Transcript First Presentation

Study and Analysis of Cholera
Transmission
MALLESSA YEBOAH
OLIVIA HYLTON
Cholera
Diarrhoeal disease caused by an intestinal bacteria (Vibrio
cholerae).
◦ Thrives on environmental conditions such as salinity and
temperature
◦ Can persist for extended periods of time in water sources under
appropriate conditions
◦ Found in the feces of infected individuals
◦ Spreads when infected individuals shed pathogens into drinking
water or contaminates food
◦ Contact may occur from person-to-person
◦ Can be life-threatening
Outbreak in Haiti
Following the earthquake of January 2010 in Haiti the first cholera outbreak in
almost a century was announced in October of that year.
To date, over 470,000 cases of cholera have been reported in Haiti with 6,631
attributable deaths.
This marks the worst cholera outbreak in recent history, as well as the best
documented cholera outbreak in modern public health.
Since the beginning of the outbreak the CDC (Center of Disease Control) has worked
closely with the Haitian MSPP (Ministry of Public Health and Population) to combat
the cholera epidemic and reduce the impact of the disease.
Outbreak in Haiti
It is uncommon for developed countries to have large outbreaks of
this disease.
◦ Advanced water and sanitation systems prevent the occurrences of such
waterborne diseases
Controlling an outbreak remains a major challenge in both developing
and underdeveloped countries.
Mathematical Modeling
• A tool that complements statistical analysis and field epidemiology. ​
• Study the progression of infectious diseases is a significant aspect of
mathematical research. ​
• Allow us to use information about a specific disease to show the projected
outcome of an epidemic, control the spread of disease, and reduce disease
related costs.​
Modeling Cholera Outbreak
The SIWR model was developed by Tien and Earn.
S-Susceptible I-Infectious W-Waterborne Pathogen Concentration R-Recovered
• Described dynamics of a disease transmission that occurred through contaminated water
and by personal contact.
• No intervention strategies are included.
The model is modified by Tuite, Tien, Eisenberg, Earn, Ma, and Fishman to include some
interventions to identify optimal control interventions.
• This model predicted the sequence and timing of regional cholera epidemics in Haiti and
explored the potential effects of disease-control strategies.
• Based on SIR model Miller, Schaefer, Gaff, Fister, and Lenhart modeled disease dynamics
with added features.
• Less infectious, hyper infectious, symptomatic, asymptomatic, disease waning immunity.
• Considered several intervention strategies such as rehydration and antibiotic treatment,
vaccination, sanitation.
• Suggested optimal intervention strategies to regulate cholera transmission during an
outbreak.
SIR Diagram
• Black arrows: Susceptible people become infected, recover,
then become immune.
• Blue arrows: Infectious people contaminate the water
supply with bacteria and the bacteria decay.
• Red arrow: Susceptible people are exposed to contaminated
water and may become infected.
• Gray arrows: People are born into the susceptible
population; they may die as a result of cholera infection or
other reasons.
SVIR-B Model
Modified SIWR model
◦ Vaccination is added
◦ Disease waning immunity term is added
◦ Disease related death term is added
dS
 n ( S  I  R )  bWWS  bI SI  V (t ) S   S   R
dt
dI
 bWWS  bI SI   I   I  dI , dI  disease related death
dt
dW
  I  W
dt
dR
  I   V (t ) S   R   R
dt
Ghosh-Dastidar and Lenhart, JBS, June 2015
Parameters
Definition
units
Introduced
Values here are given per day
n
µ
bW
natural birth rate
natural death rate
transmission rate for water-toperson
day-1
day-1
ml cells-1 day-1
0.044/365
0.033/365
(2.14*10-5)/365
A
Amplitude of seasonality of
bW(t)
Average value of bW(t)
ml cells-1 day-1
0.88
Endemic
Values here are given per
day
0.044/365
0.033/365
(3.9*10-5)/365
(A+1)*B=(0.88+1)*(2.14*10
^(-5))/365
0.5
ml cells-1 day-1
2.14 * 10-5
3.9*10-5
Individuals-1
day-1
year-1
(3.65 x 10-4)/365
(9.12*10-4)/365

transmission rate for personto-person
immunity waning rate
-1
recovery rate
0.4/365 (Low population
density)
0.7/365 (High population
density)
3
d
disease related death rate
0.033
0.033
α
rate of pathogen shedding into
reservoir
mean pathogen lifetime in
water reservoir
disease duration
Individuals1day-1
Individuals1day-1
Cells ml-1 day-1
individuals-1
weeks
0.4/365 (Low population
density)
0.7/365 (High population
density)
3
3650/365=10
3650/365=10
7
14
B
bI
-1
T
days
300-600
Reproductive Number Equilibria,
and Stability
Mathematical results show that the control reproduction number
satisfies a threshold property with threshold value 1.
When Rv < 1, disease free equilibrium E0 is globally asymptotically stable
under sufficient conditions.
When Rv > 1, disease free equilibrium E0 is globally asymptotically
stable.
Research Objectives
New model: We combined the model proposed by Tien and Earn,
(SIWR: Susceptible S – Infectious I -Waterborne Pathogen Concentration
W – Recovered), model proposed by Neilan et. al (introducing the
symptomatic and asymptomatic classes), the model developed by Cui
et. al. (introducing a vaccination class), and the model proposed by Qiu
et. al.
Mainly we added:
◦
◦
◦
◦
Vaccination,
Hyperinfectious and less infectious class in waterborne pathogens
Symptomatic and Asymptomatic
Introduce treatment such as rehydration and antibiotics
Our model: Parameters
I A : asymptomatic state
I S : symptomatic state
p : proportion of inf ected who are asymptomatic
1  p : proportion of inf ected who are symptomatic
bWH : transmission coefficient from waterborne hyper inf ectious pathogens to humans
bWL : transmission coefficient from waterborne less inf ectious pathogens to humans
bIH : transmission coefficient from humans hyper inf ectious pathogens to humans
bLH : transmission coefficient from humans less inf ectious pathogens to humans
 A : disease re cov ery rate of asymptomatic (inf ected who do not have symptoms )
 H : disease re cov ery rate of symptomatic who has not received the treatment
 Hu : disease re cov ery rate of symptomatic who received the treatment (rehydration and antibiotic )
Assume  A   Hu   H
d A : disease related death of asymptomatic people
 : natural death rate
u (t ) : proportion of the symptomatic population who received the treatment
1  u (t ) : proportion of the symptomatic population w ho did not receive the treatment
d H : disease related death of hyper inf ectious population who did not receive the treatment
d Hu : disease related death of hyper inf ectious population who received the treatment
d A  d Hu  d H
 : proportion of the susceptible population who were vaccinated
 : rate of waning immunity of vaccination
 A : pathogens shedding rate by asymptomatic individuals
 S : pathogens shedding rate by symptomatic individuals
 : rate of hyper inf ectious pathogens moving to less inf ectious state
 : decay rate of less inf ectious pathogen rate
 : disease ralated waning immunity rate
Our Modified model
dI A
 p (bWH WH S  bWLWL S  bIH SI H  bIL SI A )   A I   I A  d A I A
dt
dI H
 (1  p )(bWH WH S  bWLWL S  bIH SI H  bIL SI A )  (1  u (t )) H I H  u (t ) Hu I H   I H  (1  u (t ))d H I H  u (t )d Hu I H
dt
dV
  S   V  V
dt
dWH
  A I A   S I H  WH
dt
dWL
 WH   WL
dt
dR
  A I  (1  u (t )) H I H  u (t ) Hu I H   R   R
dt
Research Objectives
1.
Are the solutions (S(t), I(t), W(t), R(t)) non-negeative for all t >
0 with the initial conditions?
2. Are all solutions (S(t), I(t), W(t), R(t)) bounded? (3) Does the
system have a unique endemic equilibrium when Rv > 1?
3. What happens when Rv < 1?
4. Does any disease free equilibrium exist and if it does, then
under what conditions is it locally asymptotically stable?
5. Under what conditions the endemic equilibrium is locally and
globally stable?
References (include the papers
that I gave you if it is not included)
Cholera Prevention and Control, CDC
Cholera Outbreak – Haiti, October 2010, MMWR, CDC
"Cholera Outbreak in Haiti." - International Medical Corps. N.p., n.d. Web. 04 June 2015.
"Flooding Intensifies Cholera Outbreak in Haiti - New America Media."Flooding Intensifies Cholera Outbreak in Haiti - New America Media. N.p., n.d. Web. 04 June 2015.
Fung, Isaac Chun-Hai. "Cholera Transmission Dynamic Models for Public Health Practitioners." Emerging Themes in Epidemiology. BioMed Central, n.d. Web. 04 June 2015.
Suzanne Lenhart and John T. Workman, Optimal Control Applied to Biological Models, Chapman and Hall/CRC, pp. 1-56. New York, 2007.
Albert MJ, Neira M, Motarjemi Y., The Role of Food in the Epidemiology of Cholera, World Health Stat Q., 1997; vol. 50: 111-8. [PMID:9282393]
Rachael L. Miller Neilan, Elsa Schaefer, Holly Gaff, K. Renee Fister, and Suzanne Lenhart, Modeling Optimal Intervention Strategies for Cholera, Bulletin of Mathematical Biology, vol.
72, 2004 – 2018, 2010.
S. M. Moghadas, Modeling the Effect of Imperfect Vaccines on Disease Epidemiology, Discrete and Continuous Dynamical Systems – Series B, vol. 4, no. 4, pp. 999-1012, Nov. 2004.
Joseph H. Tien and David J.D. Earn, Multiple Transmission Pathways and Disease Dynamics in a Waterborne Pathogen Model, Bulletin of Mathematical Biology, vol. 72, 1506-1533,
2010
Update: Outbreak of Cholera --- Haiti, 2010, MMWR, CDC
Tien et al., Herald Waves of Cholera in Nineteenth Century London, Journal of the Royal Society, 2010.
Tuite et al., Cholera Epidemic in Haiti, 2010: Using a Transmission Model to Explain Spatial Spread of Disease and Identify Optimal Control Interventions, Annals of Internal Medicine,
vol. 154, pp. 593-601, May 2011.