A Mathematical Approach to Characterize the Transmission Dynamics of the Varicella-Zoster Virus

Transcription

1 Proceedings of The National Conference On Undergraduate Research (NCUR) 2012 Weber State University, Ogden Utah March 29 31, 2012 A Mathematical Approach to Characterize the Transmission Dynamics of the Varicella-Zoster Virus Maria Gommel, Sam Jaros, and Donglun Liu Department of Mathematics North Central College 30 North Brainard Street Naperville, IL Faculty Advisor: Dr. Linda Q. Gao Abstract The Varicella-Zoster Virus (VZV) is the cause of chickenpox and shingles in humans. Chickenpox is generally a mild disease affecting children under thirteen, but is much more dangerous if contracted at an older age. Complications from chickenpox include encephalitis, pneumonia, bronchitis, cerebral ataxia, congenital varicella syndrome, and Reye s syndrome. Shingles generally affects older or immunocompromised people, and is characterized by a painful rash. Complications include glaucoma, blindness, post-herpatic neuralgia, or encephalitis. In the U.S., the chickenpox vaccine was added to the national immunization program in This one dose policy effectively decreased incidences, but failed to prevent outbreaks. The two-dose vaccination program was adopted in A shingles vaccine exists and is recommended for people over sixty as a booster shot, but it is only about 50% effective. The question now is whether an effective vaccination program for chickenpox would increase the incidence of shingles due to the lack of adult exposure to VZV. Mathematical modeling is used to investigate VZV transmission in the US population as well as the effectiveness of various control measures. Their effectiveness in preventing outbreaks was investigated, and four streamlined models were developed to describe the spread of chickenpox and shingles, and the models basic epidemic characteristics, such as equilibrium solutions and their stability, were analyzed. In addition, the Vaccination Reproduction Number (Rv) was calculated for each model using the Next Generation Operator Method. Numerical solutions for various values of parameters are investigated in the attempt to predict the efficacy of various vaccination strategies within a population as well as view the longterm behavior of shingles incidence in conjunction with these vaccination strategies for VZV. The result from numerical solutions shows that with 2-dose program and 90% coverage, the zoster incidence will increase significantly for about 27 years, and then decrease after that. Keywords: Epidemiology, Mathematical Modeling, Chickenpox 1. Introduction The Varicella-Zoster Virus (hereafter referred to as VZV) causes chickenpox, generally in young children between the ages of one and thirteen. After a person is infected with chickenpox, the VZV remains in that person s nervous system, dormant until it is reactivated. At this time, VZV causes herpes zoster (shingles). Shingles is a disease which generally only affects older (most commonly over 50), immunocompromised people or those with a medical history including intrauterine varicella or varicella within the first year of life, though it could technically affect anyone who has already been infected with the chickenpox 3. Varicella (chickenpox) is a common childhood illness, and is highly contagious. It can spread through direct contact with an infected person or through aerosolization of the VZV. The incubation period for chickenpox is generally between 10 and 21 days, but is most commonly on the order of 14 to 16 days. Typical symptoms of

2 chickenpox include fever and a rash characterized by 200 to 500 lesions on the body, concentrated on the face, scalp, and torso. A person is infectious two days prior to the rash appearing until the rash has completely crusted over, generally about 3-5 days. Treatment for mild cases of varicella consists of isolation, acetaminophen to control fever, and topical cream to control the itch associated with the rash. In more severe cases (in people over age thirteen, in immunocompromised people, in pregnant women, or other extreme cases), acyclovir, an effective antiviral medication that the VZV is highly responsive to, can be administered 3. The first vaccine for varicella was invented in 1974, and a one-dose vaccine was approved for the U.S. national immunization program in However, the one-dose vaccine was not effective enough in preventing outbreaks, as it prevented 95% of severe cases and was 70%-90% effective in preventing the illness entirely There have been documented outbreaks of chickenpox in areas with 90%-plus vaccination rates. So, a two-dose program was adopted in 2006, and so far it has been virtually 100% effective in preventing severe cases of chickenpox and 95% effective in preventing the illness entirely 3. The other illness caused by the VZV is herpes-zoster, otherwise known as shingles. Shingles occurs if the VZV happens to reactivate after lying dormant in a person s nervous system, and its symptoms are usually limited to a painful rash with lesions, concentrated on the torso. An infected individual feels pain 2-3 days before the rash appears, then for 3-5 days afterwards. The rash then typically disappears, but can linger for upwards of 3-4 weeks. After the rash is gone, it can take up to a month for the skin to look normal again, and in rare cases the pain never goes away (a complication known as post-herpetic neuralgia). A vaccine for zoster does exist and is recommended as a booster shot for individuals over 60 years old, but is only 51% effective at preventing shingles and 63% effective at preventing post-herpetic neuralgia 3. The vaccine for Varicella does not prevent illness entirely; breakthrough cases of chickenpox are common since it is a live vaccine. The symptoms in these breakthrough cases are the same, but are much milder; the fever may not appear and there are typically only lesions. Breakthrough cases of chickenpox pass more quickly (with an infectious period totaling 4-5 days on average as opposed to 5-7), and are less contagious, with a transmission rate of about 12% as opposed to the 80% rate associated with wild-type varicella. Since the varicella vaccine was implemented in 1995, the death rate associated with varicella is down 97%, and one-dose varicella vaccine coverage in children aged months was 89.6% nationwide in 2009, with state and city estimates ranging from 76% to 95%. The first dose of the varicella vaccine is given between 12 and 15 months of age, while the second dose is given between 4 and 6 years of age. People over 13 years of age that have not been vaccinated should get the vaccination, with the first dose coming 4-8 weeks before the second Models, Analysis, and Assumptions In our effort to mathematically model VZV and still be able to analyze the models without numerical computer simulation, we developed several models to capture the transmission dynamics. In these models, the population is divided into up to seven distinct, disjoint classes, denoted S, I, P, V, J, Z, and R. These abbreviations respectively stand for susceptible, infected/infectious (with chickenpox), partial immunity (immune to chickenpox, not zoster), vaccinated against varicella, breakthrough infected/infectious, infected with zoster, and recovered/removed. Various parameters are used in each model, denoted by Greek letters (with the exception of N), sometimes with subscripts, and these will be defined as they are introduced. 2.1 The First Model Our first and simplest model divides the population into four classes: S, I, R, and Z, as defined above. The compartmental diagram of the model is given below. Note that N denotes the entire population, while S, I, R, and Z are subgroups of that population. Further note that zoster does not occur in epidemics 3, so there is only one rate associated with catching shingles (labeled delta below), not an effective contact term in conjunction with the entire susceptible population like chickenpox (the beta times SI/N term below). 433

3 Figure 1, to the left, the compartmental diagram for model one, which yields (to the right) Figure 2, the system of nonlinear differential equations, which we use to analyze the model. There are two equilibrium points in this model, the Disease-Free Equilibrium (DFE) and the Endemic Equilibrium (EE). The DFE occurs when there are no infected persons left in the population, hence the term disease-free. The EE, however, has a nonzero value for the infected fraction i, indicating that the disease is persistent in the population. These equilibrium points are found by solving the system, which has been normalized (using susceptible, infected, and recovered fractions s, i, and r, respectively leaving z = 1-s-i-r) 4. We can do this because we are assuming a constant population. In doing this, we obtain the following equilibrium points: Figure 3. equilibrium points corresponding to model 1 Using these two equilibrium points, combined with the Vaccination Reproduction Number (denoted R v ), we can analyze our model based on parameter values, not just computer simulations 4. The Vaccination Reproduction Number is also known as the basic reproduction number, the basic reproduction ratio, or the basic reproductive rate. We deviate from the basic terminology because generally, basic reproduction numbers do not account for vaccinations. The value R v is defined as the average number of secondary infections that occur when one infective individual is introduced to a susceptible population 2. When R v <1, we know that the average infective causes less than one secondary infection, and thus the disease will die out. Therefore, when R v <1, the DFE is locally asymptotically stable and the EE is unstable. When R v >1, it follows that an infected individual will cause more than one additional infection on average, and thus the disease will spread and the EE will be stable, whereas the DFE will be unstable 4. The R v for this model, found using the Next Generation Operator Method, is given below 2 : 434

4 . Figure 4. vaccination reproduction number for model The Second Model Our second model improves upon the first one by removing the ambiguity of the class R; we are unsure whether an individual in the R class is only immune to chickenpox, or immune to both zoster and chickenpox. With this in mind, we create the class P, for individuals who have contracted and recovered from chickenpox but have not yet been infected with shingles. Also, the parameter π is added to denote the vaccination rate for shingles. The compartmental diagram and associated differential equations for the model are as follows: Figure 5. the compartmentalized model 2 and the corresponding system of differential equations As with the first model, the next step taken is to find both the DFE and the EE 4, which are as follows for the normalized system for this model: Figure 6. equilibrium points pertaining to model 2 Next, we apply the Next Generation Operator Method to find R v for this model 2. We find that: 435

5 Figure 7. vaccination reproduction number for model 2 Thus, R v for our second model is the same as R v for our first model, and the interpretation for this vaccination reproduction number is identical. The same criteria for stability of equilibrium points apply to this model as well The Third Model Our third model accounts for the potential for breakthrough cases after vaccination by adding a vaccinated class V (not just by moving vaccinated people to the class P) and a breakthrough infected/infectious class J. The compartmental diagram and associated differential equations are given below: Figure 8. compartmentalized diagram of model 3, accompanied by the resultant system of differential equations We can then find the DFE 4, which comes out to be: Figure 9. the equilibrium point found when analyzing model 3 This model is too complex to find the endemic equilibrium, either by hand or using a computer program, and thus we are unable to list the EE here. We were, however, able to use the Next Generation Operator Method to find the vaccination reproduction number 2, which is:. 436

6 = Figure 10. vaccination reproduction number corresponding to model 3 The DFE is locally asymptotically stable when R v <1, but here we can make no claims about the stability of the EE since we were unable to find it The Fourth Model Our fourth model is an extension of the third model; it adds the P class back in, and accounts for the zoster vaccine. In addition, we account for both doses of the varicella vaccine (not just one), and we provide a path for an individual to bypass both chickenpox and zoster and end in the recovered/removed class with the vaccinations and the P class. The second shot parameter, ρ 2, is something that could easily be factored in to the above three models. It is approximated between 0 and 1 and defined as 1/(4*365), given that the second shot is administered on average four years after the first. The compartmental diagram for our fourth, final, and most complex model is given below, followed by the resulting system of seven differential equations. Figure 11. a compartmentalized diagram for model 4, followed by the system of differential equations for the model Now, by solving this system, we can find the DFE for this model 4, but again we are unable to find the EE, either by hand or with computer programs. The DFE is given by: Figure 12. the equilibrium point found when analyzing model 4 437

7 Next, we again use the Next Generation Operator Method to determine the vaccination reproduction number for this model 2. The R v comes out to be: = Figure 13. vaccination reproduction number corresponding to model 4 The individual fractions comprising R v each have distinct meanings. Here, we have four separate terms to interpret, while the definition of R v as a whole is still the same as above. In this model, is the proportion of people who survive the susceptible (S) class, is the average time an individual spends in the infected (I) class, is the proportion of people who survive the vaccinated (V) class, and is the average time an individual spends in the vaccinated (V) class. As was the case in the previous three models, when R v <1, the DFE is locally asymptotically stable, and when R v >1, it follows that the DFE is unstable Methodology Parameter Estimation Now, we look at the parameters used in each model and their numerical estimates. The recovery rates (γ, γ 1, and γ 2 ) are based on data from the CDC on the average infectious period of wild-type and breakthrough varicella 3. The zoster recovery rate, ε, is also based on CDC data 3. The contact rates β, β 1, and β 2 are estimated using previous papers estimates 5, and the term f is based on the CDC figure that 12% of breakthrough cases of varicella are contagious, whereas 80% of the wild-type cases are contagious 3. Any of these three terms can theoretically fall in the range of zero to one. Vaccination rates ρ, ρ 1, and ρ 2 are varied according to coverage estimates and varicella vaccine efficacy, and can theoretically vary between 0 (no vaccine) and 1 (100% coverage and efficacy). The parameters δ, θ, and π are based on the proportion of the U.S. population that contracts shingles, the average age at which they contract it, and the average age at which they are vaccinated against it (with a 50% vaccine efficacy) 3. The birth rate and death rate are assumed to be equal to keep the total population, N, constant, so we can normalize our models. Lastly, the term µ is based on a 75 year life span, and can be altered to reflect any life expectancy. With all of our parameters, an exponential distribution is used to determine the mean values listed in our parameter tables, unless otherwise noted. Table 1. comprehensive list of estimated parameter values for all four models Parameter Estimation Meaning µ 1/(75*365) Birth and Death Rate β = β Contact Rate between susceptible and I/J classes β 2 0.1*β 1 Contact Rate between vaccinated and I/J classes γ = γ 1 1/7 Recovery rate, wild-type varicella γ 2 2/9 Recovery rate, breakthrough varicella f 0.12 Fraction of contagious breakthrough cases δ 0.2/(63*365) Zoster contraction rate after wild-type varicella ε 1/28 Zoster recovery rate π (0.5*0.51)/(57*365) Zoster vaccination rate θ 0.2/(63*365) Zoster contraction rate after vaccination, no breakthrough varicella ρ = ρ 1 (1/(12*30))(Vaccine efficacy) Varicella vaccination rate (1 st Dose) ρ 2 (1/(4*365))(Vaccine efficacy) Varicella vaccination rate (2 nd Dose) 438

8 4. Conclusions and extensions These models were developed in an effort to accurately depict the transmission dynamics of the VZV without being so complex that we could not analyze them without computer simulation. As such, more complex models, such as that of Shuette and Hethcote, predict a rise in zoster incidence with a universal vaccination program 5. Using model number four, our results seem to agree with this claim; higher zoster rates coincide with higher values for ρ 1 and ρ 2. We investigated the timing associated with the parameters as the cause of this phenomenon, as the zoster contraction rate is based on how long on average it takes an individual to become infected with shingles after they ve been infected with chickenpox, coupled with zoster incidence rates in the U.S. The age for vaccination against varicella (the second dose) is 4-6 years old, but the average age for the onset of wild-type varicella is between 11 and 13 years old. When the zoster contraction parameter is adjusted for this gap in timing, we still see an increased zoster incidence in the long run (over about years), although it is lower than with the original timing intact. With a one-dose varicella vaccine, we did not observe any long run increased zoster incidence. This tells us that our fourth model, although drastically simplified from the fifteen-dimensional model of Shuette and Hethcote 5, still produces similar results. There are several theories to explain the increase in zoster incidence, the most commonly accepted being that vaccinations prevent people from coming into contact with the VZV (by being near people with chickenpox) later in life, which would cause a boost in immunity. Rather, since people do not receive this immune system boost, they are more susceptible to zoster, so shingles incidence rises over time as the varicella vaccine continues to be administered, preventing children from getting chickenpox and thus preventing the natural immune boost against the VZV for adults. Other theories include more people having weakened immune systems due to over-dependency on vaccines and medicine, or extended life expectancy allowing people to live long enough for the latent VZV to reactivate as cellular immunity declines with age. Otherwise, some predict the zoster incidence rate to rise simply based on the fact that the U.S. population is aging as a result of the Baby-Boomer generation growing older 5. Our research data also backs the claim that conditions where the DFE is unstable do exist when the one-shot varicella vaccination is given at a 90% coverage rate, meaning that breakouts could happen given these conditions 2, which leads us to believe that the two-dose varicella vaccination program is the most effective way to prevent outbreaks and achieve maximum efficacy. Further research needs to be done, including finding the endemic equilibriums in the two models where, currently, we have not been able to. Proving the local stability of the EE in the other two is another open question, as is proving the global stability of any equilibrium point. It would be beneficial to see how complex can we make a model and still analyze it as we did the four models above. Our next steps, time permitting, would be to assume a nonconstant population in the models, and then develop an age-structured model with them. To the age structured model, immigration rates, age-specific incidence rates, age-specific death rates, and immunity rates could be added, and in general, this would provide a much more precise model to depict the transmission dynamics of the VZV. It would also be beneficial for the sake of parameter estimation to complete sensitivity analysis 4 on the models. Sensitivity analysis for the major parameters will determine which parameters have the largest impact on Rv, and may assist researchers when collecting data pertaining to specific parameters. This would also provide more accurate numerical simulation and greater accuracy in the results from the model. 5. Acknowledgements The authors wish to express their appreciation to Dr. Linda Gao, Dr. Dapeng Yin, the North Central College Mathematics Department, the North Central College research coordinators, and the various friends and family members who have helped to guide, shape, and edit this project. 6. References 1. S.M. Blower and H. Dowlatabadi, Sensitivity and Uncertainty Analysis of Complex Models of Disease Transmission: an HIV Model, as an Example, International Statistical Review 62 (1994): Carlos Castillo-Chavez, Zhilan Feng, and Wenzhang Huang, On the Computation of R 0 and its Role on Global Stability, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, ed. Carlos Castillo-Chavez et al. (New York: Springer, 2002),

9 3. Centers for Disease Control and Prevention, Prevention of Varicella: Recommendations of the Advisory Committee on Immunization Practices (ACIP), Morbidity and Mortality Weekly Report 56 No. RR-4 (2007): Gerda de Vries et al., A Course in Mathematical Biology: Quantitative Modeling with Mathematical and Computational Methods, (United States: Society for Industrial and Applied Mathematics, 2006). 5. Matthew C. Schuette and Herbert W. Hethcote, Modeling the Effects of Varicella Vaccination Programs on the Incidence of Chickenpox and Shingles, Bulletin of Mathematical Biology 61 (1999):