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Outbreaks of Hantavirus induced by seasonality

Using a model for rodent population dynamics, we study outbreaks of Hantavirus infection induced by the alternation of seasons. Neither season by itself satisfies the environmental requirements for propagation of the disease. This result can be explained in terms of the seasonal interruption of the relaxation process of the mouse population toward equilibrium, and may shed light on the reported connection between climate variations and outbreaks of the dise ase.


I. INTRODUCTION
Hantaviruses are rodent-borne zoonotic agents that may cause diseases in humans such as hemorrhagic fever with renal syndrome and hantavirus pulmonary syndrome [1, 2,3].
Hantaviruses have been identified at an increasing rate in recent years, and as of now about thirty different ones have been discovered throughout the world. One of these, the Sin Nombre virus, was not isolated until 1993 after an outbreak in the Four Corners Region in the USA [1,2]. The host of this particularly dangerous virus is the deer mouse, Peromyscus maniculatus, the most numerous mammal in North America. The virus produces a chronic infection in the mouse population, but it is not lethal to them. It is believed that transmission in the rodent population is horizontal and due to fights, and that the subsequent infection of humans, where the mortality rate can be as high as fifty percent, is produced by their contact with the excreta of infected mice. Moreover, so far there is no vaccine or effective drug to prevent or treat the hantavirus pulmonary syndrome. Therefore, a major effort has been made to understand the population dynamics of deer mice colonies in order to design effective prevention policies [1].
It has been noted that environmental conditions are directly connected to outbreaks of Hanta [2]. For instance, the 1993 and 1998 outbreaks occurring in the Four Corners Region have been associated with the so-called El Niño Southern Oscillation [2]. Related to this and other such observations, the effects of seasonality in ecological systems have been a subject of recent interest [4,5]. Multi-year oscillations of mammal populations [6], preypredator seasonal dynamics [7], and persistence of parasites in plants between seasons [8] are examples that illustrate the importance of seasonality in population dynamics.
itself fulfills the environmental requirements on the carrying capacity for spreading of the infection, may produce an outbreak of the disease. The mechanism driving this behavior is the interruption of the relaxation process that equilibrates the mouse population from season to season: if the duration of seasons becomes shorter than the relaxation time of the population, the disease spreads.
The paper is organized as follows. In Sec. II we briefly review the model for mouse populations introduced in [9]. In Sec. III we explain how seasonality is introduced in that model and analyze the conditions for Hanta outbreaks to take place due to the alternation of seasons. The exact solution of the model and a particular example that illustrates the phenomenology is given in Sec. IV. Finally, in Sec.V, we summarize the main conclusions and propose some directions for future work.

II. THE MODEL
The model introduced in [9] for the temporal evolution of a population of mice subjected to the Hantavirus infection reads: for any value of the parameters. The other two equilibria are The stability of the equilibrium points (2) and (3) depends on the value of the carrying (2) is stable and (3) unstable. If K > K c it is the other way around. That is, when the available resources, K, are below the critical value, K c , the infection does not propagate in the colony, the whole population of mice grows healthy, and its size increases proportionally with those resources. As soon as K surpasses K c the virus spreads in the colony, the susceptible mouse population saturates, and the fraction of infected mice becomes larger as K increases (see Fig. 1).

The Four Corners Region, where an important number of cases of Hantavirus Pulmonary
Syndrome have occurred, has a desert climate. The largest climate variations within this region come from the alternation between dry and rainy seasons. We will therefore assume alternation in time of these two seasons. It is important to remark that a two-season assumption is not crucial, and that the analysis with four seasons is also straightforward within the formalism introduced herein. During each of the two seasons under consideration we assume there to be no climate variations, so that each season can be characterized by a We implement square-periodic season alternation where the duration of each season is T /2. Again, this particular alternation pattern is not essential for the mechanism. Other schemes of season alternation, e.g. different duration of the seasons or random switching between seasons, do not qualitatively change the phenomenology.
Any quantity ρ(t) alternating in the way described above can be written as: where ρ ± = 1 2 (ρ 1 ± ρ 2 ) and µ(t) is a periodic square wave Let us now suppose the following conditions for the sets {ρ i } according to seasonality: where 1 stands for the rainy season and 2 for the dry one. The biological motivation for these conditions is the following. The dry season provides less resources for the colony than the rainy season (K 2 < K 1 ), and as a consequence the death rate is higher (c 2 > c 1 ), and the transmission rate is also larger since fights for the available resources are expected to increase (a 2 > a 1 ). However, notice that we consider the birth rate to be larger during the dry season There are two reasons for this. First is the assumption that the colony makes an attempt to maintain its population. Second is an implementation of the maturation process in the model: baby mice do not contribute to the propagation of the disease [1], and, therefore, even if births were actually more numerous during the rainy season, the Moreover, we assume that K 1 < min(K c1 , K c2 ), i.e., the resources are all times (during both seasons) below the minimum critical threshold that triggers the propagation of the disease. We will show that nevertheless it is possible for the infection to spread. The internal time scale is defined as the fastest relaxation process, i.e., t i = min[t r (i → j)].
"Fast" or "slow" seasonal alternation then refers to the comparison between these two time scales. If t e ≫ t i the mouse population has enough time to accommodate to the new conditions from season to season and relax to equilibrium. Moreover, since we have imposed the condition that the resources at any time of the year are below the critical thresholds K ci , there will be no infected mice. In the other limit, t e ≪ t i , seasonal changes occur too fast, the relaxation process is interrupted, and no equilibrium can be reached from season to season. Then note that an adiabatic elimination can be implemented [12], and µ(t) in Eq. (4) can be replaced by its average value, µ(t) = 0. Therefore, in the limit of fast season alternation the system is driven by the set of averaged values {ρ + } = {a + , b + , c + , (1/K) + }, and the critical carrying capacity is given by . As a consequence, it is possible to find regions of parameters where K c+ is smaller than the effective value of the carrying capacity associated with the averaged values: and the infection propagates.
General conditions leading to this behavior can be posed, but the expressions are rather cumbersome. We prefer, for the sake of simplicity, to show a particular typical case. We use the following values for the parameters: that lead to K c1 = 6 and K c2 = 73/4 respectively. The dynamics are completely determined once the value of the carrying capacity during each season is specified. According to the previous discussion, these parameters can be chosen such that the following conditions hold: (1/K) + −1 > K c+ , K 1 > K 2 , K 1 < min(K c1 , K c2 ).
These conditions lead to the points (K 1 , K 2 ) that fulfill the seasonal requirements given by Eq. (5), so that slow alternation of seasons leads to infection-free states while fast alternation leads to Hanta outbreaks. This region is plotted in Fig. 2. Notice in particular that the point (K 1 = 4, K 2 = 1) lies inside the region and that K i < K ci . In the next section we illustrate the seasonality-induced propagation of the disease for this particular point.

IV. THE CRITICAL PERIOD
So far we have determined that outbreaks of Hanta induced entirely by seasonal changes can occur if the duration of the seasons are short enough. Now we establish the meaning of "short enough" quantitatively. Since K c+ is strictly smaller than the effective value of the carrying capacity, there should be a finite value of T c such that for any T < T c the population of infected mice is greater than zero, but for periods above this critical period the infected population goes to zero.
In order to obtain the value of the critical period we need to solve the system of equations (1a) and (1b). In spite of its nonlinearities the system can be solved analytically by means of a reciprocal transformation [13] and the following exact solution is obtained: (Ω(t)e ct ) where I 0 and S 0 are the initial conditions for M I and M S respectively, and the following definitions have been introduced, Because the external forcing due to the alternation of seasons is periodic, Floquet theory ensures the existence of a periodic solution. The values of I 0 and S 0 compatible with the nonequilibrium periodic solution can be obtained by evolving the system during the first half of a period under dynamics 1 and the second half under dynamics 2, and forcing periodity on the solutions after a whole period of evolution, that is, The striking feature of that behavior lays in the fact that neither season satisfies the conditions for the infection to spread in terms of the availability of resources. The mechanism responsible for the phenomenon is the competition between two time scales: an external one, the duration of a season, and an internal one, the relaxation time for the mouse colony to equilibrate its population from season to season. We have shown that if the duration of the seasons is longer than the relaxation time, no propagation of Hantavirus occurs. On the other hand, if the relaxation process is interrupted by a fast seasonal alternation, the disease spreads. We have analyzed the general conditions for which the phenomenon occurrs.
Moreover, we have illustrated the mechanism with a particular example that can be solved exactly.
This work may help to clarify the reported relation between climate and propagation of Hanta in mice populations. However, to elucidate whether the proposed phenomenon actually takes place in nature we depend on data that unfortunately are not available in the literature. One can envision further modifications of the model that may improve its features, such as, for example, the inclusion of spatial dependence or of noisy contributions to the dynamics. Finally, we stress that the general idea underlying the mechanism is modelinsensitive and can therefore be extended to other systems where seasonality plays a relevant role. Work along these directions is in progress.