Mathematical Modeling of Hand-Foot-Mouth Disease: Quarantine as a Control Measure

Transcription

1 E S Scholars T Knowledge is Power April 2012 Volume 1, Issue 2 Article #04 IJASETR Research Paper ISSN: Mathematical Modeling of Hand-Foot-Mouth Disease: Quarantine as a Control Measure Nandita Roy 1 1B.Sc. graduate, Khulna University, Khulna, Bangladesh *Corresponding author s nandita_roy_26@yahoo.com Abstract Children are mostly infected by hand, foot and mouth disease (HFMD) in their early age. HFMD is considered as a mild to moderate contagious disease, but recent outbreaks are threatening to child health and also caused social and economic disturbance along with causalities in many countries. In this study we build a mathematical model of HFMD to understand the dynamics and analytically evaluate the effectiveness of quarantine as a control strategy. Our analyses show that disease could be controlled by quarantine of more actively infected individuals. We also qualitatively showed that disease transmission depends more on the number of actively infected individuals in the population at the initial time and also on the disease transmission coefficient at a given time. Keywords: compartmental modeling, differential equation, mathematical modeling Citation: N. Roy (2012), Mathematical Modeling of Hand-Foot-Mouth Disease: Quarantine as a Control Measure. IJASETR 1(2): p Received: Accepted: Roy, N. This is an open access article distributed under the terms of the Creative Common Attribution 3.0 License. 1. Introduction Hand, foot, and mouth disease (HFMD) is a contagious viral illness that commonly affects infants and children. HFMD occurs often in the countries with temperature climates especially in summer and early autumn. HFMD is caused by enteroviruses from the family called Picornoviridae. It is most commonly caused by Coxsackie virus (A16), human enteroviruses (HEV71) or other enteroviruses including Coxsackie virus A (CAV) 4, 5, 9 and 10 or Coxsackie virus B (CBV) 2 and 5 [1] [2] [3] [4]. Fever, poor appetite, malaise (feeling vaguely unwell), and often a sore throat that the common symptom of this disease. A couple of days after the fever starts, painful sores can develop in the mouth. A skin rash with flat or raised red spots can also develop, usually on the palms of the hands and soles of the feet and sometimes on the buttocks. This rash may blister, but it will not 34

2 itch. Some people with HFMD may only have a rash; others may only have mouth sores. Other people with HFMD does not show any symptoms. HFMD is moderately contagious and usually not a serious illness among the infected population, recent outbreak of HFMD in countries such as China, Taiwan, Singapore and Malaysia had brought the world attention to HFMD due to complications of death related cases [3] [5] [17]. The recent outbreaks of HFMD are shown in Table 1 [5]. It is certain that HFMD not only caused health problems but also it has great social and economical impacts which are not easily quantifiable. So it is important to understand the dynamics of HFMD spread among the susceptible populations and enable policy makers to take effective measure to curb the disease spread and reduce the adverse impact of the disease. It may be a general thought that statistical regression is the only modeling technique available for the health policy maker s toolkit to analyze the spread of infectious diseases; but statistical models are only one of several types of analytical models that are valuable to understand and predict transmission of infectious diseases. In addition, mathematical modeling is a set of techniques, tools and equations that define interactions between individuals or populations and other individuals, populations and environments. By defining the rules of these interactions and translating those rules into equations, a complex set of processes can be broken down into components and quantified. The model can then be used to explore relationships in the modeled populations, to examine the impact of changes of rules on the system and its components, and the outcomes of various events that might have an effect on the population. Table 1: recent outbreaks of HFM Disease Year Country Reported number of infected cases and deaths 1997 Sarawak in Malaysia 2626 infected children and 31 deaths Taiwan 405 children with severe complication; 78 died. Estimated cases were 1.5 million Sarawak in Malaysia 14,423 infected cases and 13 deaths China infected cases and 42 deaths Singapore 2600 infected cases Vietnam 2300 infected cases and 11 deaths Mongolia 1600 infected cases Brunei 1053 infected cases China 115,000 reported cases, 773 were severe complication and 50 were fatal Indonesia Several severe cases with fatality China Until march children were infected and 40 died from the disease Vietnam 42,000 infected cases China 1,340,259 infected cases, 437 deaths Alabama in USA 14 identified cases. The foundations of mathematical modeling of human and mammalian disease epidemics were first established by Kermarck and McKendrick [6] [7] [8]. Later, the extensive reviews [9] [10] 35

3 [11] contributed to the subject of disease dynamics opening new discussions for concepts, critical threshold, basic reproductive ratio, transmission coefficient, etc. The importance of mathematical models in disease epidemics has been discussed extensively by many researchers in [12] [13] [14]. Several studies [1] [15] [16] [17] are investigated the spread of HFMD among the young children. In the study [1], the authors established a simple deterministic SIR model of HFMD to predict disease outbreak. They estimated critical population density for the outbreak of HFMD in Sarawak, Malaysia in In [15], SIR model of HFMD has been used to analyze the occurrence of enterovirus complications among the severe ill cases in Taiwan. In [16], the authors attempted to established nonlinear mathematical models to simulate the impact of global warming on the incidences of HFMD in Tokyo. Numerical analysis has been shown to find the relationship between the outbreaks of HFMD with the weather patterns in the respective countries in those models. In [17], the authors formulated a compartmental SEIR model for HFMD using computer algebra providing the analytical results obtained from the model. In this paper, using systems of differential equations we describe the basic formulation of an SIR model for hand, foot and mouth disease. Then we extend this model to SLIR model to capture the dynamics of latent infectious individual and we also include quarantine as a control strategy into the model to evaluate its impact on disease spread. We then gave an analytical description with qualitative results of quarantine using the extend SLIQR disease model. In the following sections we illustrate the clinical characteristics of HFMD. This will be followed by mathematical formulation of SIR and SLIR model of HFMD with analytical results. Then we provide a description of quarantine associated SLIQR disease model followed by qualitative results obtained from the model. 2. Clinical Characteristics of Hand-Foot-Mouth Disease (HFMD) Clinical and pathological characteristics of hand foot mouth disease are summarized in many studies [1] [15] [16] [17] and also described on the websites [3] [5]. The most likely susceptible to HFMD are the children aged below 10 years [1] [3] [17]. An individual who is exposed to HFMD viruses will exhibit the symptoms after three to seven days. This is considered the incubation period of HFMD. Fever is usually the first symptoms followed by poor appetite, malaise and sore throat. One or two days after the fever begins, small red spots i.e. blisters develop in the mouth and often develop into ulcers. These are mostly found on the tongue, gums and inside of the cheeks. The skin rash develops over one or two days with flat or raised red spots, some with blisters on the palms of the hand and the soles of the feet. Thus, the name of the disease is hand, foot and mouth disease (HFMD). An individual with HFMD may some time have only the rash or the mouth ulcers. HFMD is considered moderately contagious among children. An individual is most contagious during the first week of the illness. As the viruses are present in the throat and stools of an infected individual, infection generally occurs via the facial-oral or via contact with skin lesions and oral secretions. The virus may continue to be excreted in the stools of infected individuals up till one month. The spread of the virus does not involve any vectors [1] [3] [5]. 36

4 At this moment there is no specific antiviral drug to cure HFMD. There is also no vaccine available for the treatment of HFMD [3] [5]. Infected person is usually given medication to provide relieve from the pain caused by fever, aches or mouth ulcers. Victims are asked to take plenty of liquid. Quarantine would be a control measure to slow down the disease spread. An infected person will fully recover after 7 to 10 days. There is no permanent immunity against HFMD as the disease is caused by a group of viruses much like the case of flu. A person who recovered from the HFMD caused by Coxsackie A is susceptible to HFMD caused by enteroviruses 71 or any other enteroviruses. 3. Basic HFMD model The purpose of this modeling study is to develop a simple deterministic model for HFMD that might capture the dynamics of the disease spread within the young children population. According to the SIR disease model (see Figure 1), described in [6] [7] [8] [12] [13], we consider the simple HFMD model which should have three basic classes: Susceptible (S), Infected (I) Recovered (R) Susceptible (S) represents the number of individuals who are not yet infected with the disease or those who are susceptible to the disease. Infected (I) is the number of symptomatic infectious individual (people who have been infected with disease and show the signs of symptoms) and Recovered (R) is the number of recovered individuals (people in this class have been infected but then recovered from the disease; they may be able to be infected again). Birth of new susceptible Birth rate(α) S βi I δ ρ D λ R Figure 1: SIR Disease model It is assumed that in case of directly transmitted infectious diseases, the dynamics proceed according to the mass action theory, such that the net rate of individuals becoming infected is the product of density of susceptible (S) individuals times the density of infected (I) individuals. Thus, the number of successful contacts made between S and I determine the magnitude of the disease outcome. To construct the disease transmission model we further assume that; a) An average member of the population makes contact sufficient to transmit infection with βn others per unit time, where N represents total population size (mass action incidence). b) Infected individuals leave the infective class to recovered class at rate ρi per unit time. c) Infected individuals are dead due to illness at rate δi per unit time. d) New susceptible arrives into the susceptible population at α rate per unit time. e) Recovered individuals will be susceptible again at rate λr per unit time. 37

5 All lists of parameters are used in this disease model are illustrated in Table 2. From the model given in Figure 1, we can derive the systems of differential equations for the dynamics of hand, foot and mouth disease. Thus, the basic model is given by, (1) (2) (3) Modeling infectious diseases demand that we investigate whether the disease spread could attain an epidemic level or it could be wiped out. The equilibrium analysis helps to achieve this. Thus two equilibriums will be considered; the disease free equilibrium and epidemic equilibrium. At equilibrium, the LHS of the three equations constituting the system of equations are zeros, i.e. equations (1) to (3) will be zero as follows, Hence, the system of equations would be, (4) (5) (6) From (5), we get either or. When, we have the disease free equilibrium. Otherwise i.e. which implies the critical density of susceptible population to occur an epidemic. If epidemic would be occurred. This is also called endemic equilibrium. then there must be an epidemic otherwise no To test for the stability of each of the two equilibriums where we examine the behavior of the model population near the equilibrium solutions (the disease free and the endemic), we linearized the system of equations (1) to (3) by taking the Jacobian matrix of the system. For the modeled system of equation, the Jacobian matrix is, [ ] (7) 38

6 At the disease free equilibrium i.e. at the total fraction of susceptible S = 1), the Jacobian matrix would be as follows (assuming [ ] (8) From (8), we have characteristic polynomial, ( ) ( ) Hence, the eigenvalues are as follows Theorem 1: The disease free equilibrium (DFE) is stable if. The disease free equilibrium for the model only exists if all the eigenvalues are non-positive. This holds if At the endemic equilibrium i.e. at the Jacobian matrix would be as follows, [ ( ) ] (9) From (9), we have characteristic polynomial as, ( ) ( ) The eigenvalues of could all be negative, positive, zero or any combinations of the three alternatives. Therefore, the endemic equilibrium could be stable, unstable or saddle depending on the values of the various model parameters. 4. HFM disease model with Latent infection We now revise the model to include a latent class of infected individuals. When susceptible contact with infected individuals, it is assumed that the susceptible individuals do not get the disease immediately. There is time gap between a susceptible individual to become an infected individual. This time period is known as latent period of an infectious disease. We can model this phenomenon of the disease by simply introducing a new class of individual (L class or latent class) in our basic model and assuming γ as the rate at which individual in latent class would become infected. The revised model is shown in Figure 2 with latent class. 39

7 Birth of new susceptible Birth rate(α) S βi L γ I δ ρ D λ R Figure 2: HFM disease model with Latent infection Then the model is described by the following set of differential equations. (10) (11) (12) (13) At equilibrium, the LHS of the equations constituting the disease model are zeros, i.e. equations (10) to (13) will be zero as follows, (14) (15) (16) (17) Hence, ( ) Putting the value of in equation (16) we get, or 40

8 Now substituting the value of and from equation (17) into equation (14) we get, ( ) Therefore, at epidemic equilibrium, The basic reproductive ratio is the number of secondary infectives per index case in a naïve population of susceptible. If we look at equation (12), we get. The parameter is the transmission rate per infective and the negative terms imply that each infectious individual spends an average time units in the infectious class. Therefore, if we assume the entire population is susceptible ( ), then the average number of new infection per infectious individual is determined as follows, ; which is the basic reproductive ratio of the disease. 5. HFM disease model with Quarantine There is no available treatment and vaccination for Hand, foot and mouth disease among the young children. Quarantine is the best approach for not to spread the disease among the other susceptible. Here we revise our previous HFMD model to evaluate the effect of quarantine in controlling the disease. We assume that the probability of an infected individual to be quarantined is σ and average period for quarantine is. The modified model is shown in Figure 3. Birth of new susceptible Birth rate (α) S βi L γ I δ ρ D λ σ Q τ R Figure 3: HFM disease model with Quarantine The systems of equations that governs the dynamics of HFMD with quarantine are given as follows, (18) (19) 41

9 (20) (21) (22) In this case, in order to analyze stability, we determined the basic reproductive ratio, using the next generation method [13] [18]. The basic reproductive ratio has the property that if, then the outbreak will persist, whereas if, then the outbreak will be eradicated. If we were to determine the Jacobian and evaluate it at disease free equilibrium, we would have to evaluate a nontrivial 5 by 5 system and a characteristic polynomial of degree of at least 3. With the next generation method, we only need to consider the infective differential equations. The infective differential equations are as follows, Let be the matrix of new infections and be the matrix of transfers between components for the infective equations. Hence, using the next generation method [18], at disease free equilibrium we get, [ ] and [ ] [ ( ) ( ) ( ) ( ) ] [ ] (23) Thus, we get by finding the spectral radius of the matrix,. If we assume that no deaths occur then (24) 42

10 From (24), it is clear that for a particular enterovirus strain, β and infectious period 1/ρ are constant. Therefore, R 0 could be maintained at less than 1 if σ is increased i.e. an outbreak of a HFMD could be invade due to increasing Quarantine. 6. Conclusion and Future works In this work, we presented the dynamics of the hand, foot and mouth disease (HFMD) using a differential equation based mathematical model. The resulting model equations were solved analytically; the equilibriums and their stability were also presented. Our results show that disease transmission depends more on the number of actively infected individual in the population at the initial time and also on the disease incidence transmission rate at a given time. We also showed that the disease free equilibrium is stable while the endemic equilibrium may or may not be stable depending on the various values of the model parameters. Most importantly, we qualitatively showed that the disease could be controlled if increased number of infectious host could be quarantined. In future works, we will do an extensive numerical analysis of the assumed model parameters and also show the effectiveness of quarantine as a control measure with different compliance level in the population. References 1. Felix Chuo Sing Tiing and Jane Labadin, A Simple Deterministic Model for the Spread of Hand, Foot and Mouth Disease (HFMD) in Sarawak. Sceond Asia International Conference on Modelling & Simulation, 2008, pp Y. Podin, E.L.M. Gias, F. Ong, Y.W. Leong, S.F. Yee, M.A. Yusof, D. Perera, B. Teo, T.Y. Wee, S.C. Yao, S.K. Yao, A. Kiyu, M.T. Arif and M.J. Cardosa, Sentinel Surveillance for Human Enterovirus 71 in Sarawak, Malaysia: Lessons from the first 7 years. BMC Public Health Hand, Foot, and Mouth Disease from Centers for Disease Control and Prevention: 4. P. McMinn, K. Lindsay, D. Perera, H.M. Chan, K.P. Chan and M.J. Cardosa, Phylogenetic Analysis of Enterovirus 71 Strains Isolated during Linked Epidemics in Malaysia, Singapore, and Western Australia. Journal of Virology, 2001, pp Hand, Foot, and Mouth Disease from Wikipedia: 6. W.O. Kermack, and A.G. McKendrick, Contribution to the mathematical theory of epidemics, part I. Proceedings of the Royal Society of London Series A, 115: [Reprinted as Bulletins of Mathematical Biology, 53, 1991, pp ] 7. W.O. Kermack, and A.G. McKendrick, Contribution to the mathematical theory of epidemics, II- The problem of endemicity. Proceedings of the Royal Society of London Series A, 138: [Reprinted as Bulletins of Mathematical Biology, 53, 1991, pp ] 43

11 8. W.O. Kermack, and A.G. McKendrick, Contribution to the mathematical theory of epidemics, III- Further studies of the problem of endemicity. Proceedings of the Royal Society of London Series A, 141: [Reprinted as Bulletins of Mathematical Biology, 53, 1991, pp ] 9. R.M. Anderson, The population dynamics of infectious diseases: Theory and applications. Chapman and Hall, London, R.M. Anderson, and R.M. May, Population biology of infectious diseases Part I. Nature, 280: , R.M. May, and R.M. Anderson, Population biology of infectious diseases: Part II. Nature, 280: , D. J. Bradley, Epidemiological Models -theory and reality. In Population Dynamics of Infectious Diseases. Roy M. Anderson (Ed.), Chapman and Hall, London, 1982, pp R.M. Anderson, and R.M. May, Infectious Diseases of Humans, Dynamics and Control. Oxford University Press, Oxford, J.A.P. Heesterbeek, and M.G. Roberts, Mathematical models for microparasites of wildlife. B.T. Grenfell, A.P. Dobson (Eds.), In Ecology of infectious diseases in natural populations, Cambridge University Press, Cambridge, , Y.C. Wang and F.C. Sung, Modeling The Infections for Enteroviruses in Taiwan. From M. Urashima, N. Shindo, and N. Okabe, Seasonal Models of Herpangina and Hand-Foot-Mouth Disease to Simulate Annual Fluctuations in Urban Warming in Tokyo. Japan Journal of Infectious Disease, 2003, pp Roy N. and Halder N. Compartmental modeling of Hand, Foot and Mouth Disease (HFMD), Research Journal of Applied Sciences, 5(3), 2010, pp Pauline D. and James W. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180:29 48,