Rev. Saúde Pública vol.30 número3

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 λ (the spectral radius of a certain monodromy matrix) one can state that either the infection peters out naturally) (λ≤ 1) or if λ > 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 popu-lações de indivíduos e mosquitos infectados quando as popupopu-lações de mosqui-tos são periódicas no tempo. Pela determinação de um parâmetro λ (o raio espectral de uma matriz de monodromia) pode-se estabelecer que a infecção termina naturalmente (λ ≤ 1) ou se λ > 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 recen-tes estudos sobre mosquitos numa estação experimental de arroz irrigado, na região Sudeste do Brasil.

Malária, epidemiologia. Culicidae. Dinâmica populacional. Ecologia de vetores.

Periodic dynamic systems for infected hosts

and mosquitoes

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.)

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: [email protected]

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 λ was established to predict whether the infection will peter out naturally (when λ ≤ 1) or, if λ > 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 λ 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 λ.

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 system5, 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

dS_{= − εS} (H − S)

+ (b'₁f₁I₁ + b'₂f₂I₂)

dt S

dI₁ b₁f₁S = − δ₁I₁ + (V₁ − I₁)

dt VI

dI₂ b₂f₂S = − δ₂I₂ + (V₂ − I₂)

dt V₂

dS (H − S)

= − εS + (b'₁f₁I₁ + b'₂f₂I₂)

dt H

dI₁ b'₁f₁S = − δ₁I₁ + (V₁ − I₁)

dt H

dI₂ b'₂f₂S = − δ₂I₂ + (V₂ − I₂)

dt H

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;

δ i : death rate of mosquitoes of type i; ξ: 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 ’_iI_i : population of infected mosquitoes of type i which are infective;

f_iS : population of infected hosts which are infective.

Since b_i(t)H = b’_iV_i(t), the system above can be written as:

(1)

where H, ξ, b’_i, δ_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 (Lotka12_{) and of that of}

Dye-Hasibeder2, 3 _{for malaria. Moreover it is also a}

variation of the Aronson-Mellander1 _{model that}

describes the dynamics of gonorrhea.

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 ξ, δ, 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) = e , the fundamental solution of

(2)’

If we make U = − δS − b’fI, (2)’ becomes

(3)

Since A_i= MA_iM−1_{, with M =}

one obtains

(4) ø_i(1) = e = Me M−1_.

Let ø(t) be the matricial solution of = A(t)

such that ø(0) = I_d . Then one can write

ø(T) = ø₁₂(1)ø₁₁(1)ø₁₀(1)...ø₂(1)ø₁(1)

(5) = Me e e ...e e M−1_.

So from (2)’ we have

(6) S− (trace A_i)S + (det A_i)S = 0

and so, from (6)

(7) S(t) = αe + βe

where the are the real eigenvalues of A(i) = A_i, j = 1, 2, i = 1, ..., 12. From (3),

S = (trace A_i)S − U

Theorem A

Let (S(t), I₁(t), I₂(t)) be a non zero solution of (1) such that 0 ≤S(t₀) ≤H, 0 ≤I_i(t₀) ≤V_i(t₀), i = 1, 2, and some t₀≥ 0. Then 0 < S(t) < H, 0 < I_i(t) < V_i(t), i = 1, 2, for all t > t₀.

We may write (1) in a matricial form

˙

y = A(t)y + N(t, y), y = (S, I₁, I₂), where

Let ø(t) be the matricial solution of ˙y = A(t)y such that ø(0) = I_d. The monodromy matrix C = ø(T) is positive (Aronson, Mellander1_{, Lemma 2) and, by}

Perron’s theorem (Gantmacher10_{), it has a simple}

positive eigenvalue λ such that

λ = max{Reλ_i : det(C− λ_iI_d) = 0}.

Theorem B

There are two possibilities for the non zero solutions (S(t), I(t)) = (S(t), I₁(t), I₂(t)) of (1) such that for some t₀≥ 0, 0 ≤S(t₀) ≤H and 0 ≤I_i(t₀) ≤V_i(t₀), i = 1, 2: a) If λ ≤ 1 then (S(t), I(t)) tends to the zero solution

as t → ∞ ;

b) If λ > 1, then there exists a unique T -periodic solution (S*, I*) such that for any t > t₀ 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 λ ≤ 1; or, if λ > 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 λ.

THE PERIODIC POPULATIONS OF MOSQUITOES AS STEP FUNCTIONS

One patch of hosts and one patch of mosquitoes

Let us consider the following system:

(2)

A_it

S S

= A_i where A_i = A(i)

I I

A

i

A

12 A11 A10 A2 A1

λi

1t λ

i

2t

λi

j

.

S S trace A_i − 1

= A_i , A_i= .

U U det A_i 0

A

i

1 0

δ 1

− −

b'f b'f

S I − ξ b'₁f₁ b'₂f₂

b'₁f₁V₁

A(t) = − δ₁ 0 .

H b'₂f₂V₂

0 − δ₂ H

dS b'f I

= − ξS + (H − S)

dt H

dI b'f S

= − δI + (V(t) − I)

dt H

S _S − ξ _b'f

= A(t) + N(S, I) whereA(t) = .

I _{I b'f V(t)}

− δ H

.

.

.

¨

.

.

.

S I

. .

For each i = 1, 2, ..., 12, one considers ø_i(t) = etA(i)

which is the fundamental solution of

(9)

If we make

(10) W = − δS − b'₂f₂I₂

U = − 2δS − b'₁f₁I₁ − b'₂f₂I₂

system (9) becomes

S = TrA(i)S − U (11) U = [δ 2 ₊ ∆

2A(i)]S + δU

(b'₂f₂)2

W = δξ + δ2− _V

2 S + δU− δW

H

or

whereA(i) =

From (10) we see that A(i) = MA(i)M− 1 _{with M}− 1 ₌

Let Φ(t) be the matricial solution of

such that I(0) = I_d. Then, as above, one can write

(12) Φ(T) = Φ₁₂(1)Φ₁₁(1)...Φ₂(1)Φ₁(1)

= MeA(12)_eA(11) _...eA(2)_eA(1)_M− 1_.

Let us write the matrix A(i) of system (11) in the form

B(i) 0 A(i) =

* − δ

TrA(i) − 1

where B(i) = .

δ2₊ ∆

2A(i) δ

so we have U = αλi

2e + βλ

i

1e and then

(8)

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 δ₁ = δ2 = δ and Vi=Vi (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) = − ξδ 2₊δϕ_(t)

TrA(t) = − ξ − 2δ (TrA(t) does not depend on t) ∆₂A(t) = 2ξδ +δ 2− ϕ_(t)

where

It is easy to check that (− δ ) is a eigenvalue of A(t), say λ₃ = − δ . From the expressions of the invariants, the other two eigenvalues λ₁ and λ₂ satisfy.

λ₁ + λ₂ = − δ − ξ λ₁λ₂ = δξ − ϕ(t).

These two last equations imply that

Note that the three eigenvalues of A(t) are real and distinct; moreover, we have λ₂ <− δ < λ₁.

λi

1t λ

i

2t

1 λ2

12 − i_e − λ

1

12 − i_{e e} − _e

λ2 12 − i_λ

1 12 − i _λ

1 12 − i_λ

2

12 − i _{e − e λ}

2

12 − i_e − λ

1 12 − i_e

λ2

12 − i _λ 1 12 − i

11

Π

i=0 λ2

12 − i λ 1

12 − i _λ 1

12 − i _λ 2 12 − i

λ1 12 − i _λ

2 12 − i

− ξ b'₁f₁ b'₂f₂ b'₁f₁

V₁ − δ 0

A(t) = H .

b'₂f₂

V₂ 0 − δ

H

(b'₁f₁)2 _(b' 2f2)

2

ϕ (t)= V₁ + V₂.

H H

(δ + ξ) 1

λj= − ± (δ − ξ)

2 ₊₄ϕ _{(t), j = 1, 2.}

2 2

S S

U = A(i) U

W W

TrA(i) − 1 0 δ2₊ ∆

2A(i) δ 0 .

(b'₂f₂)2

δξ + δ 2 − _V

2 δ − δ

H

1 0 0

− 2δ − b'₁f₁ − b'₂f₂ . − δ 0 − b'₂f₂

S S

I₁ = A(t) I₁ .

I₂ I₂

1 λi

2e − λ

i

1e e − e

e = λi

2 – λ

i

1 λ

i

1 λ

i

2 (e − e ) λ

i

2e − λ

i

1e

λi

2 λi1 λi1 λ

i 2

λi

2 λi2 λ

i 2 λi 1 Ai .

√

I₁ = A(i) I₁

I₂ I₂

. .

For x = (x₁, ..., x_n) and y = (y₁, ..., 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, Mellander1 _{one can state:}

I) If y(t) and z(t) are non-zero solutions of (*) such that for some t₀ ≥ 0 we have 0 ≤y(t₀) ≤z(t₀) ≤c(t₀) with y(t₀) ≠z(t₀), then 0 < y(t) < z(t) < c(t) for all t > t₀.

For fixed t₀ ≥ 0 one considers the map f (y₀) = y(t₀ + T, t₀, y₀) for 0 ≤y₀≤c(t₀). When t₀ = 0 we take the sequence of positive numbers c(0) = c₀ > c₁ > c₂ > ... > c_n > ... > 0, where c_n = f₀(c_{n- 1}) and Q = lim c_n≥ 0. When Q = 0 we have lim y(t, t₀, y₀) = 0 for 0 ≤y₀≤c(t₀). If Q≠ 0 we have y(t) = y(t, 0, Q) positive, periodic with period T > 0 and, by I), Q > 0.

Let Φ(t) the matrix solution of ˙y = A(t)y, Φ(0) = I_d; since A(t) = (a_ij(t)) is irreducible with a_ij(t) ≥ 0 for i≠j, then the matrix C = Φ(T) is positive and so, by Perron's theorem, it has a simple positive eigenvalue λ = max{Reλ_i; det(C − λ_iI_d) = 0}.

As in theorem 1 of Aronson, Mellander1 _{one has}

analogously:

II) If λ < 1, there exists K > 0 such that | y(t)| ≤ Kλ |y(t₀)|for all t ≥t₀ ≥ 0 and any solution y(t) = y(t, t₀, y₀) of (*) such that 0 ≤y₀≤c(t₀).

Consider now the case λ > 1. Let E(t₀) = {y ∈ IRn : 0 ≤y≤c(t₀)}, ω > 0 eigenvector of Ct_{corresponding} to λ, v(t₀)t₌ ωtΦ_(t

0)

− 1 _{and E}

θ(t0) = {y ∈E(t0): v(t0)

t _y ≥θ}. We claim that for θ > 0 sufficiently small we have f (E_θ(t₀)) 傺E_θ(t₀) where 0 ≤t₀≤T. In fact

v(t₀)t_f

(y) = λv(t0)

t_{y +} ϕ_(t

0, y) where

ϕ(t₀, y) = λwt

∫

0, y))ds ; and

v(t₀)t_f

(y) − v(t0)

t_{y = (}λ − _1)v(t

0)

t_{y +} ϕ_(t

0, y) ≥ (λ −

1)vt_{y +} ϕ_(t

0, y), where 0 < v≤v(t0) for all t0 ∈ [0, T].

Since

is continuous on the compact set

{(t₀, y) | t₀∈ [0, T], y ∈E(t₀)}, then there exist δ > 0 and N > 0 such that v(t₀)t_f

(y)

≥v(t₀)t_{y +} N | y | _{for t}

0 ∈ [0, T] and | y | ≤ δ.

Using the Brower fixed point theorem (Hönig11_{) one}

concludes that f has a fixed point in E_θ(t₀). Then

eB(i) ₀

eA(i)_{= ,}

* e− δ

and also

eB(12)_eB(11)_...eB(1) ₀

M− 1Φ_{(T)M= .}

* e− 12δ

The eigenvalues of Φ(T) are e−Tδ _{together with} the eigenvalues of eB(12)_eB(11)_...eB(1)_{. So we only need}

to compute eB(i) _{for i = 1, ..., 12. For that one has to}

solve the system

or, equivalently, to solve the second order equation

(13) S− TrB(i)S + detB(i)S = 0.

Since the characteristic roots of (13) are the eigenvalues of B(i), that is,

δ + ξ 1

λi

j = λj(i) = − ± (δ − ξ)

2 _{+ 4}ϕ_(i),

2 2

j = 1, 2, and i = 1, ..., 12, then, as we did in section 3.1, eB(i) _{is given by the second member of equality (8).}

APPENDIX

Let us consider the following system

y1 = − α1y1 + (c1(t) − y1)(β11(t)y1 + β21(t)y2 + ...+ βn1(t)yn)

(*)

yn = − αnyn+ (cn(t) − yn)(β1n(t)y1 + β2n(t)y2 + ...+ βnn(t)yn)

which can be written as

y' = A(t)y + N(t, y) where

− α1 + c1β11 c1β21 ... c1βn1

A(t) = c2β12 − α2 + c2β22 ... c2βn2

c

nβ1n cnβ2n ... − αn+ cnβnn Assume α_i to be positive constants; c_i = c_i(t) to be positive continuous periodic functions of period T > 0; β_ji = β_ji(t) to be non negative continuous periodic functions of period T > 0 such that A(t) is an irreducible matrix (Gantmacher10_{), for all t.}

Moreover, assume there exists ε > 0 such that β_ji(t) ≥ ε for all t provided that β_ji(t) are not identically zero.

S S

= B(i)

U U

t0

n→∞

t→∞

t − t0 T

t₀

t₀

t

0

t + T

ϕ(t₀, y)

if y ≠ 0 Φ(t₀, y) = |y |

0 if y =0 t0

t₀

t0 .

√

…

.

.

.

¨

are bounded functions (see Aronson, Mellander1_{, Th. 2).}

Given a compact set K傺 E(t₀), one considers a point x₀, 0 < x₀ <c(t₀), such that x₀ < y(t₀ + T, t₀, y₀)< c(t₀) for all y₀ ∈K. Since there are constants M(x₀) > 0 and N > 0 such that for all y₀ ∈K we have

|y(t, t₀, y₀) − y(t, t₀, Q)|≤M(x₀)e− a(t − )_{, t}≥ t 0 + T

and

|y(t, t₀, y₀) − y(t, t₀, Q)|≤N, t₀≤ t≤ t₀ + T, then one has the following result.

III) For λ > 1, system (*) admits a unique non zero periodic solution y(t, t₀, Q), which has period T, and a constant a > 0 such that for each

compact set K 傺 E(t₀) there corresponds a constant M_K> 0 and we have

| y(t, t₀, y₀) − y(t, t₀, Q) |≤M_ke− a(t − ) _{for all t}≥ t 0 ≥0.

Moreover, as in theorem 3 of Aronson, Mellander1

we state:

IV) For λ = 1 there is a constant L > 0 such that for any solution y(t, t₀, y₀) = y(t) of (*) with 0 ≤y₀ ≤ c(t₀), t₀ ≥0, we have

| y(t) |≤ provided t ≥ t₀. For any two solutions y(t) = y(t, t₀, y₀) and z(t) =

y(t, t₀, z₀) of (*) such that 0 < y₀, z₀ < c(t₀),one has

D+_u(t) ≤− _(u− _{1) min(}ψ

z, ψy) < 0 for t ≥ t0 ≥ 0

where u(t) = max {max { , }} and ψ(y) = k

(see Aronson and Mellander1_{; Lemma 5, [1]).}

So, one can conclude that for θ > 0 sufficiently small, f : E_θ(t₀) →E_θ(t₀) has only one fixed point Q, and then f : E(t₀) →E(t₀) 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, t₀, y₀) with 0 ≤y₀ ≤c(t₀) and z(t) = y(t, t₀, Q) we have

u(t) − 1 ≤eu( )_(u(t 0) − 1)e

− p(t)_e− a(t − )

where a = ψ_z(t)dt and p is continuous and T

-periodic, T > 0. So, for each x₀ > 0 sufficiently close to the origin, there exists a constant M(x₀) > 0 such that

|y(t, t₀, y₀) − y(t, t₀, Q)| ≤ M(x₀)e− a(t − )

for all y₀, x₀ ≤ y₀ ≤c(t₀) and all t ≥ t₀ ≥ 0 because the c_k(t) y_k z_k

z_k y_k

j = 1 n

min ∑ β_jm(t)y_j(t)

1 ≤m ≤ n

t0

t0

t0

T

∫

1 T

0

REFERENCES

1. ARONSON, G. & MELLANDER, I. A deterministic model in biomathematics. Asymptotic behavior and threshold

conditions. Math. Biosci., 49:207-22, 1980.

2. DYE, C & HASIBEDER, G. Population dynamics of mosquito-borne deseases: effects of flies which bite some peoples more frequently than others. Trans. R. Soc. Trop. Med. Hyg., 80-:69-77, 1986.

3. DYE, C. & HASIBEDER, G. Population dynamics of mosquito-borne diseases: persistence in a completely heterogeneous

environment. Theoret. Pop. Biol., 33:31-53, 1988.

4. FORATTINI, O.P.; KAKITANI, I.; MASSAD, E.; GOMES, A. de C. Studies on mosquitoes (Diptera: Culicidae) and anthropic environment. 1. Parity of blood seeking Anopheles (Kerteszia)

in South-Eastern Brazil. Rev. Saúde Pública, 27:1-18, 1993.

5. FORATTINI, O.P.; KAKITANI, I.; MASSAD, E.; MARUCCI, D. Studies on mosquitoes (Diptera: Culicidae) and anthropic environment. 2. Immature stages research at a rice irrigation system location in South-Eastern Brazil. Rev.

Saúde Pública, 27:227-36, 1993.

6. FORATTINI, O.P.; KAKITANI, I.; MASSAD, E.; MARUCCI, D. Studies on mosquitoes (Diptera: Culicidae) and anthropic environment. 3. Survey of adult stages at the rice irrigation system and the emergence of Anopheles

t₀

albitarsis in South-Eastern Brazil. Rev. Saúde Pública,

27:318-411, 1993.

7. FORATTINI, O.P.; KAKITANI, I.; MASSAD, E.; MARUCCI, D. Studies on mosquitoes (Diptera: Culicidae) and anthropic environment. 4. Survey of resting adults and synanthropic behavior in South-Eastern Brazil. Rev. Saúde

Pública, 27:398-411, 1993.

8. FORATTINI, O.P.; KAKITANI, I.; MASSAD, E.; MARUCCI, D. Studies on mosquitoes (Diptera: Culicidae) and anthropic environment. 5. Breeding of Anopheles albitarsis in flooded rice fields in South-Eastern Brazil. Rev.

Saúde Pública, 28:329-31, 1994.

9. FORATTINI, O.P.; KAKITANI, I.; MASSAD, E.; MARUCCI, D. Studies on mosquitoes (Diptera: Culicidae) and anthropic environment. 6. Breeding in empty conditions of rice fields in South-Eastern Brazil. Rev. Saúde Pública,

28:395-9, 1994.

10. GANTMACHER, F. R. The theory of matrices. Vol. 2. New York, Chelsea, 1959.

11. HÖNIG, C. S. Aplicações da topologia à análise. IMPA, Rio de Janeiro, 1976.

12. LOTKA, A. J. Contrib. to the analysis of malaria

epidemiology. Am. J. Hyg., 3, 1923.

t0 t₀

K

1 + (t − t0)