Sistemas dinâmicos periódicos para hospedeiros e mosquitos infectados
W. M. Oliva and E. M. Sallum
Departamento de Matemática Aplicada do Instituto Superior Técnico da Universidade Técnica de Lisboa. Portugal (W.M.O.), Instituto de Matemática e Estatística da Universidade de São Paulo. São Paulo, SP - Brasil (E.M.S.)
| ABSTRACT
A mathematical model for the purpose of analysing the dynamic of the populations of infected hosts anf infected mosquitoes when the populations of mosquitoes are periodic in time is here presented. By the computation of a parameter l (the spectral radius of a certain monodromy matrix) one can state that either the infection peters out naturally) (l £ 1) or if l > 1 the infection becomes endemic. The model generalizes previous models for malaria by considering the case of periodic coefficients; it is also a variation of that for gonorrhea. The main motivation for the consideration of this present model was the recent studies on mosquitoes at an experimental rice irrigation system, in the South-Eastern region of Brazil. Malaria, epidemiology. Culicidae. Population dinamics. Ecology, vectors. |
| RESUMO
Desenvolveu-se um modelo matemático para analisar a dinâmica das populações de indivíduos e mosquitos infectados quando as populações de mosquitos são periódicas no tempo. Pela determinação de um parâmetro l (o raio espectral de uma matriz de monodromia) pode-se estabelecer que a infecção termina naturalmente (l £ 1) ou se l > 1 que a infecção torna-se endêmica. O modelo generaliza, para o caso de coeficientes periódicos, modelos anteriores para malária; como também é uma variação de modelo para a gonorréia. A principal motivação para a consideração do modelo proposto foram os recentes estudos sobre mosquitos numa estação experimental de arroz irrigado, na região Sudeste do Brasil. Malária, epidemiologia. Culicidae. Dinâmica populacional. Ecologia de vetores. |
INTRODUCTION
A model was constructed for the purpose of analysing the dynamic of the populations of infected hosts and infected mosquitoes, when the populations of mosquitoes are periodic in time. A deterministic estimator l was established to predict whether the infection will peter out naturally (when l £ 1) or, if l > 1, the infection will become endemic (see Appendix). In the model, incubation and immunity are neglected.
The special case of one patch of hosts and two patches of mosquitoes are considered and the main results concerning the dynamics are given (see Theorems A and B). The estimator l is the spectral radius of the monodromy matrix C, corresponding to the periodic matrix A(t) of the linear system associated with the complete system (1). It is a non trivial matter to compute the matrix C as also to discover its spectral radius l.
The case of one patch of hosts and one (resp. two) patch(es) of mosquitoes is studied and it is assumed that the populations of mosquitoes behave according to periodic step functions. The monodromy matrix C is the product of exponentials of computable matrices; moreover, using suitable changes of coordinates, it was possible to obtain explicitly all the exponentials that appear in the expression of C, so that its spectral radius can be easily determined.
The general case (system (*)) that can be used for any number of patches of hosts and mosquitoes is presented in the Appendix.
The main motivation for the construction of the model here presented as well as the applications made was the recent studies by Forattini et al.^{4} on mosquitoes at an experimental rice irrigation system^{5, 6, 7, 8, 9}.
The constant and periodic coefficients that appear in the equations of the mathematical model should be determined experimentally. If they are available, reasonable predictions about possible mosquito-borne diseases that can appear at irrigation systems can be made.
ONE PATCH OF HOSTS AND TWO PATCHES OF MOSQUITOES
A deterministic model is here presented with a view to describing the dynamic of the population of hosts and mosquitoes infected by malaria when there is a homogeneous group of hosts with a constant population H and two groups of mosquitoes of different types i = 1, 2, with population V_{i}_{}= V_{i}(t) periodic in the time t with periodic T > 0.
Let us considere the following system of ordinary differential equations:
where:
H : population of hosts;
V_{i}_{}= V_{i}(t) = V_{i}(t + T) : population of mosquitoes at instant t, periodic of periodic T > 0;
S = S(t) : population of infected hosts at instant t;
I_{i}_{}= I_{i}(t) : population of infected mosquitoes of type i at instant t;
d _{i} : death rate of mosquitoes of type i;
x : cure rate of sick hosts;
b'_{i} : bites by one mosquito of type i on hosts per unit of time;
b_{i} = b_{i}(t) : bites by mosquitoes of type i taken on one person, per unit of time, which is a periodic function with period T > 0;
f'_{i}I_{i} : population of infected mosquitoes of type i which are infective;
f_{i}S : population of infected hosts which are infective.
Since b_{i}(t)H = b'_{i}V_{i}(t), the system above can be written as:
where H, x, b'_{i}, d_{i }, f_{i }are positive constants and V_{i} = V_{i}(t) are continuous periodic functions of period T > 0.
That system (1) corresponds to a generalization of the Ross model (Lotka^{12}) and of that of Dye-Hasibeder^{2, 3} for malaria. Moreover it is also a variation of the Aronson-Mellander^{1} model that describes the dynamics of gonorrhea.
The main results A and B, stated below, for system (1), are special cases of general results proved in the Appendix.
Theorem A
Let (S(t), I_{1}(t), I_{2}(t)) be a non zero solution of (1) such that 0 £ S(t_{0}) £ H, 0 £ I_{i}(t_{0}) £ V_{i}(t_{0}), i = 1, 2, and some t_{0}³ 0. Then 0 < S(t) < H, 0 < I_{i}(t) < V_{i}(t), i = 1, 2, for all t > t_{0}.
We may write (1) in a matricial form
Let ø(t) be the matricial solution of = A(t)y such that ø(0) = I_{d}. The monodromy matrix C = ø(T) is positive (Aronson, Mellander^{1}, Lemma 2) and, by Perron's theorem (Gantmacher^{10}), it has a simple positive eigenvalue l such that
Theorem B
There are two possibilities for the non zero solutions (S(t), I(t)) = (S(t), I_{1}(t), I_{2}(t)) of (1) such that for some t_{0}³ 0, 0 £ S(t_{0}) £ H and 0 £ I_{i}(t_{0}) £ V_{i}(t_{0}), i = 1, 2:
a) If l £ 1 then (S(t), I(t)) tends to the zero solution as t ® ¥ ;
b) If l > 1, then there exists a unique T -periodic solution (S*, I*) such that for any t > t_{0} we have 0 < S*(t) < H and 0 < I*_{i}(t) < V_{i}(t), i = 1, 2. In this case (S(t) - S*(t), I(t) - I* (t)) tends to zero as t ® ¥ .
In other words, Theorem B says that the infection peters out naturally when l £ 1; or, if l > 1, the infection becomes endemic if the initial number of infected cases in at least one group is positive.
As usual, it is a non trivial matter to obtain the matrix C = ø(T) and an expression for l.
THE PERIODIC POPULATIONS OF MOSQUITOES AS STEP FUNCTIONS
One patch of hosts and one patch of mosquitoes
Let us consider the following system:
This describes the dynamic of malaria when we have one group of mosquitoes with a periodic population V = V(t) interacting with a homogeneous group of individuals of a fixed population H. It will be shown that for V = V(t) periodic with period T = 12 (months) with V(t) = V_{i} positive and constant, i < t < (i + 1), i = 1, ..., 12, the eigenvalues of the fundamental solution of the associated linear system can be computed directly showing their dependence on the data of system (2) that is, on the parameters x, d, H, b', f and on V =_{}V(t).
Let us write (2) in the following form
For each i = 1, ..., 12 one considers ø_{i}(t) = ^{}, the fundamental solution of
If we make U = - dS - b'fI, (2)' becomes
one obtains
Let ø(t) be the matricial solution of such that ø(0) = I_{d }. Then one can write
So from (2)' we have
and so, from (6)
where the are the real eigenvalues of A(i) = A_{i}, j = 1, 2, i = 1, ..., 12. From (3),
so we have and then
Finally we are able to obtain the eigenvalues of ø(T) that are those of the following matrix:
One Patch of Hosts and Two Patches of Mosquitoes
In this section we consider system (1) with d_{1} = d_{2} = d and V_{i} =_{}V_{i} (t) as a periodic step function, which is positive and constant during the month i, i = 1, 2, ..., 12, and periodic with period equal to T = 12 (months). We write
The three invariants (coefficients of the characteristic polynomium) of A(t) are:
det A(t) = - xd ^{2} +dj(t)
TrA(t) = - x - 2d (TrA(t) does not depend on t)
D_{2}A(t) = 2xd +d ^{2}- j(t)
where
It is easy to check that (- d ) is a eigenvalue of A(t), say l_{3} = - d . From the expressions of the invariants, the other two eigenvalues l_{1} and l_{2} satisfy.
These two last equations imply that
Note that the three eigenvalues of A(t) are real and distinct; moreover, we have l_{2} <- d < l_{1}.
For each i = 1, 2, ..., 12, one considers ø_{i}(t) = e^{tA}^{(i)} which is the fundamental solution of
If we make
_{}
_{}system (9) becomes
or
where
From (10) we see that A(i) = M(i)M^{-}^{ 1} with M^{-}^{ 1 }=
Let F(t) be the matricial solution of
such that I(0) = I_{d}. Then, as above, one can write
Let us write the matrix (i) of system (11) in the form
Then
and also
The eigenvalues of F(T) are e^{-}^{Td }together with the eigenvalues of e^{B}^{(12)}e^{B}^{ (11)}...e^{B}^{(1)}. So we only need to compute e^{B}^{(i)} for i = 1, ..., 12. For that one has to solve the system
or, equivalently, to solve the second order equation
Since the characteristic roots of (13) are the eigenvalues of B(i), that is,
j = 1, 2, and i = 1, ..., 12, then, as we did in section 3.1, e^{B}^{(i)} is given by the second member of equality (8).
APPENDIX
Let us consider the following system
which can be written as
y' = A(t)y + N(t, y) where
Assume a_{i} to be positive constants; c_{i}= c_{i}(t) to be positive continuous periodic functions of period T > 0; b_{ji} = b_{ji}(t) to be non negative continuous periodic functions of period T > 0 such that A(t) is an irreducible matrix (Gantmacher^{10}), for all t. Moreover, assume there exists e > 0 such that b_{ji}(t) ³ e for all t provided that b_{ji}(t) are not identically zero.
For x = (x_{1}, ..., x_{n}) and y = (y_{1}, ..., y_{n}) we denote x £ y (x < y) if, for all i, x_{i }£ y_{i} ( x_{i }< y_{i} ). With the same arguments used in Aronson, Mellander^{1} one can state:
I) If y(t) and z(t) are non-zero solutions of (*) such that for some t_{0}³ 0 we have 0 £ y(t_{0}) £ z(t_{0}) £ c(t_{0}) with y(t_{0}) ¹ z(t_{0}), then 0 < y(t) < z(t) < c(t) for all t > t_{0}.
For fixed t_{0} ³ 0 one considers the map f_{}(y_{0}) = y(t_{0} + T, t_{0}, y_{0}) for 0 £ y_{0}£ c(t_{0}). When t_{0} = 0 we take the sequence of positive numbers c(0) = c_{0} > c_{1} > c_{2} > ... > c_{n} > ... > 0, where c_{n} = f_{0 }(c_{n}_{- 1}) and Q = c_{n} ³ 0. When Q = 0 we have y(t, t_{0}, y_{0}) = 0 for 0 £ y_{0}£ c(t_{0}). If Q 0 we have y(t) = y(t, 0, Q) positive, periodic with period T > 0 and, by I), Q > 0.
Let F(t) the matrix solution of = A(t)y, F(0) = I_{d}; since A(t) = (a_{ij}(t)) is irreducible with a_{ij}(t) ³ 0 for i ¹ j, then the matrix C = F(T) is positive and so, by Perron's theorem, it has a simple positive eigenvalue l = max{Rel_{i}; det(C - l_{i}I_{d}) = 0}.
As in theorem 1 of Aronson, Mellander^{1} one has analogously:
II)If l < 1, there exists K > 0 such that | y(t)| £ |y(t_{0})| for all t ³ t_{0}_{}³ 0 and any solution y(t) = y (t, t_{0},y_{0}) of (*) such that 0 £ y_{0}£ c(t_{0}).
Consider now the case l > 1. Let E(t_{0}) = {y Î IR^{n}: 0 £ y £ c(t_{0})}, w > 0 eigenvector of C^{t}corresponding to l, v(t_{0})^{t}= w^{t}F(t_{0})^{- 1 }and E_{q}(t_{0}) = {y Î E(t_{0}) : v(t_{0})^{t} y ³ q}. We claim that for q > 0 sufficiently small we have _{}(E_{q}(t_{0} )) Ì E_{q}(t_{0}) where 0 £ t_{0}£ T. In fact
Since
is continuous on the compact set
{(t_{0}, y) | t_{0}Î [0, T], y Î E(t_{0})},
then there exist d > 0 and N > 0 such that ³ v(t_{0})^{t}y + N | y | for t_{0 }Î [0, T] and | y | £ d. Using the Brower fixed point theorem (Hönig^{11}) one concludes that has a fixed point in E_{q}(t_{0}).
For any two solutions y(t) = y(t, t_{0}, y_{0}) and z(t) = y(t, t_{0}, z_{0}) of (*) such that 0 < y_{0}, z_{0 }< c(t_{0}), one has
D^{+}u(t) £ - (u - 1) min(y_{z}, y_{y}) < 0 for t ³ t_{0 }³ 0
(see Aronson and Mellander^{1}; Lemma 5, [1]).
So, one can conclude that for q > 0 sufficiently small, f_{} : E_{q}(t_{0}) ® E_{q}(t_{0}) has only one fixed point Q, and then f_{} : E(t_{0}) ®E(t_{0}) has only one fixed point Q > 0, besides the origin, that corresponds to a periodic orbit of period T > 0 for system (*).
For y(t) = y(t, t_{0}, y_{0}) with 0 £ y_{0 }£_{}c(t_{0}) and z(t) = y(t, t_{0}, Q) we have
where and p is continuous and T-
periodic, T > 0. So, for each x_{0} > 0 sufficiently close to the origin, there exists a constant M(x_{0}) > 0 such that
for all y_{0}, x_{0 }£ y_{0 }£_{}c(t_{0}) and all t ³ t_{0} ³ 0 because the c_{k}(t) are bounded functions (see Aronson, Mellander^{1}, Th. 2).
Given a compact set K Ì E(t_{0}), one considers a point x_{0}, 0 < x_{0 }<_{}c(t_{0}), such that x_{0 }< y(t_{0 }+ T, t_{0}, y_{0}) <_{}c(t_{0}) for all y_{0}Î K. Since there are constants M(x_{0}) > 0 and N > 0 such that for all y_{0} Î K we have
then one has the following result.
III) For l > 1, system (*) admits a unique non zero periodic solution y(t, t_{0}, Q), which has period T, and a constant a > 0 such that for each compact set K Ì E(t_{0}) there corresponds a constant M_{K}> 0 and we have
Moreover, as in theorem 3 of Aronson, Mellander^{1} we state:
IV) For l = 1 there is a constant L > 0 such that for any solution y(t, t_{0}, y_{0}) = y(t) of (*) with 0 £ y_{0 }£_{}c(t_{0}), t_{0} ³ 0, we have
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Correspondence to:
Elvia Mureb Sallum
Departamento de Matemática Aplicada do Instituto de Matemática e Estatística da Universidade de São Paulo.
R. do Matão, 1010. Cidade Universitária - 05508-900 São Paulo, SP - Brasil.
Fax: (011) 814.4135
E-mail: elvia@ime.usp.br
The publication of this article was sponsored by FAPESP (Process 95/2290-6).
Received in 5.17.1995. Approved in 1.29.1996.