Impact of the Latent State in the R 0 and the Dengue Incidence

Transcription

1 Applied Mathematical Sciences, Vol. 12, 2018, no. 32, HIKARI Ltd, Impact of the Latent State in the R 0 and the Dengue Incidence Angie J. Osorio R. 1, Oscar A. Manrique A. 2, Julián A. Olarte G. 3 and Anibal Muñoz L. 4 1,2,3 Estudiante de Maestría en Biomatemáticas 1 Seminario interdisciplinario Grupo de Matemática Aplicada (SIGMA) 1,2,3 Grupo de Modelación Matemática en Epidemiología (GMME) Faculta de Ciencias Básicas y Tecnologías, Facultad de Educación Universidad del Quindío Armenia, Quindío, Colombia Copyright c 2018 Angie J. Osorio R. 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 A mathematical model that interprets the host-vector dynamic for a serotype of dengue is presented. The model includes four different hypotheses that consist of considering latent states in the dynamics in order to analyze the effect of these states on the basic reproduction number and changes in the evolution of the disease. Keywords: Dengue, host, vector, basic reproduction number, latency 1 Introduction Dengue is an acute infection transmitted by the bite of the female Aedes aegypti mosquito carrying a serotype of the dengue RNA virus to a susceptible person. It is one of the pathologies of great impact on public health in the world, particularly in Latin American countries, for which there is no effective prevention method such as a vaccine, in such a way that efforts are directed to the control of the mosquito, through chemical applications (adulticides, larvicides) and manual activities aimed at interrupting the biological cycle of the vector [17] Recently, biological control has also been applied using the bacteria

2 1710 Angie J. Osorio R. et al. Wolbachia [16]. Dengue is transmitted to humans through the bite of infected mosquitoes of the species Aedes [18]. The virus multiplies in the intestinal epithelium of the infected female mosquito, nervous ganglia, fatty body and salivary glands; approximately 7 to 14 days (latent period known as extrinsic incubation) can infect man with a new bite [19, 20, 21, 22]. When the virus is inoculated to the organism, the incubation period (latent period) lasts between 4 to 7 days on average, after this time it may or may not present symptoms, depending on the viral strain, age, immune status or other factors [19, 21, 18]. In the mathematical modeling of epidemics there are several thresholds that help to understand the evolution of the epidemic, one of them is the basic reproduction number, R 0, introduced by Ross in the year 1909 [3, 4, 2, 1]; the concept of basic reproduction number supposes that an infectious case is introduced in a totally susceptible population; in the modeling of epidemics it is established that if R 0 < 1, the disease is extinguished, while if R 0 > 1 the disease spreads in the population. In the literature there are several researches focused to calculate R 0 in different epidemiological models of infectious diseases [5, 6, 7, 8, 9, 10, 11, 12]. In mathematical models, the basic reproduction number is given by the radio spectrum of the next generation matrix in continuous models and, in particular, is determined by the dominant eigenvalue of the matrix evaluated in the infection-free equilibrium for models in a finite dimension space [10]. In this article a mathematical model is formulated to interpret the host-vector dynamic with a dengue serotype; In the model, four different hypotheses are considered, consisting of considering latent states in the dynamics in order to analyze the effect of these states on the basic number of reproduction and changes in the evolution of the disease, for this, simulations of the dynamic system of each model are carried out in the Maple software. The article is composed of three sections, including the introduction: the first contains the formulation of the mathematical model and the different estimates of the basic reproduction number; in the second section the simulations of the dynamic system are shown; finally in section 3 are the conclusions. 2 The model In this section we present the formulation of four models based on ordinary nonlinear differential equations that differ only in the inclusion or not of the

3 Impact of the latent state in the R latent population in the dynamics. In each model, the basic reproduction number is determined using the next generation matrix. The general assumptions for the models are: 1. The flow of the constant human population is considered. 2. In the dynamics are considered entomological variables in mosquitoes, these are the recruitment rate (η) and the mortality rate (ɛ). 3. Only one dengue serotype is considered. 4. Transovarial transmission is not considered in the vector. 2.1 Formulation of Model 1 The formulation of model 1 will be the basis for the model 2, model 3 and model 4 that will be shown in this section. A mathematical model type host-vector for dengue disease is considered, the state variables that intervene in the model are x 1 (t): average number of susceptible people, x 3 (t): average number of infectious people by a dengue serotype, x 4 (t): average number of people recovered, y 1 (t): average number of noncarrier mosquitoes and y 3 (t): average number of carrier mosquitoes. The parameters considered are θ: recovery rate, µ: natural death rate in the human population, η: constant increase in the average number of non-carrier mosquitoes, ɛ: rate of vector mortality, σ v : transmission rate of the vector virus to the host and σ h : transmission rate of the virus from the host to the vector. The compartment diagram of the host-vector infectious process is shown in Figure 1. µn x 1 (t) σ v y 3 M x 1 x 3 (t) θx 3 (t) x 4 (t) µx 1 (t) µx 3 (t) µx 4 (t) η y 1 (t) σ h x 3 N y 1 y 3 (t) ɛy 1 (t) ɛy 3 (t) Figure 1: Compartment diagram of model 1.

4 1712 Angie J. Osorio R. et al. To calculate R 0, we find the spectral radius of the next generation matrix; considering the infectious populations, that is x 3 and y 3, the next generation matrix is established as y σ 3 v x ( ) M 1 (θ + µ)x3 F = y V = x σ 3 h y ɛy 3 H 1 Now we have that and ( ) Γ(X) DF = (E 0 ) = X DV 1 = ( ) 1 Φ(X) (E 0 ) = X ( ) 0 σ vnɛ η σ h η 0 ɛn ( 1 ) 0 θ+µ 1 0 ɛ where E 0 indicates the infection free equilibrium, in this case E 0 = ( N, 0, 0, η ɛ, 0) So the next generation matrix is given by M = DF ( DV 1 ) = ( 0 σ vn ) η σ h η 0 ɛn(θ+µ) Finally, the basic reproduction number is given by the spectral radius of said matrix, σh σ v R 01 = ρ(m) = ɛ(θ + µ) 2.2 Formulation of model 2 In the construction of model 2, the same parameters and state variables are considered as in model 1, but the intrinsic stage that occurs in the human organism during viremia is considered, for which a state variable denoted by x 2 (t) is included, representing the average number of people in latent state. To calculate R 0, we find the spectral radius of the next generation matrix; considering the infectious populations, that is x 2, x 3 and y 3, the process shown above is repeated, obtaining: σ h σ v γ R 02 = ρ(m) = 3 ɛ(γ + µ)(θ + µ) (1) (2)

5 Impact of the latent state in the R Formulation of model 3 In the construction of model 3, the same parameters and state variables are considered as in model 1, but the extrinsic state of the vector where the virus is multiplied is considered, for which a state variable denoted by y 2 (t) is included, representing the average number of mosquitoes in latent state. To calculate R 0, we find the spectral radius of the next generation matrix; considering the infectious populations, that is x 2, x 3, y 2 and y 3, the basic reproduction number is given by σ h σ v ω R 03 = ρ(m) = 3 ɛ(ɛ + ω)(θ + µ) 2.4 Formulation of model 4 The formulation of model 4 preserves parameters and state variables that intervene in the models presented above and includes a state variable denoted by y 2 (t), representing the average number of mosquitoes in latent state. To calculate R 0, we find the spectral radius of the next generation; considering the infectious populations, that is x 2, x 3, y 2 and y 3, the basic reproduction number is given by 3 Simulations σ h σ v γω R 04 = ρ(m) = 4 ɛ(ɛ + ω)(γ + µ)(θ + µ) Simulations of the formulated models are carried out (Fig. 1) in the Maple software with the purpose of observing the effect of the latent states in the evolution of the dynamics of the average number of infected people (x 3 (t)) and the average number of carrier mosquitoes (y 3 (t)). Figure 2 shows the variation of the dynamics of the average number of people infected for each proposed model. (3) (4)

6 1714 Angie J. Osorio R. et al. Figure 2: Dynamics of the population of infected people for the different models with: σ h = 0.375, β h = 0.75, b s = 0.5, µ = 0.034, ɛ = 0.25, σ v = 0.75, β v = 0.75, b I = 1, θ = 3.96, η = 500, γ = 0.04 and ω = 1/12. Figure 3 shows the variation of the dynamics of the average number of carrier mosquitoes for each proposed model. Figure 3: Dynamics of the population of carrier mosquitoes with: σ h = 0.375, β h = 0.75, b s = 0.5, µ = 0.034, ɛ = 0.25, σ v = 0.75, β v = 0.75, b I = 1, θ = 3.96, η = 500, γ = 0.04 and ω = 1/12. Table 1 shows the value of the basic reproduction number of each formulated model, this value was calculated with the values used in the simulations. Número R 01 R 02 R 03 R 04 Valor Table 1: Value of the basic reproduction number for the same values of the parameters. If a variable f denoting the fraction of people taking preventive measures is introduced into the dynamics, the inclusion of this factor modifies the basic

7 Impact of the latent state in the R reproduction numbers by the appearance of the factor 1 f in them, the expression 1 f expresses the fraction of people who do not take any measure of prevention. The dynamics of the basic reproduction numbers previously calculated when they are a function of the parameter f are shown below. Figure 4: Dynamics of R 0i versus f. 4 Results and conclusions The consideration of the latent states in the proposed dynamic is a factor that directly affects the basic number of reproduction, this can be seen in the equations (1), (2), (3) and (4) representing said threshold regarding each of the considerations discussed in section 2. The latent states both in the host population and in the population of the vector produce a significant change in the evolution of the infection dynamics of the disease, this is evidenced in the simulations of the figures 2 and 3 where the maximum peak reached by the average number of infected people (x 3 (t)) varies, indicating that the inclusion or not of latent states has an impact on the increase of this population. With respect to the effect of the latent states in the dynamics of the disease versus the dynamics represented by the variable y 3 (t), it is shown in the simulation process (see Fig. 3) that the dynamics of this population increases when the latent population is considered. The value of the basic reproduction number tells us how to make the process of the evolution of the disease; considering the values of the parameters implemented in the simulations, the value of this threshold was calculated (see

8 1716 Angie J. Osorio R. et al. Table 1) and the variability of R 0 was shown under small changes in the assumptions considered, indicating the sensitivity that presents this threshold under different considerations that can be considered in the study of a disease, in this case, the dengue virus. Acknowledgements. The authors thank the Grupo de Modelación Matemática en Epidemiología (GMME) of the Universidad del Quindío, Colombia, for making this work possible. References [1] Favier, C. et al., Early determination of the reproductive number for vector-borne diseases: the case of dengue in Brazil, Tropical Medicine & International Health, 11(3) (2006), [2] Chaharborj, S. S., and Gheisari, Y., Study of Reproductive Number in SIR-SIS Model, Advanced Studies in Biology, 3(7), (2011), [3] Dietz, K., The estimation of the basic reproduction number for infectious diseases, Statistical Methods in Medical Research, 2(1) (1993), [4] Greenland, S., and Frerichs, R. R., On measures and models for the effectiveness of vaccines and vaccination programmes, International Journal of Epidemiology, 17(2), (1988), [5] Sharpe, F. R., and Lotka, A. J., A problem in age-distribution, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 21(124) (1911), [6] Dublin, L. I., and Lotka, A. J., On the true rate of natural increase: As exemplified by the population of the United States, 1920, Journal of the American Statistical Association, 20 (151) (1925), [7] Macdonald, G., The analysis of equilibrium in malaria, Tropical Diseases Bulletin, 49(9) (1952), 813. [8] Neyman, J., and Scott, E. L., A stochastic model of epidemics, Stochastic Models in Medicine and Biology, 45 (1964), 83.

9 Impact of the latent state in the R [9] Jacob, C., Branching processes: their role in epidemiology, International Journal of Environmental Research and Public Health, 7(3) (2010), [10] Becker, N. G., The use of mathematical models in determining vaccination policies, Bulletin of the International Statistics Institute, 46 (1975), [11] Dietz, K., Transmission and control of arbovirus diseases, Epidemiology, 104 (1975), 121. [12] Hethcote, H. W., The mathematics of infectious diseases, SIAM Review, 42(4) (2000), [13] Secretaría de Salud, Manual para la vigilancia epidemiológica del dengue, [14] Halstead, S. B., Pathogenesis of dengue: challenges to molecular biology, Science, 239(4839) (1988), [15] SSA, Secretaría de Salud, Manual para la Vigilancia, Diagnóstico, Prevención y Control del Dengue, [16] Hedges, L. M., Brownlie, J. C., O Neill, S. L., and Johnson, K. N., Wolbachia and virus protection in insects, Science, 322(5902) (2008), [17] Gustavo, K., El dengue, un problema creciente de salud en las Américas, [18] Ávila Agüero, M. L., Dengue: Una enfermedad que vino para quedarse, Revista Médica del Hospital Nacional de Niños Dr. Carlos Sáenz Herrera, 39(1), (2004), [19] Hayes, E. B., and Gubler, D. J., Dengue and dengue hemorrhagic fever, The Pediatric Infectious Disease Journal, 11(4) (1992), [20] Herrera-Basto, E., Prevots, D. R., Zarate, M. L., Silva, J. L., and Sepulveda-Amor, J., First reported outbreak of classical dengue fever at 1,700 meters above sea level in Guerrero State, Mexico, June 1988, The American Journal of Tropical Medicine and Hygiene, 46(6) (1992), [21] Kautner, I., Robinson, M. J., and Kuhnle, U., Dengue virus infection: epidemiology, pathogenesis, clinical presentation, diagnosis, and prevention, The Journal of Pediatrics, 131(4) (1997),

10 1718 Angie J. Osorio R. et al. [22] Kouri, G. P., Guzmán, M. G., Bravo, J. R., and Triana, C., Dengue haemorrhagic fever/dengue shock syndrome: lessons from the Cuban epidemic, 1981, Bulletin of the World Health Organization, 67(4) (1989), 375. Received: November 5, 2018; Published: December 28, 2018