Understanding the effect of vector dynamics in epidemic models using center manifold analysis

In vector borne diseases the human hosts' epidemiology often acts on a much slower time scales than the one of the mosquitos which transmit the disease as a vector from human to human, due to their vastly different life cycles. We investigate in a model with susceptible (S), infected (I) and recovered (R) humans and susceptible (U) and infected (V) mosquitoes in how far the fast time scale of the mosquito epidemiology can be slaved by the slower human epidemiology, so that for the understanding of human disease data mainly the dynamics of the human time scale is essential and only slightly perturbed by the mosquito dynamics. This analysis of the SIRUV model is qualitatively in agreement with a previously investigated simpler SISUV model, hence a feature of vector-borne diseases in general.


INTRODUCTION
Modelling vector-borne diseases and as one of the main examples dengue fever has attracted the attention of biomathematics researchers since a long time. Dengue fever and dengue hemorrhagic fever are vector-borne diseases caused by an arbovirus with 4 serotypes, which infects millions of people every year in tropical and subtropical areas. The main vector of the disease is the mosquito Aedes aegypti which is well adapted to live in human environments. This mosquito species lives in tropical and subtropical areas, but it is spreading its habitat thanks to the global changes and also the globalization of the humans societies. The fast spreading that has been observed in recent years is expanding the risk areas so, in order to avoid new outbreaks, research interest is increasing to prevent disease spreading by e.g. vaccine development and modelling and predicting the spread of disease.
In this paper we investigate the interplay between mosquitos and humans dynamics in vector-born diseases in the SIRUV setting, hence modelling transitions between susceptible (S), infected (I) and recovered (R) humans and susceptible (U) and infected (V) mosquitoes, after breifly reviewing earlier results in a simplified SISUV setting, in which geometric insights are easier obtained than in the SIRUV model.

THE SIMPLEST MODEL FOR VECTOR-BORNE DISEASES
For the human disease epidimiology we assume in the simplest case an SIS model (susceptibles, infected and again susceptibles) for the human epidemiological system coupled with the dynamic of the vectors, which can be susceptible (U) or infected (V). This gives the SISUV model as simplest coupled human-vector system where α is the recovery rate, β is the infection rate, ϑ is the infection rate of mosquitos, ψ is the number of mosquitos that are born and ν is the death rate of mosquitos. If we consider constant population sizes for the humans N = S(t) + I(t) and for the mosquitos M = U(t) + V (t) it is possible to simplify the model to a two dimensional dynamical system, given by represents a fixed relationship between the state space variables as will be analyzed below. A closer look at these parameters shows that the parameters for the mosquito dynamics ν = 36.5y −1 and ϑ = 73y −1 are much larger than the parameters for the human dynamics α = 0.1y −1 and β = 0.2y −1 . To describe this separation of time scales we set ν =:ν ε withν in the scale of α and β , e.g. we can set exactlyν := α giving ε =ν ν = 0.1 36.5 . Then we set ϑ =:θ ε givinḡ ϑ = ϑ ν α. With these parameters we obtain for the ODE system the form with different time scales, slow for the I dynamics and fast for the V dynamics. This system has been analyzed in [1] as slow-fast dynamics, giving as result a relation between the fast variable V as function of the slow variable I as In order to understand the general set-up also in systems where a simple separation into fast and slow ODEs is not possible, like it will be the case of the SIRUV-model, we now apply an approximate center manifold analysis around the endemic fixed point, which again gives the information of the manifold in which the slow dynamics finally evolves. In order to obtain the center manifold for the slow dynamics we first analyze the endemic fixed point and the eigenvectors and eigenvalues of the Jacobian matrix around the fixed point given by Eq. (4) and From the Jacobian Matrix we calculate the eigenvalues and the eigenvectors for i = 1, 2 Inspecting the numerical values it is possible to verify that one eigenvalue is close to 0 ( λ 1 ≈ 0 ) and another one is largely negative ( λ 2 ≈ d << 0 ). This approximation implies that the second eigenvector is In figure 1 a) we plot the eigenvector directions around the fixed point. It is possible to see that the second eigendirection is almost in V direction and the other one is tangent to the curve V (I).
In order to perform the center manifold analysis the (I,V ) coordinate system was shifted into a (x, y) system with the global fixed point at the origin and the eigenvectors as coordinate axis. Thsi transformation in done considering To find the transformation y = h(x) along the center manifold, the functional N (h (x)) has to vanish with h having vanishing value and vanishing first derivative at the origin. The functional Eq. (8) can be solved e.g. by comparison of coefficients of a polynomial expansion of h. The polynomial approximation · a j · a j− (10) for j = 4, 5, ..., ∞. The polynomial approximation (shown in the figure 1 b)) fits well to the manifold V (I). From the polynomial approximation it is possible to calculate the backtransformation into the original (I,V ) coordinate system ( figure 1 c)), thanks to the equation In a stochastic version of this SISUV system we observe qualitatively the same behaviour as in the deterministic system, namely fast initial trajectory in V direction followed by a slower convergence along the center manifold towards the fixed point [1]. Around the fixed point the stochastic system keeps fluctuating mainly along the center manifold. the stochastic system is formulated in the framework of time continuous Markov processes, in physics often also called master equations [2].

THE SIRUV MODEL FOR COUPLED HUMAN AND MOSQUITO EPIDEMIOLOGY
The SIRUV model is based on a more complex epidemic model (SIR) which has an additional state (recovered humans). This case was created to consider diseases, such as dengue fever, where humans after recovery become immune to the pathogen. The vectors dynamic remains as before because the mosquitos donot have an immune system, hence cannot recover from the infection. The correspondent ODE system is given by d where γ and μ are the birth and death rates of the humans, respectively. For the simulations, these parameters are set as γ = 1 7 d = 365 7 y −1 and μ = 1 65 y . Considering constant population sizes for humans (N(t) = S(t) + I(t) + R(t)) and for mosquitos (M(t) = U(t) +V (t)), the ODE system can be symplified to a three dimensional SRV system Once again, the dynamic of the vector is faster than the human one, so it is necessary to separate the time scales and change the parameters considering ν =ν ε and γ =γ ε , and changing as well the connected parameters ϑ and β . A trajectory is plotted in state space projections in figure 2. The stationary states of the system are given by . In order to verify the stability of the system we found the Jacobian matrix for the model The plots of figure 2 suggest that the system has two complex conjugate eigenvalues λ 1/2 = a ± ib and another one which is real λ 3 = c. As the plot shows a fast approach to an invariant manifold, and then in this manifold a slow spiraling into the fixed point it is to be expected that a ≈ 0 and c << 0, and the numerical simulation show that (λ 1/2 = a ± ib = −0.029909 ± i · 0.99478 and λ 3 = c = −88.68). In the same way, we expect two complex eigenvectors (u 1/2 = v ± i · w) and the other one real (u). The vectors needed to calculate the eigenvectors of the model are given by The real eigenvector indicates that the model goes in V direction. But the complex eigenvectors with real vectors indicate a very small angle between strongly contracting direction and the invariant manifold. This seems to reflect the close correlation between infected mosquito dynamics and human disease dynamics. Still a slaving of the fast mosquito dynamics via the slow human dynamics is observed due to the huge difference between real eigenvalue with c = −88.68 and the near zero real part of the other two eigenvalues a = −0.029909, leaving the mosquito dynamics to matter little in the approach towards the endemic fixed point.