A Simulation Model Including Vaccination and Seasonality for Influenza A-H1N1 Virus

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1 Applied Mathematical Sciences, Vol. 10, 2016, no. 26, HIKARI Ltd, A Simulation Model Including Vaccination and Seasonality for Influenza A-H1N1 Virus Maria E. Cardenas P., Irma Pérez C., C. L. V. Pérez Facultad de Medicina Benemérita Universidad Autónoma de Puebla Puebla, Puebla, México Gonzaga Ospina P. and Anibal Muñoz L. Facultad de Educación Grupo de Modelación Matemática en Epidemiología (GMME) Universidad del Quindío Armenia, Colombia Copyright c 2015 Maria E. Cardenas P. et al. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We have described a SIR simulation model for the influenza A- H1N1 virus including variable population, periodic transmission coefficient and constant vaccination rate for any age and time. Also, it has been considered the natural mortality rate due to the infection. We analyze the model by means of simulations using several scenarios. The epidemic threshold, that is time and seasonality dependent, has been obtained. Keywords: SIR model, variable population, epidemic threshold, seasonality 1 Introduction A-H1N1 flu is a severe respiratory disease caused by some of the three known influenza viruses: A, B, and C. Type A has a sub-classification depending

2 1270 Maria E. Cardenas P. et al. on their surface proteins (hemagglutinin (HA) and neuraminidase (N)), from which, it is dependent their capacity to generate severe forms of the disease. From a public health care point of view the most important virus is type A. This one may drive to pandemics associated with high mobility, high number of deaths due to the virus, and social and economic disruptions. This infection causes substantial morbidity and mortality around the world. Also, it is extremely difficult to find a solution non-limited to prevention with vaccination or medication to control the symptoms. For the prophylaxis are employed some antivirals and annual vaccination, that are adjusted each year to the circulating virus. In special circumstances antivirals are prescribed to enhance the vaccination efficacy. The OMS recommends trivalent or inactivated vaccines that are constituted by purified superficial antigens. These ones are the most important prevention method for influenza virus, being of particular importance for infections with highest risk of complication [7]. Vaccines against influenza virus are essential components of an adequate response to the pandemic. The scientific and technological advances in the creation of vaccines motivate to set and model the A-H1N1 transmission dynamics. These models are applied to the susceptible and vaccinated populations as an optimal measure to the disease control. Aron and Schwartz (1984) reported a SEIR model including a periodic incidence rate, limit cycles and bifurcations [5]. Barnes, Glass and Becker (2007) have determined the role of the healthcare workers and the antiviral drugs in the influenza pandemic control, using the basic reproduction number of the dynamics [1]. Also, Becker, Glass, Li and Aldis (2005) have studied the emergency control of infectious diseases (as example, the severe acute respiratory syndrome) by using the basic reproduction number [6]. On the other hand, Browman, Gumel, Driessche, Wu and Zhu (2005) have proposed and analyzed a mathematical model to establish control strategies for the virus del Nilo [2]. Iwami, Takeuchi, Liu and Nakaoka (2009) have realized studies on the resistance to the vaccine in the geographic propagation of the influenza epidemic. Also, it has been described a deterministic model structured in patches for the avian influenza in heterogeneous zones and their control using a vaccination program in the context of the resistance-chains existence [9]. Klinkenberg, Van der Wind, Graat, and Jong (2003) have been applied an ordinary differential equations system including stochastic variables to the quantification of the effect on the control strategies in classic-fever epidemics [3].

3 A simulation model including vaccination and seasonality for Model We propose a simulation model based on coupled nonlinear differential equations that interpret the A-H1N1 virus transmission dynamics, including constant preventive control by using vaccination, periodic transmission coefficient due to the stationarity of the virus appearance, and death caused by infection. To construct this model we have considered the following assumptions: it has been modeled the A-H1N1 virus transmission with stationarity, considering a periodic transmission coefficient. The human population is variable, since the birth rate is constant but different to the natural death rate. Also, it is considered the death rate due to the infection, immunity to a virus serotype and constant vaccination rate for any age and time. The variables and parameters of the model are: x average number of susceptible people, y is the average number of infected people, z average number of immune people to a serotype of the influenza virus by medic treatment and vaccination, N is the variable-total population at time t. is the susceptible people mobility, µ the natural mortality rate, σ the vaccinated people rate at any age and time, β(t) is the periodic transmission coefficient, θ is the infected people recovery rate, ω is the death rate due to the infection, and ɛ is the stationarity degree [5, 10, 8]. The following diagram represents the dynamics behavior. δ σx x y β(t) x+y+z x y θy z µx (µ + ω)y µz The variation of each population as function of the time is interpreted by means of a differential equation, this one results of the difference between inflows and outflows. So then, dx indicates the susceptible population variation as function of the time. This one is defined as the susceptible people mobility δ minus the vaccinated population at any age and time σx minus the incidence term y β(t) x minus the population death due to natural causes µx. Namely, we x+y+z have the following differential equation: dx = δ σx β(t) y x µx (1) N

4 1272 Maria E. Cardenas P. et al. The variation of the infected population in the time dy is defined as: the y presence of new cases (incidence) β(t) x minus the infected population x+y+z that recovers and becomes immune to an A-H1N1 virus serotype θy minus the death people due to the disease and by natural causes (ω + µ)y. Such that, dy = β(t) y x θy (ω + µ)y (2) N dz indicates the variation of an immune population to an A-H1N1 virus serotype as function of time, this one having the following form. The positive term of the vaccinated people against A-H1N1 σx plus people that becomes immune to an A-H1N1 virus serotype θy minus the death of immune people by natural causes µz. In mathematical therms we have the following equation: dz = σx + θy µz (3) N(t) = x(t) + y(t) + z(t) drives to the following differential equation dn = δ µn ωy (4) In a compact way, the A-H1N1 virus transmission dynamics with constant preventive control by vaccination is interpreted by the following dynamic system of nonlinear differential equations: dx dy dz dn = δ σx β(t) y x µx (5) N = β(t) y x (µ + θ + ω)y (6) N = σx + θy µz (7) = δ µn ωy (8) All considered parameters are constant and positive, δ, µ, σ, β 0, θ > 0. The initial conditions are: x(0) = x 0, y(0) = y 0, z(0) = z 0, N(0) = N 0 and where 0 ɛ Epidemic threshold β(t) = β 0 (1 + ɛ cos 2πt) The basic reproduction number represents the average number of cases in which an infected person, in the infectious period, produces a susceptible population

5 A simulation model including vaccination and seasonality for [4]. This epidemic threshold interprets the infection evolution. If R 0 < 1, the A-H1N1 virus is extinguished, when R 0 1, there is presence of the A-H1N1 virus in the population. where, R 0 (T ) = βµ (σ + µ)(µ + θ + ω) β = 1 T T 0 β(t) being β the average transmission coefficient in the interval of time [0, t] [5]. So then, β = β T 0 (1 + ɛ cos 2πt) = β 0 T 0 T (T ɛ sin 2πT ) 2π (9) namely, R 0 (T ) = β 0 µ T (T ɛ sin 2πT ) 2π (σ + µ)(µ + θ + ω) 3 Results and conclusions Table 1: Variables and parameters Parameters Simbol Value Initial susceptible population x(0) 1000 Initial infected population y(0) 5 Initial immune population z(0) 0 Total population N(0) 1005 Susceptible people flux (mobility) δ 2 Natural mortality rate µ β(t) coefficient β 0 1 Stationarity degree ɛ 0.9 Infected people recovery rate θ 0.5 Death rate due to the infection ω 0.1 According to the hypothetic assigned values to the parameters, see Table 1., we have realized plots of the R 0 (T ) and simulations of the model using the MAPLE code. In figure 1(A) we plot the basic reproduction number behavior in a time period without vaccination, where we can observe an oscillatory decrement in

6 1274 Maria E. Cardenas P. et al. Figure 1: Behavior of basic reproduction number R 0 (T ) without vaccination σ = 0 and with vaccination σ = 0.1. the range from 1.4 to 1.9. This fact indicates that the disease remains endemic and periodic over the time. On the other hand, figure 1(B) includes results with a 10% of vaccination. In this image is observed an oscillatory decrement but with a threshold in the range from 0.4 to 0.6, this result implies that the disease remains in an endemic and periodic way, showing an effect of efficient control by vaccination as a measure of primary prevention. In figure 2 we depict the behavior of the infected population in a period Figure 2: Behavior of infected people y(t)( ), with σ = 0, σ = 0.1 and σ = 0.7. of 50 days, without and with different percentage of vaccination (10%, 70%). As it is observed from the simulations with σ = 0, there exist an epidemic with periodic and variable infection, having their maximum value around 15 days. Comparing these results with the ones for 10% of vaccination, we can observe a less number of infected persons in less days. Finally, for a 70% of

7 A simulation model including vaccination and seasonality for vaccination the infected persons are even less in number in around 2 days. In general, the impact of the vaccination is efficient for small coverages. Acknowledgements. We thank to Facultad de Medicina of the Benemérita Universidad Autónoma de Puebla, México. Also, thank to the Grupo de Modelación Matemática en Epidemiología (GMME) from Facultad de Educación (Universidad del Quindío, Colombia). References [1] B. Barnes, K. Glass, N. G. Becker, The role of health care workers and antiviral drugs in the control of pandemic influenza, Mathematical Biosciences, 209 (2007), [2] C. Bowman, A. B. Gumel, P. Van den Drissche, J. Wu, H. Zhu, A matematical model for assessing control strategies against West Nile virus, Bulletin of Mathematical Biology, 67 (2005), [3] D. Klinkenberg, A. Everts-van der Wind, E. A. M. Graat, M. C. M. Jong Quantification of the effect of control strategies on classical swine fever epidemics, Mathematical Biosciences, 186 (2003), [4] H. Heesterbeek, R 0, Centrum Voor Wiskunde en Informatica, Amnsterdam, [5] J. L. Aron, Ira B. Schwartz, Seasonality and period-doubling bifurcations in an epidemic model, J. Theor. Biol., 110 (1984), [6] N. G. Becker, K. Glass, Z. Li, G. K. Aldis, Controlling emerging infectious diseases like SARS, Mathematical Biosciences, 193 (2005), [7] Organizacion Mundial de la Salud, Nota informativa (2009). [8] S. F. Dowell, Seasonal variation in host susceptibility and cycles of certain infectious diseases, Emerging Infectious Diseases, 7 (2001), no. 3,

8 1276 Maria E. Cardenas P. et al. [9] S. Iwami, Y. Takeuchi, X. Liu, S. Nakaoka, A geographical spread of vaccine-resistance in avian influenza epidemics, Journal of Theoretical Biology, 259 (2009), [10] Y. A. Kuznetsov, C. Piccardi, Bifurcation analysis of periodic SEIR and SIR epidemic models, J. Math. Biol., 32 (1994), Received: November 22, 2015; Published: April 2, 2016