a e age numbe o seconda y in ec ions
caused by a ypical in ec ious indi idual in a ully suscep ible popula ion. When R0 < 1, each gene a ion o in ec ion is
smalle han he p e ious one and he disease e en ually dies ou ; when R0 > 1, he disease can pe sis o g ow.
Pa ame e alues and simula ion se ings
Pa ame e alues (bi h a e, mo ali y, p og ession a es, eco e y a es, accine waning, e c.) we e aken om published
TB and HBV modelling s udies and clinical li e a u e. T ansmission a es β we e a ied o e a plausible ange (0.8–
1.98), and di e en le els o he con ols u1 and u2 we e explo ed.
Because acu e HBV in ec ion ypically esol es o s abilises wi hin six mon hs in immunocompe en adul s, he main
simula ions we e un o e a one-yea pe iod. This ime ame allows us o cap u e he sho - e m impac o in ensi ied
in e en ions while s ill obse ing longe - e m ends in ch onic in ec ion and complica ions.
D Shailend a Vashis ha e al. OPTIMAL CONTROL, VACCINATION AND AWARENESS IN TUBERCULOSIS AND HEPATITIS
B VIRUS INFECTION: A QUALITATIVE CLINICAL MODELLING STUDY. In . J Med. Pha m. Res., 6 (6): 561‐566, 2025
564
RESULTS (QUALITATIVE SUMMARY)
Baseline scena io wi hou in ensi ied con ol
In he absence o any addi ional accina ion o awa eness e o s (u1 = 0, u2 = 0), he model p edic s ha he numbe o
acu ely in ec ed indi iduals A( ) inc eases om low baseline le els. A p opo ion o hese indi iduals ansi ion in o
ch onic asymp oma ic o symp oma ic ca ie s a es (Cn, Cs). O e ime, he e is a g adual accumula ion o pa ien s wi h
disease complica ions Dc( ). The eco e ed compa men R( ) inc eases s eadily, bu no enough o p e en ongoing
ansmission.
These beha iou s co espond o an R0 g ea e han 1, indica ing ha he in ec ion can pe sis and po en ially become
endemic.
Impac o op imal accina ion and awa eness
When bo h con ols a e allowed o a y o e ime and a e op imised using he objec i e unc ion, he model sugges s an
ea ly, high-in ensi y phase o bo h accina ion and awa eness du ing oughly he i s h ee mon hs. Du ing his pe iod,
he numbe o suscep ibles alls as mo e people a e accina ed, and beha iou al changes educe e ec i e ansmission.
As a esul , he numbe o acu ely in ec ed indi iduals d ops, and ewe people en e ch onic ca ie s a es o de elop
complica ions. The eco e ed compa men g ows mo e apidly, e lec ing bo h immunologically eco e ed pa ien s and
hose who bene i om accina ion.
A e he ini ial pe iod, he op imal s a egy g adually educes he in ensi y o bo h con ols, as he pool o suscep ibles
sh inks and he e ec i e ep oduc ion numbe mo es below 1. A his s age, main aining maximal e o p o ides
diminishing e u ns ela i e o cos .
E ec on he basic ep oduc ion numbe (R0)
By examining R0 as a unc ion o u1 and u2, he model shows ha wi h no addi ional con ol (u1 = 0, u2 = 0), R0 is
app oxima ely 2.39 unde he chosen pa ame e alues. As ei he u1 o u2 inc eases, R0 dec eases, bu he slope o
educ ion is s eepe o u1 ( accina ion) han o u2 (awa eness).
When bo h con ols a e used oge he a mode a e o high le els, R0 can be d i en below 1, implying ha he in ec ion
can be b ough unde con ol and e en ually elimina ed in he model popula ion. In o he wo ds, accina ion has a
s onge di ec impac on ansmission, while awa eness campaigns ac as a aluable complemen by educing isky
con ac s and encou aging ea ly diagnosis and ea men .
Sensi i i y o ansmission a e
Fo highe ansmission a es (β = 1.49–1.98), he model indica es ha wi hou in ensi ied con ol, in ec ion quickly
becomes mo e widesp ead and he bu den o ch onic ca iage and complica ions inc eases. E en a hese highe β alues,
app op ia e combina ions o accina ion and awa eness can s ill educe o al in ec ion o e ime, al hough sus ained
e o s a e equi ed.
The highe he unde lying β, he longe and mo e in ense he con ol measu es need o be o main ain R0 < 1.
DISCUSSION
This modelling s udy illus a es, in a simpli ied and clinically in e p e able way, how combining accina ion and public-
awa eness in e en ions can e icien ly educe he bu den o TB o HBV in ec ion in a popula ion.7-10
Se e al poin s a e pa icula ly ele an o clinicians and p og amme manage s:
1. Vaccina ion is he co ne s one o con ol. In he model, inc easing accina ion co e age (u1) has he la ges e ec on
educing R0 and he numbe o newly in ec ed indi iduals. This suppo s cu en ecommenda ions o widesp ead HBV
accina ion, especially a bi h and in ea ly childhood, and emphasises he impo ance o p e en i e he apy and
ea men comple ion in TB con ol.11-13
2. Heal h-educa ion and awa eness campaigns a e aluable adjunc s. Al hough awa eness ini ia i es alone a e less po en
han accina ion in he model, hey signi ican ly enhance he o e all impac when combined wi h accina ion. Such
p og ammes can p omo e ea ly es ing, imp o e adhe ence o ea men , and educe isky beha iou s ha acili a e
ansmission.14
3. Ea ly, in ensi e con ol is mo e e icien han delayed o uni o mly low-le el con ol. The op imal s a egy
concen a es esou ces in he ea ly phase o he in e en ion, when he e a e s ill many suscep ibles and he po en ial o
p e en new in ec ions is g ea es . Fo clinicians and planne s, his a gues o s ong ini ial scale-up phases when new
p og ammes a e in oduced.15,17
4. Ch onic ca ie s equi e a en ion e en when hey a e asymp oma ic. Asymp oma ic ch onic ca ie s (Cn) con ibu e
subs an ially o ansmission in he model, because hey o en emain undiagnosed and un ea ed. This unde sco es he
alue o sc eening p og ammes ( o example an ena al HBV es ing and con ac acing o TB) and linkage o ca e.18-20
D Shailend a Vashis ha e al. OPTIMAL CONTROL, VACCINATION AND AWARENESS IN TUBERCULOSIS AND HEPATITIS
B VIRUS INFECTION: A QUALITATIVE CLINICAL MODELLING STUDY. In . J Med. Pha m. Res., 6 (6): 561‐566, 2025
565
5. P e en ing complica ions is bo h clinically and economically impo an . Indi iduals wi h disease complica ions (Dc)
a e weigh ed hea ily in he objec i e unc ion, e lec ing he high clinical bu den and cos o ci hosis, li e cance o
ad anced TB. By p e en ing p og ession o hese s a es h ough ea lie accina ion, diagnosis and e ec i e ea men , he
model sugges s subs an ial long- e m bene i s.21-23
CLINICAL AND PUBLIC-HEALTH IMPLICATIONS
T ansla ing he ma hema ical esul s in o p ac ice, he s udy suppo s he ollowing b oad ecommenda ions:
• Scale up accina ion p og ammes as a p io i y, aiming o high co e age in newbo ns, child en and high- isk adul s ( o
example household con ac s, heal hca e wo ke s, people li ing wi h HIV).
• Combine accina ion wi h s uc u ed awa eness campaigns, especially du ing he ini ial yea s o p og amme expansion,
o accele a e beha iou change and up ake o es ing and ea men .
• Use sho , in ensi e in e en ion pe iods ( o example he i s 3–6 mon hs o a campaign) o apidly educe
ansmission, ollowed by main enance-le el e o s a he han cons an maximal in ensi y.
• Ta ge ch onic ca ie s and high- isk g oups o sc eening and ea ly ea men , ecognising hei ole in sus aining
ansmission.
• Moni o p og amme pe o mance using epidemiological indica o s, including es ima es o R0 whe e easible, o assess
whe he in e en ions a e su icien o push he epidemic below he h eshold o sus ained ansmission.
LIMITATIONS (FROM A CLINICAL PERSPECTIVE)
As wi h any ma hema ical model, se e al simpli ica ions should be kep in mind:
• The model assumes a homogeneously mixing popula ion and does no explici ly ep esen age s uc u e, co-mo bidi ies
(such as HIV) o social ne wo ks, all o which can in luence TB and HBV ansmission.
• T ea men adhe ence, d ug esis ance and ein ec ion a e no modelled in de ail.
• Pa ame e alues we e d awn om published li e a u e and may no pe ec ly ep esen speci ic local con ex s.
• Cos s a e ep esen ed in abs ac e ms (weigh s b1 and b2) a he han ac ual inancial uni s.
The e o e, he esul s should be in e p e ed as quali a i e guidance on he ela i e alue and iming o in e en ions, no
as exac quan i a i e p edic ions.
CONCLUSION
Using a clinically o ien ed ansmission model wi h op imal con ol, we show ha accina ion is he mos powe ul
single le e o educing TB o HBV ansmission. Awa eness and heal h-educa ion campaigns complemen accina ion
and imp o e ou comes, pa icula ly when implemen ed ea ly and a high in ensi y.
The combina ion o hese s a egies can educe he basic ep oduc ion numbe below 1 and subs an ially lowe he
numbe o acu e in ec ions, ch onic ca ie s and pa ien s wi h complica ions, e en in se ings wi h ela i ely high baseline
ansmission.
Fo clinicians and public-heal h p o essionals, he key message is ha in es ing in obus accina ion p og ammes,
suppo ed by a ge ed awa eness ac i i ies, is bo h medically and economically jus i ied as pa o comp ehensi e
s a egies o con ol and e en ually elimina e TB and HBV.
ACKNOWLEDGEMENT
The au ho s wish o acknowledge he depa men and ins i u ion heads o all ou ins i u es, (GMC, Ko a; Uni e si y o
Ko a; GSH, Tonk; and IIMT Enginee ing College, Mee u ) o hei suppo . The au ho s also wish o hank The VAssis
Resea ch eam (www. he assis .com) o hei con ibu ion in manusc ip edi ing, quali a i e analysis and jou nal
submission p ocess.
CONFLICT OF INTEREST: None.
SOURCE OF FUNDING: Nil.
BIBLIOGRAPHY
1.
WHO, Hepa i is B Fac Shee No. 204, The Wo ld Heal h O ganiza ion, Gene a,
Swi ze land, 2013,
h p://www.Who.in /mediacen e/ ac shee / s204/en/
2.
Cicik Al iniyah, Wanwha Sonia Pu i A ha Soe jian o, Ahmadin, Muhamad Hi zhudin Noo Azizand Si i
Maisha ah Sheikh Ghadzi (2024), “Ma hema ical modeling and op imal con ol o ube culosis sp ead among
smoke s wi h case de ec ion.”, AIMS Ma hema ics, 9(11):30472–30492.
3.
He hco e, He be W. (2000). The ma hema ics o in ec ious diseases. SIAM Re- iew, 42(4),pp.599-653.
4.
Tunde T. Yusu , A eezAbidemi (2023), E ec i e s a egies owa ds e adica ing he ube culosis epidemic: An
op imal con ol heo y al e na i e, Heal hca e An- aly ics, Volume 3, No embe 2023, 100131
5.
F. B. Hollinge and D. T. Lau, “Hepa i is B: he pa hway o eco e y h ough
ea men ”, Gas oen e ology
D Shailend a Vashis ha e al. OPTIMAL CONTROL, VACCINATION AND AWARENESS IN TUBERCULOSIS AND HEPATITIS
B VIRUS INFECTION: A QUALITATIVE CLINICAL MODELLING STUDY. In . J Med. Pha m. Res., 6 (6): 561‐566, 2025
566
clinics o No h Ame ica, ol. 35, no.4, pp.895-931,
2006.
6.
Acu e Hepa i is B In es iga i e Guidelines(Sep embe 2024), PUBLIC HEALTH DIVISION Acu e and
Communicable Disease P e en ion
7.
Muhammad Fa han, Zahi Shah, Zhi Ling (2024), Global dynamics and compu a ional modelling o analysing
and con olling Hepa i is B: A no el epidemic app oach, 2024 Jun 27;19(6):e0304375.
doi:10.1371/jou nal.pone.0304375
8.
Mannand M. Robe s, “Modelling he epidemiology o hepa i is Bin New Zealand,” Jou nal o Theo e ical Biology,
ol.269, no.1, pp.266–272, 2011.
9.
S. Tho nley, C. Bullen, M. Robe s, “Hepa i is B in a high p e alence New Zealand popula ion: A
ma hema ical model applied o in ec ion con ol policy,” Jou nal o Theo e ical Biology, ol. 254, no. 3, pp.
599–603, 2008.
10.
K.Wang, W.Wang, S.Song,“Dynamics o an HBV model wi h di usion and delay,” Jou nal o Theo e ical
Biology, ol. 253, no. 1, pp. 36–44, 2008.
11.
R. Xu and Z. Ma, “An HBV model wi h di usion and ime delay,” Jou nal o Theo e ical Biology, ol. 257,
no. 3, pp. 499–509, 2009.
12.
L. Zou, W. Zhang, and S. Ruan, “Modeling he ansmission dynamics and con ol o hepa i is B i us in
China,” Jou nal o Theo e ical Biology, ol. 262, no. 2, pp. 330–338, 2010.
13.
J. Pang, J.A. Cui, and X. Zhou, “Dynamical beha io o a hepa i is B i us ansmission model wi h
accina ion,” Jou nal o Theo e ical Biology, ol. 265, no. 4, pp. 572–578, 2010.
14.
S. Bha acha yya and S. Ghosh, “Op imal con ol o e ically ansmi ed dis- ease,” Compu a ional and
Ma hema ical Me hods in Medicine, ol. 11, no. 4, pp. 369–387, 2010.
15.
T.K. Ka and S. Jana, “A heo e ical s udy on ma hema ical modelling o an in ec ious disease wi h applica ion
o op imal con ol,” Bio sys ems, ol. 111, no. 1, pp. 37–50, 2013.
16.
S. Lenha and J. Wo kman, Op imal Con ol Applied o Biological Models, Chap-
man and Hall/CRC
Ma hema ical and Compu a ional Biology, 2007.
17.
E. A. Baka e, A. Nwagwo, E. Danso-Addo. Op imal con ol analysis o an SIR epidemic model wi h cons an
ec ui men . In e na ional Jou nal o Applied Ma h- ema ical Resea ch, 3(3) (2014):pp 273-285.
18.
Dam ew Bewke Ki a o, Boka Kumsa Bole, Koya Pu nachand a Rao (2024), Op i- mal Con ol S a egies and
Cos -E ec i eness Analysis o Tube culosis Dynamics Wi h Mul id ug-Resis an Compa men ,
h ps://doi.o g/10.1155/2024/7235040.
19.
Tunde Tajudeen Yusu , F acis Benyah. Op imal Con ol o accina ion and ea men o an SIR epidemiological
model. Wo ld Jou nal o Modeling and Simula ion, Vol.8(2012)No.3:pp.194-204.
20.
Michael, T. H., 2002. Scien i ic Compu ing: An in oduc o y su ey. Second edi ion, The McG aw-Hill, New
Yo k, 2002.
21.
Jung,E.O., Lenha ,S., Feng,Z., Op imal con ol o ea men s in a wo s ain ube culosis model, Disc e e and
Con inuous Dynamical Sys ems-Se ies B 2-4 (2002):pp. 473-482.
22.
P. an den D iessche and J. Wa mough, Rep oduc ion numbe s and sub- h esold endemic equilib ia o
compa men al models o disease ansmission, Ma h. Biosci. 180 (2002), 29-48.
23.
Jing Zhang, Suxia Zhang. Applica ion and Op imal Con ol o an HBV Model wi h Vaccina ion and
T ea men . Disc e e Dynamics in Na u e and Socie y, Vol.2018.
