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Mathematical modeling of diphtheria transmission in Thailand
Accepted Manuscript Mathematical modeling of diphtheria transmission in Thailand Kan Sornbundit, Wannapong Triampo, Charin Modchang PII:
S0010-4825(17)30153-1
DOI:
10.1016/j.compbiomed.2017.05.031
Reference:
CBM 2686
To appear in:
Computers in Biology and Medicine
Received Date: 31 January 2017 Revised Date:
8 May 2017
Accepted Date: 29 May 2017
Please cite this article as: K. Sornbundit, W. Triampo, C. Modchang, Mathematical modeling of diphtheria transmission in Thailand, Computers in Biology and Medicine (2017), doi: 10.1016/ j.compbiomed.2017.05.031. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Mathematical Modeling of Diphtheria Transmission in Thailand
a
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Kan Sornbundita, Wannapong Triampob,c, and Charin Modchangb,c,d,*
Faculty of Science, Energy and Environment, King Mongkut’s University of
Technology North Bangkok, Rayong campus, Rayong, 21120, Thailand
Biophysics group, Department of Physics, Faculty of Science, Mahidol University,
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b
Bangkok, 10400 Thailand
ThEP Center, CHE, 328 Si Ayutthaya Road, Bangkok 10400,Thailand
d
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c
Centre of Excellence in Mathematics, CHE, 328, Si Ayutthaya Road, Bangkok 10400,
Thailand
*
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Corresponding author. Biophysics group, Department of Physics, Faculty of Science,
Mahidol University, Bangkok, 10400 Thailand.
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Fax: +66 2354 7159
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E-mail address: [email protected] (C. Modchang).
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ACCEPTED MANUSCRIPT Abstract In this work, a mathematical model for describing diphtheria transmission in Thailand is proposed. Based on the course of diphtheria infection, the population is divided into 8 epidemiological classes, namely, susceptible, symptomatic infectious,
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asymptomatic infectious, carrier with full natural-acquired immunity, carrier with partial natural-acquired immunity, individual with full vaccine-induced immunity, and individual with partial vaccine-induced immunity. Parameter values in the model
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were either directly obtained from the literature, estimated from available data, or estimated by means of sensitivity analysis. Numerical solutions show that our model
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can correctly describe the decreasing trend of diphtheria cases in Thailand during the years 1977 – 2014.
Furthermore, despite Thailand having high DTP vaccine
coverage, our model predicts that there will be diphtheria outbreaks after the year 2014 due to waning immunity. Our model also suggests that providing booster doses
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to some susceptible individuals and those with partial immunity every 10 years is a potential way to inhibit future diphtheria outbreaks.
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immunity
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Keywords: diphtheria, mathematical model, Thailand, booster vaccination, waning
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ACCEPTED MANUSCRIPT 1. Introduction Diphtheria is an acute, toxin-mediated disease caused by the bacterium Corynebacterium diphtheria (1). Transmission between persons is caused by the spread of the bacteria from the respiratory tract of infected individuals (1). Diphtheria
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was a major disease worldwide due to its high morbidity and mortality rates, especially in persons younger than 5 and older than 40 years, with a mortality rate of up to 20% (1). However, diphtheria can be prevented using a toxoid. At present, the
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diphtheria toxoid is usually packaged with the tetanus and pertussis vaccines. This package is known as DTP and is a primary vaccine course for infants (1). After the
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World Health Organization (WHO) launched the expanded program on immunization (EPI) in 1974, the incidence of diphtheria reported worldwide has declined from 2 per 100,000 individuals in 1974 to approximately 0.2 per 100,000 individuals in 2012. Even in the post-vaccine era, diphtheria outbreaks have still been reported in
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Post-Soviet states and developing countries, including Thailand (2, 3, 7, 26). Very recently, in 2015, the first outbreak in 28 years was reported in Spain (4). Many of the incidences that occur today involve teenagers and middle-aged adults who had been
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given vaccination at birth (5). This raises questions regarding how long the vaccine
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against diphtheria confers protection and how often individuals need to receive the diphtheria toxoid.
In Thailand, diphtheria was also once considered a major disease, with
approximately 5.0 incidences per 100,000 individuals in 1971 (3, 6). The EPI in Thailand started in 1977 and resulted in the decline of confirmed cases to below 0.05 cases per 100,000 individuals at present. Despite high childhood DTP coverage (19), which has been higher than 90% since 2000, sporadic outbreaks of diphtheria have been reported repeatedly in Thailand for the last 20 years (3). The most recent
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ACCEPTED MANUSCRIPT outbreak occurred in 2012, with 41 confirmed cases and 101 carriers (7). The outbreak began with a 40-year-old in Loei Province and then spread to neighboring provinces. Unlike the outbreaks in the 1980s and 1990s, the majority of cases during the 2012 outbreak in the northeast region were middle-aged adults (15 – 44 years old).
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The antitoxin-level survey conducted in some provinces in Thailand (7) during the years 2011 – 2013 revealed the lowest antibody level in the 30-39 age group, with the 40 – 44 age group having the second lowest (7). It has been suggested that the
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immunity of people in these age groups wanes because they have not been exposed to natural C. diphtheriae (natural immunity). To eradicate outbreaks, the Ministry of
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Public Health of Thailand announced a mass diphtheria vaccination program for adults aged 20 to 50 years, which corresponds to approximately 40% of the population. At present, the WHO recommends a booster vaccine (diphtheria and tetanus vaccine, dT) every 10 years after the completion of the primary dose (the first
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three doses, DTP3) (1). However, booster dose vaccination is not commonly carried out in most of Thailand, despite the evidence in the literature studying diphtheria in Thailand that suggests booster doses for middle-aged adults (22, 26).
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One way to understand the epidemiology of diphtheria in Thailand is to use a
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mathematical model to explain its transmission dynamics. This method is commonly used for many diseases, such as pertussis and influenza A (8, 23). Unfortunately, a transmission dynamics model that is designed specifically for diphtheria has been lacking. Most of the available models were designed to describe or predict dynamics of diphtheria toxin and antitoxic antibodies in an infected individual (28, 29). None of them can be used to describe the transmission dynamics of diphtheria in the level of population.
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ACCEPTED MANUSCRIPT In this work, we propose a mathematical model for explaining the diphtheria transmission dynamics in Thailand. The aims of the model are (i) to reproduce the diphtheria cases reported by the bureau of epidemiology, Ministry of Public Health, during the years 1977 – 2014, (ii) to predict diphtheria outbreak patterns that might
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occur in the near future, and (iii) to investigate booster vaccination strategies.
2. Material and Methods
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2.1 Disease characteristics
The toxigenic bacterium produces a toxin that inhibits cellular protein
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synthesis and causes local tissue damage and pseudomembrane formation (1). The basic reproduction number of diphtheria, which can be considered as the number of cases that one infectious individual can generate in an otherwise uninfected population, is approximately 6-7 (9). The incubation period is 2-5 days and ranges
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from 1-10 days. The disease usually persists for 2-4 weeks (1, 24).
2.2 The compartmental model for diphtheria transmission and vaccination
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Based on the diphtheria course of infection, the scheme of the model is presented in Fig. 1. In this model, the population is classified into 8 epidemiological
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classes: susceptible , and , symptomatic infectious , asymptomatic infectious , carrier with full natural-acquired immunity , carrier with partial naturalacquired immunity , individual with full vaccine-induced immunity , and individual with partial vaccine-induced immunity . To simplify the analysis, we did not explicitly consider exposed states in the model. This simplification is reasonable because the addition of a latent period is essentially akin to introducing a slight time delay into the system; however, it does not change the equilibrium
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ACCEPTED MANUSCRIPT behavior if the basic reproduction number and average infectious period are identical. Moreover, since the basic reproduction number for diphtheria is approximately 6.5, this very high reproduction number will make the transmission dynamics reach
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slow natural birth and dead rates are ignored.
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equilibrium in less than one year (the shortest time between two data points) if the
Fig. 1. Schematic diagram of the model structure. The epidemiological transition
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pathways are represented by solid lines. The dotted line (from to ) represents the vaccination pathway. Newborns are always in the susceptible class, . For clarity, the natural death pathways of all compartments are not shown in the diagram.
According to the international classification for diphtheria antitoxin level, an antitoxin titer greater than 0.1 IU mL-1 is considered to be a full protective level, an antitoxin titer between 0.01 IU mL-1 and 0.1 IU mL-1 is a partial protective level, and 6
ACCEPTED MANUSCRIPT an antitoxin titer less than 0.01 IU mL-1 is unprotected (10, 24). In our model, a susceptible individual refers to an individual who has an antitoxin titer lower than 0.01 IU mL-1. An individual can be susceptible, , because he/she has never been given the vaccine or has never contracted the infection. It is known that the vaccine-
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induced antitoxin levels wane over time (6, 10–12). Thus, we defined another susceptible class, , for an individual who lost the protective level of the antitoxin. Individuals in and can be infected by individuals who are in infectious states. For
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diphtheria, an infectious population may or may not be symptomatic. Individuals who have disease symptoms, e.g., fever, sore throat and white membrane in the upper
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respiratory tract, are classified as symptomatic infectious individuals, . On the other hand, individuals who become infected but have no signs or symptoms are identified as the asymptomatic infectious individuals, . After the contagious state, infectious individuals will develop antitoxin, which is called naturally acquired antitoxin. An
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individual who recovered from the disease with a full or partial level of antitoxin is denoted as or , respectively. Although individuals who have just recovered from the disease will have antitoxin, they may also still have bacterium Corynebacterium
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diphtheria in their body. Therefore, recovered individuals can still transmit the
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pathogens to susceptible individuals. In other words, individuals in the or state are also disease carriers (7, 27). Since the EPI program in Thailand was launched in 1977, the incidences of diphtheria have been decreasing dramatically (1), as infants that were born since then have been given the DTP vaccine. Individuals with antitoxins that developed in this way are classified as having vaccine-induced antitoxins. Infants who completed the primary vaccination program (3 doses of DTP) are denoted by . The partial protective level, due to vaccine waning, of vaccineinduced antitoxins is defined in the model and is denoted as . 7
ACCEPTED MANUSCRIPT The rates of change of individuals in each epidemiological state are described by the following set of ordinary differential equations:
= + , − − = + , − −
, = + ! ,
+ +
, =− ",# $
=
! ,
−
,
− ,
! ,
− ,
− , − ,
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, =
! ,
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, = −
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= − − + 1 −
,
,
−
− , + ,
− , ,
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where % 1 ≤ % ≤ 77 indicates the % $( province in Thailand. The descriptions and values of all parameters used in the models are shown in Table 1. According to the equations mentioned above, susceptible and individuals
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can be infected through contact with and from the same or other provinces. In the 2012 diphtheria outbreak in Thailand, the confirmed symptomatic cases were 41, and
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101 carriers were reported (7). This implies that should be larger than . In the model, we chose a value of 0.8 for . The force of infection, , which is the rate at which a susceptible or individual in compartment % acquires an infection from
, , or carriers in any compartment, is defined as follows: 4
= + ,- . - + / - + 01 ,- + ,- 23 , -56
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ACCEPTED MANUSCRIPT Table 1 Epidemiological parameters and their values used in the model. Values
Basic reproduction number 7
6.5
Fraction of asymptomatic individuals from primary infection
0.8
Fraction of symptomatic individuals from primary infection
0.2
Fraction of asymptomatic individuals from secondary infection Fraction of symptomatic individuals from secondary infection
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Parameters
Fraction of recovered individuals who are protected by full natural immunity
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Fraction of recovered individuals who are protected by partial natural immunity
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0.8
0.2
0.9
0.1
Fraction of infants who receive diphtheria antitoxin at birth
See supplementary files
Rate of recovery from infection
1⁄14 day :6
Rate of vaccinated immunity loss Rate of natural immunity loss
!
1/15 × 365 IU day :6 1/15 × 365 IU day :6 0.1
The average reduction in transmissibility of and 0
10-6
The characteristic length constant C7
115 km
The death rate
1⁄70 × 365 day :6
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The average reduction in transmissibility of /
See supplementary files
Rate of booster vaccination (D )
See section 3.2
Rate of vaccination for people who have never obtained the DTP vaccine D
See section 3.2
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The birth rate
where ,- is the transmission rate from and in the F $( compartment to and in the % $( compartment. The coefficient / 0 < / < 1 indicates the average reduction in the transmissibility of . This parameter was included to ensure that asymptomatic infectious individuals have a lower ability to transmit the disease than symptomatic infectious individuals, . It has been documented that even individuals in recovered 9
ACCEPTED MANUSCRIPT states , can be carriers for the disease (7, 27). Thus, a similar concept was applied to the parameter 0 , which indicates the average reduction in the transmissibility of and compared to . The fraction of symptomatic infectious, , that changes to the recovered state with full or partial protective
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antitoxin levels is denoted by or , respectively. To the best of our knowledge, these fractions have never been reported. However, from sensitivity analysis, we found that the numerical solution of the model is not very sensitive to these
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parameters (see sensitivity analysis in the supplementary files). We therefore chose = 0.9 , which provided a reasonable fitting performance. The rate =
the infectious period of 2-3 weeks.
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1⁄14 day :6 was used for the transformation from to , which is consistence with
The rate of waning immunity has been documented in the literature (10, 13– 16). The values in these reports vary due to the different diphtheria epidemiologies
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observed in each country. In 1992, Bottiger and Petterson calculated the rate of antitoxin loss in Swedish people under 23 years of age (16) and estimated the rate of antitoxin loss to be 0.005-0.01 IU year-1. Swart and co-workers (10) published the
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loss rate that was analyzed from 1995/1996 and 2006/2007 serosurveys in the Netherlands. The estimated rates of antitoxin loss for the years 1995/1996 and
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2006/2007 were -1.20 and -1.19 lnIU⁄ml per lnyear , respectively. Another report by Amanna IL et al. found that the rate was approximately 1⁄30 IU year-1(13). Very recently, Hammarland et al. estimated the duration of seroprotective antitoxins (> 0.01 IU mL-1) and obtained a value of 42 years (14), which is equivalent to the waning rate of 1⁄42 IU year-1.
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ACCEPTED MANUSCRIPT All infants are required to complete primary doses of DTP. However, only a fraction ( ) of infants receive the completed doses. In Thailand, the childhood vaccine coverage has been higher than 90% since 1990 (17).
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2.3 Contact matrix and transmission matrix
The contact matrix, K = LM- N, describes the number of contacts that may
M- =
O# OP
S#P W V TU
6QR
,
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occur for any two individuals in the % $( and F $( provinces. It is given by
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where XY∈,- is the population density in province [ (the population in province [ divided by the area of province [). The distance between the provinces indexed as % and F is described by the distance matrix \- . The characteristic length scale of the distances is C7 = 115 km. The contact matrix is built in this way because diphtheria
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can be transmitted faster in a dense population. Further, the ability of transmission should decrease with distance because the disease requires close contact for transmission. The transmission matrix, _, is obtained by multiplying the contact
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matrix with the scaling factor `, as follows: _ = `K.
The scaling factor can be obtained by using the following expression: 7 = abc1d%efghbijf`K⁄ 2,
where 7 is the basic reproduction number, and k = _⁄ = `K⁄ is the next generation matrix (18). Values of population density XY , and ` are summarized in the supplementary files.
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2.4 DTP vaccination schedules and coverage in Thailand In 1977, a nationwide vaccine program for infants was launched in Thailand. The DTP vaccine recommendation started with 2 doses. In 1982, the recommended
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number was 3 doses. From 1992 to the present, the recommended number rose to 5 doses, given at ages 2, 4, 6, and 18 months and 4-6 years (1). Since 2008, a diphtheria and tetanus toxoid (dT) booster dose has been introduced to 12-year-old children who
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had received the first 5 doses before leaving primary school (21). The DTP coverage rate was at 12.3% in 1980 but rose dramatically to 92.2% in 1993 (19). Since 2003,
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the coverage rates have been as high as 98% (19). Note that the DTP coverage rates mentioned above refer to 1-3 doses of DTP (DTP1-3), which is considered as a primary dose. As such, individuals under 23 years should have acquired 5 doses of DTP (coverage of DTP5 is 54.5-90.3%), whereas individuals aged 24-37 years should
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have acquired 2-4 doses (coverage of DTP3 and DTP2 are 45.3% and 12.3%) (19). Persons aged more than 37 years did not received any doses because they were born
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before the initiation of the EPI program.
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2.5 Population and initial conditions The population birth and death rates were chosen to reflect Thailand’s
population between 1960 and 2010. The death rate used in the model is = 1⁄70 × 365 day :6 , which reflects a life span of 70 years. The birth rates vary over time and are summarized in the supplementary files. The simulations were initiated by introducing 10 incidents in 6 different provinces, province IDs = 2, 24, 73, 74, 75 and 76 (Fig 2). These provinces usually report the greatest number of incidences. Note that although the number of initial infectious individuals (60 in our work) was
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ACCEPTED MANUSCRIPT arbitrarily chosen, this will not affect the simulation results. This is because the system was allowed to evolve and reach equilibrium before we started simulation at the year 1977 (see sensitivity analysis in the supplementary files). Initially, the systems included only susceptible (), infectious ( ) and asymptomatic infectious ( )
with the constant population of 1960.
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individuals. Then, the system was allowed to evolve to reach equilibrium for 50 years,
Provinces 24 and 2 have borders with Myanmar and Lao PDR, respectively.
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Interactions between Thais people and individuals in these two countries may lead to infection because these countries may have lower vaccine coverage than Thailand,
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and Thais people in some of these regions have relatively low immunity compared to the average (26). In contrast, the southern part of Thailand has a border with Malaysia, which has high vaccine coverage. Outbreaks in this region may come from
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incomplete vaccination distribution.
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ACCEPTED MANUSCRIPT Fig 2. A map of Thailand indicating the positions of province IDs = 2, 24, 73, 74, 75 and 76 (20).
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3. Results
3.1 Mathematical model for diphtheria transmission in Thailand
The proposed mathematical equations for describing diphtheria transmission
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in Thailand were solved numerically. The simulations were divided into two time intervals, the years 1977 – 2014 and 2015 - 2060. The purpose of the simulations in
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the first time interval was to compare the number of incidences simulated from our model with the number of reported cases. The goal of the simulations in the second time interval was to predict whether there would be an outbreak in the near future. The simulations in the second time interval were conducted by allowing the model to
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evolve after the year 2014 with the same population growth rate as in the year 2014. A comparison of the number of incidences per 100,000 individuals from simulation and reported cases is shown in Fig. 3. The fractions of population for each
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epidemiological state are shown in Fig. 4.
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Fig. 3 Incidence per 100,000 individuals from reports, simulation and vaccine uptake (right y-axis) in Thailand from 1978 to 2060. This simulation predicts two outbreaks
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during the years 2014 - 2060.
Fig. 4 Fractions of susceptible , secondary susceptible , carrier with full natural-acquired immunity , carrier with partial natural-acquired immunity ,
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ACCEPTED MANUSCRIPT individual with full vaccine-induced immunity , individual with partial vaccineinduced immunity , infectious and asymptomatic infectious ) from 1978 to 2060.
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The internationally-accepted classifications of diphtheria protection levels are (1) susceptible (antitoxin titer < 0.01 IU ml-1), (2) basic protection (antitoxin titer 0.01-0.1 IU ml-1) and (3) full protection (antitoxin titer > 0.1 IU ml-1). Fig. 5 shows
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the fractions of each protective level. We defined (i) full protection as a combination of and , (ii) basic protection as a combination of and and (iii) susceptible
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(no protection) as a combination of and .
Fig. 5 Distribution of antitoxin level: No protection, basic protection and full protection of the total population from 1977 to 2060.
3.2 Booster vaccination The WHO recommendation for diphtheria booster frequency is every 10 years after completing the primary doses. Ten-year booster vaccination frequencies were tested with our model and were administered in the years 2014, 2024, 2034, 2044 and 16
ACCEPTED MANUSCRIPT 2054. Note that the nationwide booster vaccination in Thailand was done in the year 2014 for adults aged 20 – 50 years. Individuals within that age group were targeted because they have lower antitoxin levels than others. In this work, the booster doses were provided randomly to people regardless of their epidemiological state, which is
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the most practical way to operate a vaccination campaign. This strategy was implemented in our model by giving the booster vaccination to , and at rate D and giving the booster vaccination to
at rate D (see equations in
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supplementary files). People in the S state, who never had the DTP vaccination, needed to receive 3 shots of the primary doses within a year. The rate of vaccination
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for this class is denoted by D . The booster doses were given to targeted classes at equal rates, D = D . The duration of each vaccination is one year. The results of implementing the booster vaccination with different numbers of booster doses are shown in Fig. 6, and the corresponding fractions of protective levels are shown in Fig.
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7. The numbers of booster doses and the corresponding rates of booster vaccination
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used in the simulations are shown in Table 2.
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Fig. 6 The simulation results when the booster vaccination strategy was implemented. The number of incidences per 105 individuals are shown for five booster vaccination scenarios: No booster, scenario 1, scenario 2, scenario 3 and scenario 4. The rate of
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booster vaccination and the corresponding numbers of the booster doses provided in
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the years 2014, 2024, 2034, 2044 and 2054 for each scenario are shown in Table 2.
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Fig. 7 Distribution of antitoxin level: No protection, basic protection and full protection of the total population from 1977 to 2060. The results are obtained by implementing booster vaccination scenario 2.
Table 2. The rate of booster vaccination (lm ) and the corresponding numbers of
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booster doses provided in each booster vaccination scenario. lm
2024
2034
2044
2054
no × pqr −n
4.9
5
5.3
5.7
6.1
nr × pqr −n
3.3
3.5
3.6
3.9
4.2
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scenario 2
2014
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scenario 1
Numbers of booster doses (million)
scenario 3
so × pqr −n
2.5
2.6
2.8
3
3.2
scenario 4
po × pqr −n
1.7
1.8
1.9
2
2.2
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ACCEPTED MANUSCRIPT 4. Discussion In this work, we proposed a mathematical model for describing diphtheria transmission in Thailand. The aims of the simulations were to describe diphtheria transmission in Thailand during the years 1977 – 2014, to predict possible outbreaks
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in the near future, and to investigate the impact of providing booster vaccinations every 10 years. Unlike previous models (28-30), our model was designed specifically for diphtheria transmission dynamics in the level of population.
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The results in Fig. 3 shows that the number of reported cases has decreased dramatically since the introduction of the national vaccine program in 1977. After
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1995, few confirmed cases have been reported each year, which indicates the efficacy of the national DTP vaccine program. However, the sporadic outbreaks that occurred in 1994, 1996, 1999 and 2012 suggest the need for vaccine program improvement to fully eradicate diphtheria in Thailand (3, 7, 26). The results from the simulations
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demonstrate a very similar trend in diphtheria case reduction. Therefore, when our model is used together with the estimated parameter values, it could help us better understand diphtheria transmission in Thailand. The simulations also predict
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outbreaks in approximately the years 2020 and 2045 (Fig 3), which would be caused
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by an increase in the number of secondary susceptible individuals ( ) due to immunization waning (Fig 4a). This result suggests the need for vaccination boosting in adults, as recommend by the WHO. Additionally, and exhibited a downward trend since the immunization program was launched due to a decrease in infectious individuals (Fig 4b). The vaccinated classes and increased rapidly between 1980 and 2000 due to the larger infant vaccination coverage. However, the growth rate slowed after 2005 because the births rates were lower and vaccine administration
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ACCEPTED MANUSCRIPT reached its saturation. The number of infectious individuals ( ) and asymptomatic infectious individuals ( ) revealed the same downward trends (Fig 4c). In Fig. 5, the fractions of protective levels (basic and full) are lower than that of no protection. However, when the vaccine uptake is at a higher level (after the year
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2000), the fractions of protective levels are also higher. This is because most of the newborns entering the system are transformed to , and only a very small portion will remain in . In 2016, for example, the fraction of individuals with no protective
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level was approximately 0.60, which is still high. Thus, there is a risk for outbreaks to occur. This may explain why sporadic outbreaks have been detected in Thailand,
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despite the high vaccination rate. Our results also indicate that the fraction of individuals with full protective levels is higher than that with basic protective levels, which is consistent with Wanlapakorn et al. (7) and Bansiddhi et al. (6). Our results suggest that most Thais are still susceptible to diphtheria, which leads to the
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possibility of future outbreaks. In terms of herd immunity, the value of 1 − 1⁄ 7 can be used to estimate the required minimum fraction of the population with a protective immunity level to prevent an outbreak (25). For 7 = 6.5, the herd immunity was
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approximately 85%. In Fig. 5, the portion of the population with a protective
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immunity level in the year 2015 was approximately 40%. Therefore, it is reasonable that our simulation predicted outbreaks after the year 2015. To investigate the booster vaccination strategy, we implemented the booster
vaccinations every 10 years in our model. It is clearly demonstrated in Fig. 6 that providing appropriate booster doses every 10 years (Table 2) can prevent subsequent outbreaks. The results in Fig. 7 show that the protective antitoxin levels (basic or full) in the population increased in the year that the booster doses were provided.
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ACCEPTED MANUSCRIPT 5. Conclusions We have proposed a mathematical model for describing diphtheria epidemiology in Thailand after the EPI program was launched in 1977. The model considers Thailand to be a system of 77 compartments, which corresponds to its 77
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provinces. The populations in each province are further classified into 8 epidemiological classes: primary susceptible ( ), symptomatic infectious ( ), asymptomatic infectious ( ), carrier with full natural-acquired immunity ( ), carrier
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with partial natural-acquire immunity ( ), secondary susceptible ( ), individual with full vaccine-induced immunity ( ) and individual with partial vaccine-induced
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immunity ( ). The force of infection is also proposed in a way that corresponds to the nature of diphtheria transmission.
The infectious cases that resulted in our model before the year 2014 are consistent with real cases from reports. The model suggested that there would be
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recurring outbreaks after the year 2014 because a large portion of the population is susceptible to diphtheria. Those susceptible are secondarily susceptible because they lost the toxin antibody. The rates of antibody loss are estimated to be
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1⁄15 × 365 day :6 , which is in agreement with the literature [14]. Booster
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vaccinations every 10 years were shown to be sufficient for eradicating diphtheria in Thailand.
Conflict of Interests
The authors declare that there are no conflicts of interest.
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ACCEPTED MANUSCRIPT Acknowledgments This project was financially supported by the Health Systems Research Institute, Thailand, the National Science and Technology Development Agency, Thailand, and the Thailand Research Fund and Mahidol University (Grant No. TRG5880157). The
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funders had no role in the study design or preparation of the manuscript.
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epidemiology in Thailand. Proc Natl AcadSci U S A. 2013;110(23):9595–600. Eradication GS. History and Epidemiology of Global Smallpox. CDC World Heal Organ. 2011;17.
Swart EM, van Gageldonk PGM, de Melker HE, van der Klis FR, Berbers
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GAM, Mollema L. Long-Term Protection against Diphtheria in the
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Netherlands after 50 Years of Vaccination: Results from a Seroepidemiological Study. PLoS One. 2016;11(2):e0148605. 11.
Kurugöl Z, Midyat L, Türkoĝlu E, Işler A. Immunity against diphtheria among children and adults in Izmir, Turkey. Vaccine. 2011;29(26):4341–4. Wu Y, Gao Y, Zhu B, Zhou H, Shi Z, Wang J, et al. Antitoxins for diphtheria
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and tetanus decline more slowly after vaccination with DTwP than with DTaP: A study in a Chinese population. Vaccine. Elsevier Ltd; 2014;32(22):2570–3. Amanna IJ, Carlson NE, Slifka MK. Duration of Humoral Immunity to
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Common Viral and Vaccine Antigens. 2007; 14.
Hammarlund E, Thomas A, Poore EA, Amanna IJ, Rynko AE, Mori M, et al. Durability of Vaccine-Induced Immunity Against Tetanus and Diphtheria Toxins: A Cross-sectional Analysis. Clin Infect Dis. 2016;62(9):1111–8.
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Hasselhorn H, Matthias N, Tiller FW, Hofmann F. Factors influencing immunity against diphtheria in adults. 1998;16(1):70–5.
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Böttiger M, Pettersson G. Vaccine Immunity to Diphtheria: A 20-Year Followup Study. Scand J Infect Dis. 1992;24(6):753–8.
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Bcg C. Chad: WHO and UNICEF estimates of immunization coverage: 2013 revision. 2015;1–20.
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2010;7(November 2009):873–85.
Wanlapakorn N, Ngaovithunvong V, Thongmee T, Vichaiwattana P, Vongpunsawad S, Poovorawan Y. Seroprevalence of Antibodies to Pertussis
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Toxin among Different Age Groups in Thailand after 37 Years of Universal Whole-Cell Pertussis Vaccination. PLoS One. 2016;11(2):1–9. The
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modified
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from
https://en.wikipedia.org/wiki/Provinces_of_Thailand#/media/File:Thailand_ad a_location_map.svg 21
Outbreak, surveillance and investigation report, Ministry of public health
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Thailand, March 2015; 8(1): 13-21 Wiboonchutikul,
S.,
Manosuthi,
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Sangsajja,
C.,
Thientong,
V.,
Likanonsakul, S., Srisopha, S., …Puthavathana, P. (2014). Baseline immunity
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to diphtheria and immunologic response after booster vaccination with reduced
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diphtheria and tetanus toxoid vaccine in Thai health care workers. American Journal of Infection Control, 42(7), e81–e83.
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Modchang, C., Iamsirithaworn, S., Auewarakul, P., &Triampo, W. (2012). A modeling study of school closure to reduce influenza transmission: A case
study of an influenza A (H1N1) outbreak in a private Thai school. Mathematical and Computer Modelling, 55(3–4), 1021–1033 24
Begg, N. (1994). Diphtheria. Manual for the Management and control of Diphtheria in the European Region. World Health Organisation.
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Serological Study of Re-emerging Diphtheria in Dansai District , Loei Province , Thailand , June to October 2012, 8(1), 13–21. 27
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antibody levels and the outcome of illness during an outbreak of diphtheria among alcoholics. Scand J Infect Dis, 18(3), 235–239.
Kolibo, D.V., Romanyuk, S.I. (2001). Mathematical model of the infection
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Cheuvart, B., Burgess, M., Zepp, F., Mertsola, J., Wolter, J., Schuerman, L.
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(2004). Anti-diphtheria antibody seroprotection rates are similar 10 years after vaccination with dTpa or DTPa using a mathematical model. Vaccine, 23(3), 336-342.
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S0010-4825(17)30153-1
DOI:
10.1016/j.compbiomed.2017.05.031
Reference:
CBM 2686
To appear in:
Computers in Biology and Medicine
Received Date: 31 January 2017 Revised Date:
8 May 2017
Accepted Date: 29 May 2017
Please cite this article as: K. Sornbundit, W. Triampo, C. Modchang, Mathematical modeling of diphtheria transmission in Thailand, Computers in Biology and Medicine (2017), doi: 10.1016/ j.compbiomed.2017.05.031. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Mathematical Modeling of Diphtheria Transmission in Thailand
a
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Kan Sornbundita, Wannapong Triampob,c, and Charin Modchangb,c,d,*
Faculty of Science, Energy and Environment, King Mongkut’s University of
Technology North Bangkok, Rayong campus, Rayong, 21120, Thailand
Biophysics group, Department of Physics, Faculty of Science, Mahidol University,
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b
Bangkok, 10400 Thailand
ThEP Center, CHE, 328 Si Ayutthaya Road, Bangkok 10400,Thailand
d
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c
Centre of Excellence in Mathematics, CHE, 328, Si Ayutthaya Road, Bangkok 10400,
Thailand
*
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Corresponding author. Biophysics group, Department of Physics, Faculty of Science,
Mahidol University, Bangkok, 10400 Thailand.
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Fax: +66 2354 7159
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E-mail address: [email protected] (C. Modchang).
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ACCEPTED MANUSCRIPT Abstract In this work, a mathematical model for describing diphtheria transmission in Thailand is proposed. Based on the course of diphtheria infection, the population is divided into 8 epidemiological classes, namely, susceptible, symptomatic infectious,
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asymptomatic infectious, carrier with full natural-acquired immunity, carrier with partial natural-acquired immunity, individual with full vaccine-induced immunity, and individual with partial vaccine-induced immunity. Parameter values in the model
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were either directly obtained from the literature, estimated from available data, or estimated by means of sensitivity analysis. Numerical solutions show that our model
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can correctly describe the decreasing trend of diphtheria cases in Thailand during the years 1977 – 2014.
Furthermore, despite Thailand having high DTP vaccine
coverage, our model predicts that there will be diphtheria outbreaks after the year 2014 due to waning immunity. Our model also suggests that providing booster doses
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to some susceptible individuals and those with partial immunity every 10 years is a potential way to inhibit future diphtheria outbreaks.
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immunity
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Keywords: diphtheria, mathematical model, Thailand, booster vaccination, waning
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ACCEPTED MANUSCRIPT 1. Introduction Diphtheria is an acute, toxin-mediated disease caused by the bacterium Corynebacterium diphtheria (1). Transmission between persons is caused by the spread of the bacteria from the respiratory tract of infected individuals (1). Diphtheria
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was a major disease worldwide due to its high morbidity and mortality rates, especially in persons younger than 5 and older than 40 years, with a mortality rate of up to 20% (1). However, diphtheria can be prevented using a toxoid. At present, the
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diphtheria toxoid is usually packaged with the tetanus and pertussis vaccines. This package is known as DTP and is a primary vaccine course for infants (1). After the
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World Health Organization (WHO) launched the expanded program on immunization (EPI) in 1974, the incidence of diphtheria reported worldwide has declined from 2 per 100,000 individuals in 1974 to approximately 0.2 per 100,000 individuals in 2012. Even in the post-vaccine era, diphtheria outbreaks have still been reported in
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Post-Soviet states and developing countries, including Thailand (2, 3, 7, 26). Very recently, in 2015, the first outbreak in 28 years was reported in Spain (4). Many of the incidences that occur today involve teenagers and middle-aged adults who had been
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given vaccination at birth (5). This raises questions regarding how long the vaccine
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against diphtheria confers protection and how often individuals need to receive the diphtheria toxoid.
In Thailand, diphtheria was also once considered a major disease, with
approximately 5.0 incidences per 100,000 individuals in 1971 (3, 6). The EPI in Thailand started in 1977 and resulted in the decline of confirmed cases to below 0.05 cases per 100,000 individuals at present. Despite high childhood DTP coverage (19), which has been higher than 90% since 2000, sporadic outbreaks of diphtheria have been reported repeatedly in Thailand for the last 20 years (3). The most recent
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ACCEPTED MANUSCRIPT outbreak occurred in 2012, with 41 confirmed cases and 101 carriers (7). The outbreak began with a 40-year-old in Loei Province and then spread to neighboring provinces. Unlike the outbreaks in the 1980s and 1990s, the majority of cases during the 2012 outbreak in the northeast region were middle-aged adults (15 – 44 years old).
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The antitoxin-level survey conducted in some provinces in Thailand (7) during the years 2011 – 2013 revealed the lowest antibody level in the 30-39 age group, with the 40 – 44 age group having the second lowest (7). It has been suggested that the
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immunity of people in these age groups wanes because they have not been exposed to natural C. diphtheriae (natural immunity). To eradicate outbreaks, the Ministry of
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Public Health of Thailand announced a mass diphtheria vaccination program for adults aged 20 to 50 years, which corresponds to approximately 40% of the population. At present, the WHO recommends a booster vaccine (diphtheria and tetanus vaccine, dT) every 10 years after the completion of the primary dose (the first
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three doses, DTP3) (1). However, booster dose vaccination is not commonly carried out in most of Thailand, despite the evidence in the literature studying diphtheria in Thailand that suggests booster doses for middle-aged adults (22, 26).
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One way to understand the epidemiology of diphtheria in Thailand is to use a
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mathematical model to explain its transmission dynamics. This method is commonly used for many diseases, such as pertussis and influenza A (8, 23). Unfortunately, a transmission dynamics model that is designed specifically for diphtheria has been lacking. Most of the available models were designed to describe or predict dynamics of diphtheria toxin and antitoxic antibodies in an infected individual (28, 29). None of them can be used to describe the transmission dynamics of diphtheria in the level of population.
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ACCEPTED MANUSCRIPT In this work, we propose a mathematical model for explaining the diphtheria transmission dynamics in Thailand. The aims of the model are (i) to reproduce the diphtheria cases reported by the bureau of epidemiology, Ministry of Public Health, during the years 1977 – 2014, (ii) to predict diphtheria outbreak patterns that might
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occur in the near future, and (iii) to investigate booster vaccination strategies.
2. Material and Methods
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2.1 Disease characteristics
The toxigenic bacterium produces a toxin that inhibits cellular protein
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synthesis and causes local tissue damage and pseudomembrane formation (1). The basic reproduction number of diphtheria, which can be considered as the number of cases that one infectious individual can generate in an otherwise uninfected population, is approximately 6-7 (9). The incubation period is 2-5 days and ranges
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from 1-10 days. The disease usually persists for 2-4 weeks (1, 24).
2.2 The compartmental model for diphtheria transmission and vaccination
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Based on the diphtheria course of infection, the scheme of the model is presented in Fig. 1. In this model, the population is classified into 8 epidemiological
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classes: susceptible , and , symptomatic infectious , asymptomatic infectious , carrier with full natural-acquired immunity , carrier with partial naturalacquired immunity , individual with full vaccine-induced immunity , and individual with partial vaccine-induced immunity . To simplify the analysis, we did not explicitly consider exposed states in the model. This simplification is reasonable because the addition of a latent period is essentially akin to introducing a slight time delay into the system; however, it does not change the equilibrium
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ACCEPTED MANUSCRIPT behavior if the basic reproduction number and average infectious period are identical. Moreover, since the basic reproduction number for diphtheria is approximately 6.5, this very high reproduction number will make the transmission dynamics reach
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slow natural birth and dead rates are ignored.
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equilibrium in less than one year (the shortest time between two data points) if the
Fig. 1. Schematic diagram of the model structure. The epidemiological transition
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pathways are represented by solid lines. The dotted line (from to ) represents the vaccination pathway. Newborns are always in the susceptible class, . For clarity, the natural death pathways of all compartments are not shown in the diagram.
According to the international classification for diphtheria antitoxin level, an antitoxin titer greater than 0.1 IU mL-1 is considered to be a full protective level, an antitoxin titer between 0.01 IU mL-1 and 0.1 IU mL-1 is a partial protective level, and 6
ACCEPTED MANUSCRIPT an antitoxin titer less than 0.01 IU mL-1 is unprotected (10, 24). In our model, a susceptible individual refers to an individual who has an antitoxin titer lower than 0.01 IU mL-1. An individual can be susceptible, , because he/she has never been given the vaccine or has never contracted the infection. It is known that the vaccine-
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induced antitoxin levels wane over time (6, 10–12). Thus, we defined another susceptible class, , for an individual who lost the protective level of the antitoxin. Individuals in and can be infected by individuals who are in infectious states. For
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diphtheria, an infectious population may or may not be symptomatic. Individuals who have disease symptoms, e.g., fever, sore throat and white membrane in the upper
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respiratory tract, are classified as symptomatic infectious individuals, . On the other hand, individuals who become infected but have no signs or symptoms are identified as the asymptomatic infectious individuals, . After the contagious state, infectious individuals will develop antitoxin, which is called naturally acquired antitoxin. An
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individual who recovered from the disease with a full or partial level of antitoxin is denoted as or , respectively. Although individuals who have just recovered from the disease will have antitoxin, they may also still have bacterium Corynebacterium
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diphtheria in their body. Therefore, recovered individuals can still transmit the
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pathogens to susceptible individuals. In other words, individuals in the or state are also disease carriers (7, 27). Since the EPI program in Thailand was launched in 1977, the incidences of diphtheria have been decreasing dramatically (1), as infants that were born since then have been given the DTP vaccine. Individuals with antitoxins that developed in this way are classified as having vaccine-induced antitoxins. Infants who completed the primary vaccination program (3 doses of DTP) are denoted by . The partial protective level, due to vaccine waning, of vaccineinduced antitoxins is defined in the model and is denoted as . 7
ACCEPTED MANUSCRIPT The rates of change of individuals in each epidemiological state are described by the following set of ordinary differential equations:
= + , − − = + , − −
, = + ! ,
+ +
, =− ",# $
=
! ,
−
,
− ,
! ,
− ,
− , − ,
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, =
! ,
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, = −
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= − − + 1 −
,
,
−
− , + ,
− , ,
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where % 1 ≤ % ≤ 77 indicates the % $( province in Thailand. The descriptions and values of all parameters used in the models are shown in Table 1. According to the equations mentioned above, susceptible and individuals
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can be infected through contact with and from the same or other provinces. In the 2012 diphtheria outbreak in Thailand, the confirmed symptomatic cases were 41, and
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101 carriers were reported (7). This implies that should be larger than . In the model, we chose a value of 0.8 for . The force of infection, , which is the rate at which a susceptible or individual in compartment % acquires an infection from
, , or carriers in any compartment, is defined as follows: 4
= + ,- . - + / - + 01 ,- + ,- 23 , -56
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ACCEPTED MANUSCRIPT Table 1 Epidemiological parameters and their values used in the model. Values
Basic reproduction number 7
6.5
Fraction of asymptomatic individuals from primary infection
0.8
Fraction of symptomatic individuals from primary infection
0.2
Fraction of asymptomatic individuals from secondary infection Fraction of symptomatic individuals from secondary infection
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Parameters
Fraction of recovered individuals who are protected by full natural immunity
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Fraction of recovered individuals who are protected by partial natural immunity
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0.8
0.2
0.9
0.1
Fraction of infants who receive diphtheria antitoxin at birth
See supplementary files
Rate of recovery from infection
1⁄14 day :6
Rate of vaccinated immunity loss Rate of natural immunity loss
!
1/15 × 365 IU day :6 1/15 × 365 IU day :6 0.1
The average reduction in transmissibility of and 0
10-6
The characteristic length constant C7
115 km
The death rate
1⁄70 × 365 day :6
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The average reduction in transmissibility of /
See supplementary files
Rate of booster vaccination (D )
See section 3.2
Rate of vaccination for people who have never obtained the DTP vaccine D
See section 3.2
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The birth rate
where ,- is the transmission rate from and in the F $( compartment to and in the % $( compartment. The coefficient / 0 < / < 1 indicates the average reduction in the transmissibility of . This parameter was included to ensure that asymptomatic infectious individuals have a lower ability to transmit the disease than symptomatic infectious individuals, . It has been documented that even individuals in recovered 9
ACCEPTED MANUSCRIPT states , can be carriers for the disease (7, 27). Thus, a similar concept was applied to the parameter 0 , which indicates the average reduction in the transmissibility of and compared to . The fraction of symptomatic infectious, , that changes to the recovered state with full or partial protective
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antitoxin levels is denoted by or , respectively. To the best of our knowledge, these fractions have never been reported. However, from sensitivity analysis, we found that the numerical solution of the model is not very sensitive to these
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parameters (see sensitivity analysis in the supplementary files). We therefore chose = 0.9 , which provided a reasonable fitting performance. The rate =
the infectious period of 2-3 weeks.
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1⁄14 day :6 was used for the transformation from to , which is consistence with
The rate of waning immunity has been documented in the literature (10, 13– 16). The values in these reports vary due to the different diphtheria epidemiologies
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observed in each country. In 1992, Bottiger and Petterson calculated the rate of antitoxin loss in Swedish people under 23 years of age (16) and estimated the rate of antitoxin loss to be 0.005-0.01 IU year-1. Swart and co-workers (10) published the
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loss rate that was analyzed from 1995/1996 and 2006/2007 serosurveys in the Netherlands. The estimated rates of antitoxin loss for the years 1995/1996 and
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2006/2007 were -1.20 and -1.19 lnIU⁄ml per lnyear , respectively. Another report by Amanna IL et al. found that the rate was approximately 1⁄30 IU year-1(13). Very recently, Hammarland et al. estimated the duration of seroprotective antitoxins (> 0.01 IU mL-1) and obtained a value of 42 years (14), which is equivalent to the waning rate of 1⁄42 IU year-1.
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ACCEPTED MANUSCRIPT All infants are required to complete primary doses of DTP. However, only a fraction ( ) of infants receive the completed doses. In Thailand, the childhood vaccine coverage has been higher than 90% since 1990 (17).
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2.3 Contact matrix and transmission matrix
The contact matrix, K = LM- N, describes the number of contacts that may
M- =
O# OP
S#P W V TU
6QR
,
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occur for any two individuals in the % $( and F $( provinces. It is given by
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where XY∈,- is the population density in province [ (the population in province [ divided by the area of province [). The distance between the provinces indexed as % and F is described by the distance matrix \- . The characteristic length scale of the distances is C7 = 115 km. The contact matrix is built in this way because diphtheria
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can be transmitted faster in a dense population. Further, the ability of transmission should decrease with distance because the disease requires close contact for transmission. The transmission matrix, _, is obtained by multiplying the contact
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matrix with the scaling factor `, as follows: _ = `K.
The scaling factor can be obtained by using the following expression: 7 = abc1d%efghbijf`K⁄ 2,
where 7 is the basic reproduction number, and k = _⁄ = `K⁄ is the next generation matrix (18). Values of population density XY , and ` are summarized in the supplementary files.
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2.4 DTP vaccination schedules and coverage in Thailand In 1977, a nationwide vaccine program for infants was launched in Thailand. The DTP vaccine recommendation started with 2 doses. In 1982, the recommended
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number was 3 doses. From 1992 to the present, the recommended number rose to 5 doses, given at ages 2, 4, 6, and 18 months and 4-6 years (1). Since 2008, a diphtheria and tetanus toxoid (dT) booster dose has been introduced to 12-year-old children who
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had received the first 5 doses before leaving primary school (21). The DTP coverage rate was at 12.3% in 1980 but rose dramatically to 92.2% in 1993 (19). Since 2003,
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the coverage rates have been as high as 98% (19). Note that the DTP coverage rates mentioned above refer to 1-3 doses of DTP (DTP1-3), which is considered as a primary dose. As such, individuals under 23 years should have acquired 5 doses of DTP (coverage of DTP5 is 54.5-90.3%), whereas individuals aged 24-37 years should
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have acquired 2-4 doses (coverage of DTP3 and DTP2 are 45.3% and 12.3%) (19). Persons aged more than 37 years did not received any doses because they were born
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before the initiation of the EPI program.
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2.5 Population and initial conditions The population birth and death rates were chosen to reflect Thailand’s
population between 1960 and 2010. The death rate used in the model is = 1⁄70 × 365 day :6 , which reflects a life span of 70 years. The birth rates vary over time and are summarized in the supplementary files. The simulations were initiated by introducing 10 incidents in 6 different provinces, province IDs = 2, 24, 73, 74, 75 and 76 (Fig 2). These provinces usually report the greatest number of incidences. Note that although the number of initial infectious individuals (60 in our work) was
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ACCEPTED MANUSCRIPT arbitrarily chosen, this will not affect the simulation results. This is because the system was allowed to evolve and reach equilibrium before we started simulation at the year 1977 (see sensitivity analysis in the supplementary files). Initially, the systems included only susceptible (), infectious ( ) and asymptomatic infectious ( )
with the constant population of 1960.
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individuals. Then, the system was allowed to evolve to reach equilibrium for 50 years,
Provinces 24 and 2 have borders with Myanmar and Lao PDR, respectively.
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Interactions between Thais people and individuals in these two countries may lead to infection because these countries may have lower vaccine coverage than Thailand,
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and Thais people in some of these regions have relatively low immunity compared to the average (26). In contrast, the southern part of Thailand has a border with Malaysia, which has high vaccine coverage. Outbreaks in this region may come from
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incomplete vaccination distribution.
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ACCEPTED MANUSCRIPT Fig 2. A map of Thailand indicating the positions of province IDs = 2, 24, 73, 74, 75 and 76 (20).
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3. Results
3.1 Mathematical model for diphtheria transmission in Thailand
The proposed mathematical equations for describing diphtheria transmission
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in Thailand were solved numerically. The simulations were divided into two time intervals, the years 1977 – 2014 and 2015 - 2060. The purpose of the simulations in
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the first time interval was to compare the number of incidences simulated from our model with the number of reported cases. The goal of the simulations in the second time interval was to predict whether there would be an outbreak in the near future. The simulations in the second time interval were conducted by allowing the model to
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evolve after the year 2014 with the same population growth rate as in the year 2014. A comparison of the number of incidences per 100,000 individuals from simulation and reported cases is shown in Fig. 3. The fractions of population for each
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epidemiological state are shown in Fig. 4.
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Fig. 3 Incidence per 100,000 individuals from reports, simulation and vaccine uptake (right y-axis) in Thailand from 1978 to 2060. This simulation predicts two outbreaks
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during the years 2014 - 2060.
Fig. 4 Fractions of susceptible , secondary susceptible , carrier with full natural-acquired immunity , carrier with partial natural-acquired immunity ,
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ACCEPTED MANUSCRIPT individual with full vaccine-induced immunity , individual with partial vaccineinduced immunity , infectious and asymptomatic infectious ) from 1978 to 2060.
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The internationally-accepted classifications of diphtheria protection levels are (1) susceptible (antitoxin titer < 0.01 IU ml-1), (2) basic protection (antitoxin titer 0.01-0.1 IU ml-1) and (3) full protection (antitoxin titer > 0.1 IU ml-1). Fig. 5 shows
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the fractions of each protective level. We defined (i) full protection as a combination of and , (ii) basic protection as a combination of and and (iii) susceptible
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(no protection) as a combination of and .
Fig. 5 Distribution of antitoxin level: No protection, basic protection and full protection of the total population from 1977 to 2060.
3.2 Booster vaccination The WHO recommendation for diphtheria booster frequency is every 10 years after completing the primary doses. Ten-year booster vaccination frequencies were tested with our model and were administered in the years 2014, 2024, 2034, 2044 and 16
ACCEPTED MANUSCRIPT 2054. Note that the nationwide booster vaccination in Thailand was done in the year 2014 for adults aged 20 – 50 years. Individuals within that age group were targeted because they have lower antitoxin levels than others. In this work, the booster doses were provided randomly to people regardless of their epidemiological state, which is
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the most practical way to operate a vaccination campaign. This strategy was implemented in our model by giving the booster vaccination to , and at rate D and giving the booster vaccination to
at rate D (see equations in
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supplementary files). People in the S state, who never had the DTP vaccination, needed to receive 3 shots of the primary doses within a year. The rate of vaccination
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for this class is denoted by D . The booster doses were given to targeted classes at equal rates, D = D . The duration of each vaccination is one year. The results of implementing the booster vaccination with different numbers of booster doses are shown in Fig. 6, and the corresponding fractions of protective levels are shown in Fig.
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7. The numbers of booster doses and the corresponding rates of booster vaccination
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used in the simulations are shown in Table 2.
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Fig. 6 The simulation results when the booster vaccination strategy was implemented. The number of incidences per 105 individuals are shown for five booster vaccination scenarios: No booster, scenario 1, scenario 2, scenario 3 and scenario 4. The rate of
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booster vaccination and the corresponding numbers of the booster doses provided in
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the years 2014, 2024, 2034, 2044 and 2054 for each scenario are shown in Table 2.
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Fig. 7 Distribution of antitoxin level: No protection, basic protection and full protection of the total population from 1977 to 2060. The results are obtained by implementing booster vaccination scenario 2.
Table 2. The rate of booster vaccination (lm ) and the corresponding numbers of
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booster doses provided in each booster vaccination scenario. lm
2024
2034
2044
2054
no × pqr −n
4.9
5
5.3
5.7
6.1
nr × pqr −n
3.3
3.5
3.6
3.9
4.2
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scenario 2
2014
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scenario 1
Numbers of booster doses (million)
scenario 3
so × pqr −n
2.5
2.6
2.8
3
3.2
scenario 4
po × pqr −n
1.7
1.8
1.9
2
2.2
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ACCEPTED MANUSCRIPT 4. Discussion In this work, we proposed a mathematical model for describing diphtheria transmission in Thailand. The aims of the simulations were to describe diphtheria transmission in Thailand during the years 1977 – 2014, to predict possible outbreaks
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in the near future, and to investigate the impact of providing booster vaccinations every 10 years. Unlike previous models (28-30), our model was designed specifically for diphtheria transmission dynamics in the level of population.
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The results in Fig. 3 shows that the number of reported cases has decreased dramatically since the introduction of the national vaccine program in 1977. After
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1995, few confirmed cases have been reported each year, which indicates the efficacy of the national DTP vaccine program. However, the sporadic outbreaks that occurred in 1994, 1996, 1999 and 2012 suggest the need for vaccine program improvement to fully eradicate diphtheria in Thailand (3, 7, 26). The results from the simulations
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demonstrate a very similar trend in diphtheria case reduction. Therefore, when our model is used together with the estimated parameter values, it could help us better understand diphtheria transmission in Thailand. The simulations also predict
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outbreaks in approximately the years 2020 and 2045 (Fig 3), which would be caused
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by an increase in the number of secondary susceptible individuals ( ) due to immunization waning (Fig 4a). This result suggests the need for vaccination boosting in adults, as recommend by the WHO. Additionally, and exhibited a downward trend since the immunization program was launched due to a decrease in infectious individuals (Fig 4b). The vaccinated classes and increased rapidly between 1980 and 2000 due to the larger infant vaccination coverage. However, the growth rate slowed after 2005 because the births rates were lower and vaccine administration
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ACCEPTED MANUSCRIPT reached its saturation. The number of infectious individuals ( ) and asymptomatic infectious individuals ( ) revealed the same downward trends (Fig 4c). In Fig. 5, the fractions of protective levels (basic and full) are lower than that of no protection. However, when the vaccine uptake is at a higher level (after the year
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2000), the fractions of protective levels are also higher. This is because most of the newborns entering the system are transformed to , and only a very small portion will remain in . In 2016, for example, the fraction of individuals with no protective
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level was approximately 0.60, which is still high. Thus, there is a risk for outbreaks to occur. This may explain why sporadic outbreaks have been detected in Thailand,
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despite the high vaccination rate. Our results also indicate that the fraction of individuals with full protective levels is higher than that with basic protective levels, which is consistent with Wanlapakorn et al. (7) and Bansiddhi et al. (6). Our results suggest that most Thais are still susceptible to diphtheria, which leads to the
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possibility of future outbreaks. In terms of herd immunity, the value of 1 − 1⁄ 7 can be used to estimate the required minimum fraction of the population with a protective immunity level to prevent an outbreak (25). For 7 = 6.5, the herd immunity was
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approximately 85%. In Fig. 5, the portion of the population with a protective
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immunity level in the year 2015 was approximately 40%. Therefore, it is reasonable that our simulation predicted outbreaks after the year 2015. To investigate the booster vaccination strategy, we implemented the booster
vaccinations every 10 years in our model. It is clearly demonstrated in Fig. 6 that providing appropriate booster doses every 10 years (Table 2) can prevent subsequent outbreaks. The results in Fig. 7 show that the protective antitoxin levels (basic or full) in the population increased in the year that the booster doses were provided.
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ACCEPTED MANUSCRIPT 5. Conclusions We have proposed a mathematical model for describing diphtheria epidemiology in Thailand after the EPI program was launched in 1977. The model considers Thailand to be a system of 77 compartments, which corresponds to its 77
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provinces. The populations in each province are further classified into 8 epidemiological classes: primary susceptible ( ), symptomatic infectious ( ), asymptomatic infectious ( ), carrier with full natural-acquired immunity ( ), carrier
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with partial natural-acquire immunity ( ), secondary susceptible ( ), individual with full vaccine-induced immunity ( ) and individual with partial vaccine-induced
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immunity ( ). The force of infection is also proposed in a way that corresponds to the nature of diphtheria transmission.
The infectious cases that resulted in our model before the year 2014 are consistent with real cases from reports. The model suggested that there would be
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recurring outbreaks after the year 2014 because a large portion of the population is susceptible to diphtheria. Those susceptible are secondarily susceptible because they lost the toxin antibody. The rates of antibody loss are estimated to be
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1⁄15 × 365 day :6 , which is in agreement with the literature [14]. Booster
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vaccinations every 10 years were shown to be sufficient for eradicating diphtheria in Thailand.
Conflict of Interests
The authors declare that there are no conflicts of interest.
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ACCEPTED MANUSCRIPT Acknowledgments This project was financially supported by the Health Systems Research Institute, Thailand, the National Science and Technology Development Agency, Thailand, and the Thailand Research Fund and Mahidol University (Grant No. TRG5880157). The
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funders had no role in the study design or preparation of the manuscript.
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