Abstract
Human rabies is one of the major public-health problems in China. The number of human rabies cases has increased dramatically in the last 15 years, partially due to the poor understanding of the transmission dynamics of rabies and the lack of effective control measures of the disease. In this article, in order to explore effective control and prevention measures we propose a deterministic model to study the transmission dynamics of rabies in China. The model consists of susceptible, exposed, infectious, and recovered subpopulations of both dogs and humans and describes the spread of rabies among dogs and from infectious dogs to humans. The model simulations agree with the human rabies data reported by the Chinese Ministry of Health. We estimate that the basic reproduction number 
for the rabies transmission in China and predict that the number of the human rabies is decreasing but may reach another peak around 2030. We also perform some sensitivity analysis of 
in terms of the model parameters and compare the effects of culling and immunization of dogs. Our study demonstrates that (i) reducing dog birth rate and increasing dog immunization coverage rate are the most effective methods for controlling rabies in China; and (ii) large scale culling of susceptible dogs can be replaced by immunization of them.
Introduction
Rabies is an acute and fatal zoonotic disease. The rabies virus infects the central nervous system and causes disease in the brain. Once symptoms of the disease develop, its mortality rate is 100%. Rabies can infect animals and also can be spread to humans through the bite or scratch of an infected dog or cat [1], [2]. All species of mammals are susceptible to rabies virus infection, but dogs remain the main carrier of rabies and are responsible for most of the human rabies deaths worldwide [3]. Rabies is widely distributed around the globe. More than 55,000 people die of rabies each year. About 95% of human deaths occur in Asia and Africa [2].
Human rabies in China was first reported in about 556 BC and has persisted for more than 2500 years [4]. Since 1950, the second year after the establishment of People's Republic of China, human rabies has been classified as a class II infectious disease in the National Stationary Notifiable Communicable Diseases [5], [6], and the annual data of human rabies have been archived by the Chinese Center for Disease Control and Prevention. From 1950 to 2010, 124,255 human rabies cases were reported in China [6]–[9], an average of 2,037 cases per year. Nowadays, China is second only to India worldwide in the number of people killed by rabies every year [8].
In the last 60 years, China experienced a few major epidemics of human rabies. The first peak occurred from 1956 to 1957 with about 2,000 cases in both years, followed by substantial decreases in the early 1960s. The number of cases reached 2,000 again in 1969 and increased to the historical record of 7,037 cases in 1981. During the 1980s, more then 5,000 cases were reported annually. In the 1990s, the number of cases declined rapidly from 3,520 in 1990 to 159 in 1996 [6], [8]. Since then, the number of human rabies case has increased steadily again and reached another peak in 2007 with 3,300 cases [7], [8]. From 1996 to 2010, 24,067 human rabies cases were reported [8], [9]. Though human rabies were reported in almost all provinces in China [5], nearly 60% of the total rabies cases in China were reported in the southern Guangdong, Guangxi, Guizhou, Hunan, and Sichuan provinces [8]. It is believed that the increase of rabies deaths results from a major increase in dog ownership and a very low rate of rabies vaccination [8]. In rural areas, about 70 percent of households keep dogs and low vaccination coverage of dogs is widespread, largely because of poor awareness of rabies and the high cost of vaccination. Moveover, owned dogs usually have not been registered and the number of dogs is estimated at 80–200 millions [1].
Although the recent reemergence of human rabies in China has attracted enormous attention of many researchers, the transmission dynamics of rabies in China is still poorly understood. Zhang et al. [6] analyzed the 108,412 human rabies cases in China from 1950 to 2004. They suggested that the rabies epidemics in China may be explained by dog population dynamics, untimely and inappropriate postexposure prophylaxis (PEP) treatment, and the existence of healthy carrier dogs. Si et al. [10] examined the 22,527 human rabies cases from January 1990 to July 2007 and the details of 244 rabies patients, including their anti-rabies treatment of injuries or related incidents. They concluded that the failure to receive PEP was a major factor for the increase of human cases in China. Song et al. [7] investigated the status and characteristics of human rabies in China between 1996 and 2008 to identify the potential factors involved in the emergence of rabies.
Mathematical modeling has become an important tool in analyzing the epidemiological characteristics of infectious diseases and can provide useful control measures. Various models have been used to study different aspects of rabies [11]–[28]. Anderson et al. [11] pioneered a deterministic model consisting of three subclasses, susceptible, infectious and recovered, to explain epidemiological features of rabies in fox populations in Europe. A susceptible, exposed, infectious, and recovered (SEIR) model was proposed by Coyne et al. [12], and lately was also used by Childs et al. [13], to predict the local dynamics of rabies among raccoons in the United States. Dimitrov et al. [14] presented a model for the immune responses to a rabies virus in bats. Clayton et al. [15] considered the optimal control of an SEIRS model which describes the population dynamics of a rabies epidemic in raccoons with seasonal birth pulse. Besides these deterministic models, discrete deterministic and stochastic models (Artois et al. [18], Allen at al. [19]), continuous spatial models (Källen et al. [20]), and stochastic spatial models (Smith et al. [21], Russell et al. [22]) have also been employed to study the transmission dynamics of rabies. We refer to a review by Sterner and Smith [17] and a thesis by Beyer [23] for more detailed discussions on different rabies models.
All of the above mentioned papers were about modeling wildlife rabies, recently there have been some studies on modeling canine and human rabies. Hampson et al. [24] observed rabies epidemics cycles with a period of 3–6 years in dog populations in Africa, built a susceptible, exposed, infectious, and vaccinated model with an intervention response variable, and showed significant synchrony. Carroll et al. [25] created a continuous compartmental model to describe rabies epidemiology in dog populations and explored three control methods: vaccination, vaccination plus fertility control, and culling. Wang and Lou [26] and Yang and Lou [27] used ordinary differential equation models to characterize the transmission dynamics of rabies between humans and dogs. Zinsstag et al. [28] extended existing models on rabies transmission between dogs to include dog-to-human transmission and concluded that combining human PEP with a dog-vaccination campaign is more cost-effective in the long run.
To understand the transmission dynamics of rabies in China and to explore effective control and prevention measures, in this paper we propose a deterministic SEIRS model to describe the spread of rabies among dogs and from dogs to humans. Both dogs and humans are included and are classified into susceptible, exposed, infectious, and recovered classes. We first simulate the number of human rabies cases in China from 1996 to 2010 reported by the Chinese Ministry of Health. Numerical simulations support the data reasonably well. We then estimate that the basic reproduction number 
for rabies transmission in China. We also perform some sensitivity analysis of 
in terms of the model parameters and compare the effects of culling and immunization of dogs. Our study demonstrates that (i) reducing dog birth rate and increasing the dog immunization coverage rate are the most effective methods in controlling human rabies infection in China; and (ii) culling of dogs can be replaced by immunization of dogs.
Methods
Both dogs and humans are considered in this study. We classify each of them into four subclasses: susceptible, exposed, infectious and recovered, with dog sizes denoted by 
and 
and human sizes denoted by 
and 
, respectively.
Mathematical Model
Our assumptions on the dynamical transmission of rabies among dogs and from dogs to humans are demonstrated in the flowchart (Fig. 1). The model is a system of eight ordinary differential equations:
 |
(1) |
Figure 1. Transmission diagram of rabies among dogs and from dogs to humans.
and 
represent susceptible, exposed, infectious and recovered dogs and humans, respectively.
All parameters are positive. For the dog population, 
describes the annual birth rate; 
denotes the loss rate of vaccination immunity; 
represents the incubation period of infected dogs so that 
is the time duration in which infected dogs remain infectious; 
is the risk factor of clinical outcome of exposed dogs, so 
represents those exposed dogs that develop clinical rabies and 
denotes those that do not develop clinical rabies and return to the susceptible class; 
is the natural death rate; 
is the vaccination rate; 
is the disease-related death rate; 
describes the transmission of rabies by interactions between infectious dogs and susceptible dogs. For the human population, 
is the annual birth rate; 
represents the loss rate of vaccination immunity; 
denotes the incubation period of infected individuals so 
is the time duration of infectiousness of infected persons; 
is the risk factor of clinical outcome of exposed humans, so 
represents those exposed individuals develop into the infectious class and the rest 
return to the susceptible class; 
is the natural death rate; 
is the vaccination rate; 
is the disease-related death rate. The term 
describes the transmission of rabies from infectious dogs to susceptible humans.
Basic reproduction number and stability of equilibria
Define the basic reproduction number by (see [29], [30])
 |
Equilibria are obtained by setting the right side of each of the eight differential equations equal to zero. If 
, it is easy to deduce the disease-free equilibrium:
 |
where
 |
If 
we can derive the unique endemic equilibrium:
 |
where
 |
 |
 |
in which
 |
 |
For the disease-free equilibrium point, we have the following property.
Theorem 1
If
, then (a) the disease-free equilibrium
of system (1) is locally asymptotically stable. (b) the disease-free equilibrium
of system (1) is globally asymptotically stable in the region
.
We also have the following result on the stability of the endemic equilibrium.
Theorem 2
If
, then the endemic equilibrium
of system (1) is locally asymptotically stable in the region
. All solutions in
tend toward the disease-free equilibrium
.
The proofs of Theorems 1 and 2 are given in Supporting Information S1.
Estimation of Epidemiological Parameters
In order to carry out the numerical simulations, we need to estimate the model parameters. The data concerning human rabies from 1996 to 2010 are obtained mainly from epidemiological bulletins published by the Chinese Ministry of Health [8], . However, the data involving dogs cannot be acquired easily. We have to rely on online news, our estimation or data fitting. The values of parameters are listed in Table 1. We explain the parameter values as follows: (a) The number of dogs was estimated to be 30 millions in 1996 and 75 millions in 2009 [8]. (b) The incubation period of rabies is 
months. We select the medium value: 2 months. So 
. According to the protection period of rabies vaccine, we assume that 
. The probability of clinical outcome of the exposed is 
. Here, we assume that it is 
So 
. (c) The rate of vaccination is the product of efficiency and the coverage rate of rabies vaccine. Efficiency of rabies vaccine is about 90%. However, the rates of vaccine coverage for dogs and humans are low. Considering a large number of stray dogs and the poor awareness of people in rural areas, we assume that they are equal to 10% and 60%, respectively. (d) The transmission rates 
and 
are obtained by fitting in simulations.
Table 1. Description of parameters in model (1).
| Parameters | Value | Unit | Comments | Source |
|
|
|
annual crop of newborn puppies | fitting |
|
|
|
dog loss rate of vaccination immunity | assumption |
|
0.4 |
|
risk of clinical outcome of exposed dogs | [39] |
|
6 |
|
the reciprocal of the dog incubation period | assumption |
|
|
|
dog incubation period | assumption |
|
0.08 |
|
dog natural mortality rate | assumption |
|
|
|
dog-to-dog transmission rate | fitting |
|
0.09 |
|
dog vaccination rate | [8] |
|
1 |
|
dog disease-related death rate | [8] |
|
|
|
human annual birth population | [40] |
|
1 |
|
human loss of vaccination immunity | assumption |
|
0.4 |
|
risk of clinical outcome of exposed humans | [39] |
|
6 |
|
the reciprocal of the human incubation period | [39] |
|
|
|
human incubation period | [39] |
|
0.0066 |
|
human natural mortality rate | [41] |
|
|
|
dog-to-human transmission rate | fitting |
|
0.54 |
|
human vaccination rate | [8] |
|
1 |
|
human disease-related death rate | [8] |
Results
Numerical Simulations
The numerical simulation of human rabies cases in China from 1996 to 2010 is shown in Fig. 2, indicating that our model provides a good match to the reported data. Our model does not include culling of dogs. In 2006, 50,000 dogs were slaughtered in Yunnan Province after three people died of rabies. Thousands of stray and owned dogs were killed in response to eight cases of human rabies in Hanzhong City in 2009. The awareness of rabies for people in recent years has been enhanced gradually. This may explain why the number of human rabies cases decreased in most recent years. This demonstrates further that our model has certain rationality. Moreover, our model indicates the tendency of the rabies epidemics with time, which is presented in Fig. 3. It shows that the number of human rabies cases will decrease steadily in the next 7 or 8 years, then increase again and reach another peak (about 1750) in 2030, and finally become stable. Therefore, if no further effective prevention and control measures are taken, the disease will not vanish.
Figure 2. The comparison between the reported human rabies cases in mainland China from 1996 to 2010 and the simulation of 
from the model.
The dashed curve represents the data reported by the Chinese Ministry of Health while the solid curve is simulated by using our model. The values of parameters are given in Table 1. The initial values used in the simulations were 
.
Figure 3. The tendency of human rabies cases 
in 50 years.
Basic Reproductive Number for Rabies in China
Based on the parameter values given in Table 1, we estimate that the basic reproduction number 
for rabies transmission in China. For rabies in Africa, Hampson et al. [31] obtained that 
according to the data from 2002 to 2007 when the peak of animal rabies cases was less then 30 weekly, which is far less than 393 the peak of monthly human rabies cases in China. Zinsstag et al. [28] also estimated the effective reproductive ratio to be 1.01 through a research framework for rabies in an African city. Also for the rabies in USA in the 1940s when the annual reported cases varied from 42 to 113 and sharply increased in 1948, it was estimated that 
[32]. From these, it can be seen that our estimate of 
is reasonable. More discussions of 
for outbreaks of rabies around the world can be found in [31], [32].
Sensitivity Analysis
Firstly, we look at the influence of initial conditions on the number of infected human rabies cases 
From Figs. 4 and 5, we can see that the effects of 
and 
are stronger and other initial conditions have little or almost no influence on 
. Moreover, we find that the initial conditions about dogs can influence not only the number of human rabies cases but also the time of rabies case peak. The initial conditions about humans do not have such effects. We also observe that the peak of the initial outbreak would be postponed if 
is decreasing.
Figure 4. The influence of initial conditions of dogs on the number of human rabies cases 
.
(A) 
for different values of 
. (B) 
for different values of 
. (C) 
for different values of 
. (D) 
for different values of 
.
Figure 5. The influence of initial conditions about humans on the number of human rabies cases 
.
(A) 
for different values of 
. (B) 
for different values of 
. (C) 
for different values of 
. (D) 
for different values of 
.
Next, to find better control strategies for rabies infection, we perform some sensitivity analysis of 
and the basic reproduction number 
in terms of the model parameters. First, we show variations of 
with time for different values of 
in Fig. 6. We can see that 
is really the threshold for the establishment of the disease in the susceptible pool and the number of infections increases with the increase of 
. The influences of 
and 
on 
are shown in Fig. 7. It can be observed that 
decreases as 
is declining or 
is increasing. When 
and 
, the disease can die out. Moreover, we find that the decrease of 
cannot delay the time of the first peak while an increase of 
can. Furthermore, the influences of 
on 
are given in Fig. 8. It is clear that 
changes more quickly when both 
and 
vary. When 
is very small, the disease can be eliminated even if 
. When 
the disease cannot be eliminated even if 
. From (B) and (C) in Fig. 8, it is clear that when 
or 
is very small, the disease can disappear even if 
. When 
or 
, the disease cannot be eliminated even if 
. Hence, it indicates that the influence of 
and 
on the basic reproduction number 
is greater. Fig. 8 reflects that whatever dog vaccination rate is, when the annual crop of newborn puppies is greater than 3 million and dog-to-dog transmission rate is greater than 
, 
cannot be below 1. However, it is difficult to control 
.
Figure 6. The variations of the infected human rabies cases 
for different values of 
.
Here 
, 
respectively, other parameters are as in Table 1.
Figure 7. The influence of parameters 
and 
on the number of human rabies cases 
.
(A) 
in terms of different values of 
. (B) 
in terms of different values of 
.
Figure 8. The combined influence of parameters on 
.
(A) 
in terms of 
and 
. (B) 
in terms of 
and 
. (C) 
in terms of 
and 
.
Currently, in China the annual crop of newborn puppies can exceed 5 million and the proportion of immunized dogs is only about 10%, which is too low. According to the current incidence 
, we know that if the annual crop of newborn puppies 
(
) is not reduced, it is impossible to have 
below 1; if 
, it is necessary to keep 
; if 
, it is necessary to keep 
.
The above analysis demonstrates that human rabies can be controlled with two strategies: reducing the annual crop of newborn puppies and increasing the dog immunization rate at the same time, which can also reduce the incidence rate 
.
The Equal Effect of Culling Rate and Immunization Rate
It has been found that in Europe culling as a means of rabies control was not effective once rabies became established within the fox population (MacDonald [33]) and had only limited success (Smith and Harris [34]). Recently, some studies suggest the strategy of culling dogs to control rabies (Kureishi et al. [35], Hu et al. [5], etc.) and some cities have in fact taken this measure. Here, we particularly discuss the influence of culling dogs. By adding the terms to describe the culling of dogs, the model becomes the following:
 |
(2) |
where 
is the dog culling rate. We are interested in comparing the levels of culling and immunizing that are necessary to provide the same effect. The comparison is shown in Fig. 9. It demonstrates that the culling rate must be about 10 times the immunization rate to have an equal effect. This indicates that, under the same condition, immunizing 1% of the susceptible and exposed dogs has the same effect as culling about 12.38% of dogs. Culling of infected dogs is necessary in controlling the outbreak for a short term as suggested by some studies, our results show that large scale culling of susceptible dogs can be replaced by immunization of them.
Figure 9. The equal effect of culling and immunization of dogs.
Discussion
Rabies is one of the biggest public health threats in China. Facing up to the epidemic situation in China, both the central and local governments have been seeking forceful methods to reduce rabies transmission. Various prevention and control measures have been proposed by many researchers which include: (i) strengthening the postexposure prophylaxis (PEP) schedules delivered to rabies patients [7], [10]; (ii) culling of dogs, in particular stray dogs [35]; (iii) increasing the vaccination coverage in dogs [36]. Some researches suggest that combining these methods can be more effective in controlling the rabies. For example, Hu et al. [5] came up with strategies to control and prevent human rabies that include public education and awareness about rabies, pet vaccination programs, culling of stray animals, and enhancing PEP for infected patients. However, the large-scale culling of dogs, criticized by pet owners and animal protection activists, is controversial and there is a lack of evidence of its effectiveness in controlling dog population or rabies (WHO [37]). In fact, culling may remove vaccinated dogs, increase immigration, disrupt social organization, and lose public support, which make rabies control more difficult (Carroll et al. [25]).
In this article, in order to explore effective control and prevention measures we proposed a susceptible, exposed, infectious, and recovered model to study the transmission dynamics of rabies in China. The model describes the transmission of rabies among dogs and from dogs to humans. The model simulations agreed with the human rabies data reported by the Chinese Ministry of Health and gave an estimate of the basic reproduction number 
. The sensitivity analysis of 
in terms of the model parameters and the comparison of the effects of culling and immunization of dogs demonstrate that (i) controlling dog birth rate and increasing dog immunization coverage rate are the most effective methods for controlling rabies in China; and (ii) large scale culling of susceptible dogs can be replaced by immunization of them.
The characteristics of rabies epidemics in China include the large size of the dog population, the extremely low dog vaccination rate, poor understanding of the transmission dynamics of rabies, inadequate treatment of the infected patients, and the countrywide scale of the disease [8]. WHO recommends that 70% of dogs in a population should be immunized in order to eliminate or prevent outbreaks of rabies. However, the dog immunization rate in China has yet reached 10%, which is even lower in the countryside [8]. So increasing dog vaccination coverage rate is necessary and crucial in control rabies transmission in China. We suggest decreasing the dog birth rate in order to reduce the dog population and stop culling dogs and try to vaccinate these dogs instead. For pet dogs, registration and immunization should be mandatory; the price of vaccine should be reduced; the awareness of prevention rabies for dog owners should be enhanced; and contraception measures should be taken. For stray dogs, food baits containing oral vaccine or abortifacient in capsules could be distributed (Roberts and Aubert [38]) and the fostering of stray dogs could be introduced and encouraged.
Supporting Information
Stability of the disease-free and endemic equilibria is given in this file.
(PDF)
Acknowledgments
The authors would like to thank the three reviewers and Don DeAngelis for their helpful comments and valuable suggestions.
Footnotes
Competing Interests: The authors have declared that no competing interests exist.
Funding: This research was partially supported by the National Natural Science Foundation of China, Program for New Century Excellent Talents in Universities, Program for Basic Research, Beiren Project and International and Technical Cooperation Project of Shan'xi Province, and the National Science Foundation. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. No additional external funding was received for this study.
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Associated Data
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Supplementary Materials
Stability of the disease-free and endemic equilibria is given in this file.
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