a mixed program for parasitic disease control

J. Math. Biology 10, 5 3 - 6 4 (1980) Journal of Mathematical

Biologg (~ by Springer-Verlag 1980

A Mixed Program for Parasitic Disease Control*

Jorge Gonzalez-Guzman

Departamento de Matematicas, Universidad Tecnica del Estado (UTE), Casilla 4823-Correo 2, Santiago de Chile, Chile

Summary. In this paper we are concerned with the control of a parasitic disease by a permanent, time-continuous mixed program of vector reduction (reduction of the contact rate) and drug application.

We shall use the model developed in [1] with two control functions: one for the reduction of the contact rate and another for the administration of drugs to the population. This model takes into account the possibility that there may be a certain fraction of the population which cannot be covered by any drug application. Optimal control policies for reduction of the contact rate and for the protected proportion of the population by drugs are derived by using Pontryagin's maximum principle. A cost-optimal strategy is deduced for the maintenance of the affected proportion of the population below a given level. Some numerical examples are computed.

Key words. Parasitic disease-Optimal deterministic control-Maximum principle - Drugs protection.

1. The Model

Let z(t) be the proportion of the total population which is protected at time t by drug application. The protection given by drugs is lost at rate ~ and the corresponding population returns to the susceptible state. We introduce a control parameter v(t), the proportion of the total population which receives drugs per unity of time. Let p be the accessible proportion of the population; Yl be the proportion of affected population which are accessible and sl be the proportion of susceptible population which are accessible.

We put

dz - ~Sz + vs l + v y l . (1.1)

dt

Let Y2 and s2 be the proportion of affected and susceptible populations respectively which are not accessible, y be the total proportion of affected population in the total population; /~(t) be the contact rate at time t and ~ the recovery rate. Thus

* Research supported in part by a grant of the U.T.E.

0303 - 6812/80/0010/0053/$02.40

54 J. Gonzalez-Guzman

dyl - - = f l S l y - - Y Y l - - W1, dt

dy2 - - = f l s 2 Y - - 7Y2,

d s 1

d t - 6 z + Y Y l - v s l - f l s l y ,

ds 2 - - = 7Y2 - f ls2y. (1.2) d t

We introduce the variable x -- Y2/Y , the proportion of the affected population which is not accessible with respect to the total affected population. I f we consider that

sl + y l + z = p ,

s2 + Y2 = 1 - p,

Yl + Y2 = y, (1.3)

we obtain from (1.1) and (1.2) the following equations:

d x d t = vx(1 - x ) - x f i (1 - z ) + fi(1 - p ) ,

d y d t = 8(1 - z - y ) y - Y D + v(1 - x)] ,

dz - - = - 6 z + v ( p - z ) . (1.4) d t

Since Y l = Y - x y , Y2 - - - x y , $1 - = p - y - x y - z and s2 = 1 - p x y , the solutions of 1.4 must satisfy the following constraints

O <~ y - x y <~ p ,

O<<.xy 4 1 - p ,

O < < . p - y + x y - z ~ p ,

O < ~ l - p - x y < < . l - p ,

0 ~< z ~ p. (1.5)

These constraints are automatically satisfied, if they are satisfied for the initial conditions.

We assume that

f l ( t ) = k m ( t ) , (1.6)

where re( t ) is the proport ion of the vector population with respect to the natural equilibrium of the vector-population before the reduction program starts. Thus k is the contact rate for non-reduced vector population. We consider another control

Mixed Program for Parasitic Disease Control 55

parameter u(t) which measures the quantity of insecticide administered per unit of time (spatial variables or long-time effects of the insecticide are not taken into account). For the description of the vector population we use the simple model

drn - Am(1 - m) - rnBu, (1.7)

dt

where B > 0 measures the effectivity of the insecticide and A > 0 is a constant of the vector population.

In view of (1.6) and (1.7) we have

A - f l ( k - f i ) - f l B u . (1.8)

dt k

We assume that u(t)~ [0, ~J, where ~ < ~ .

2. Optimal Control for Contact Rate Reduction

We want to investigate a cost-optimal control policy for the reduction of the contact rate. The cost is

c , = f l u ( t ) d t . (2.1)

That is, we measure the cost simply by the quantity of insecticide used during the program. The optimal control problem is: to find a function u(t) so that contact rate fl(t) will be reduced from a level flo in time 0 (start of the program) to a new level fll < flo in time T, subject to (1.8), so that the cost (2.1) is minimal. The new level fll is given, whereas the time Tis free. We shall see that the C,-cost minimal solution of the problem is also the T-time minimal solution. S. Sethi [2] and [3] solved a similar mathematical problem with a more complicated cost by using Pontryagin's maximum principle and Green's theorem. We shall use only the Pontryagin's maximum principle [-4].

We assume that u(t) is an admissible control in Pontryagin's sense, that is, u(t) is a piecewise continuous function, left-continuous on the discontinuities and also continuous at the end points, with values on the interval [0, ~]. The Hamiltonian function is

;,uf(0, fi, u ) = A O f i ( 1 - f l s (Oo-Bf lO)u , (2.2)

where 0o is a non-positive constant and 0 satisfies the adjoint equation

d~- = + Bu - A 0. (2.3)

According to the maximum principle, if u*(t) is an optimal control, then there exist a non-trivial solution 0 (0 of (2.3) so that

max Yf(0(t), fi(t), u) = Yg(O(t), tiff), u*(t)) = 0 (2.4) u~[0,~]

for each time t ~> 0.


Thus

t 0 , i f ~ o - Br(t)r < o,

u*(t) = ~ [-0, K], if r - Br(t)tl/(t) = 0, (2.5)

{.a, if 0o - Br(t)r > O.

Solving the equa t ion (2.3) we ob ta in

i)(t) Cexp[fto(2~-~ f l ( s ) + B u ( s ) - A ) d s ] , (2.6)

where C is an in tegra t ion constant . I f C > 0, then O(t) > 0 for each t ~> 0. Thus 0o - Bfl(t)r should be negat ive

and u*(t) = 0 for each t >~ 0. But the con t ro l funct ion u = 0 does not lead to the a im r l < ro, since for u = 0 the funct ion r(t) is increasing. Consequen t ly C < 0. But then Ar - r(t)/k) is also negat ive and because o f (2.4) we have

m a x , = , AO(t)r(t)( l - r~t)) + max{0, r - Br(t)O(t)} = O.

Thus max{0, ~o - Br(t)O(t)} canno t be zero and so ~o - Br(t)O(t) > 0 for each t ~> 0. Thus the op t ima l control , if it exists, mus t be u*(t) = ~ for t >~ 0.

The so lu t ion o f (1.8) is then

r(t) = ro [ [Jo B~'~ (~ ' i f 1 - - - ~ 0, (2.7)

k Bt7 r(t) - (k/rio) + A t ' if 1 A - 0. (2.8)

Since fl(t) tends mono ton i ca l l y to the value k(1 - Bgt/A) when t tends to infinity, the p rob lem has no so lu t ion if r l ~< k(1 - Bft/A) < rio.

I f r ( t ) arr ives at r l a t t ime T a n d one wants to ma in ta in this level for t > T, it is sufficient to put the new con t ro l value u' = (1 - rl/k)(A/B) ins tead o f ~i (Fig. 1).

A 86

k_ l %,,~ t ):0

u ( t ) : u'

t

Fig. 1


3. Optimal Control for Drug Administration

We state the p rob lem: to find a cont ro l funct ion v(t) so tha t the p ropor t ion z(t) o f protected popula t ion at tains a given level Zl > 0 in t ime T, subject to the equat ion (1.4) and so tha t the cost

c~=flv( t )dt (3.1)

is minimal . The initial condi t ion is z(0) = 0. The Hami l ton i an for this p rob lem is:

~4~(0, z, v) = - Oz6 + v(0 o + 0p - 0z) (3.2)

where 0o is a non-posi t ive cons tant and 0 satisfies the adjoint equat ion:

a0 - ~,(~ + 6) (3 .3) dt

The opt imal control v*(t) must satisfy:

max ~ ( O ( t ) , z(t), v) = ~(O(t), z(t), v*(t)) = 0 (3.4) v~[0,~]

for each t >~ 0. Thus:

0, if ~P0 + 0 ( P - z) < 0

V*(t) = e [0,~7], if ~Po + 0(P - z) = 0 (3.5)

~, if 0 o + 0 ( P - z ) > 0

The solution of (3.3) is:

O(t)= CexpI6t + f'ov(s)ds I (3.6)

where C is an integrat ion constant . I f C < 0, then 0(t) < 0 for each t /> 0 and thus 0o + O(t)(p - z(t)) < 0 for each

t ~> 0. But then v*(t) = 0 and z(t) = 0, for t >~ 0. Thus C > 0 and then O(t) > 0. ~9(p - z) is then monoton ica l ly increasing:

d O ( P - Z ) = 0 6 P > 0 , for each t>~0.

Thus the op t imal control v*(t) is cons tant and equal to g after the t ime ?-where 0o = - O(t-)(P - z(~)) and before that t ime v*(t) = 0. Since the cost is zero up to 7, all the controls for the different values of t a r e opt imal . We put 7 = 0. Let T b e the first t ime t where z(t) = zl. Thus

{~, for O<~t<<.T, v*(t)= 0, for t > T.

The solution of z in (1.4) with the initial condi t ion z(0) = 0 is then

z(t) = (1 - F (~ + ~'), if t ~< T, (3.7)

[z(T)F~(t-r), if t > T.


p~

Z I

o~176176176176 . . . . . . . . . . , , . . . . . . . . . . . . . . . . . . . . . . . . . . ~176

v[t)=~ ....'"

,o*~

""" v(t) =v'

Fig. 2

I f zl >~ p~/(6 + ~), the level za of protected popula t ion is not a t t a i n a b l e - the intensity o f the drug application program is not strong e n o u g h - and the problem has no solution (see Fig. 2).

If zl < p~/(6 + ~), the level z~ is at tained at time

1 log(1 6 + ~ ) T - 6 + ~ pv zl �9 (3.8)

If z(t) arrives to z~ at time T and one wants to maintain this protected level for t > T, it is enough to put the new control value v' = 6zl /(p - Zl).

4. The Behavior of y(t) and x(t)

We solve the equations (1.4) for the optimal controls u(t) = ~ and v(t) = zT. The equation for x(t) is the Riccati equat ion

dx - g ( x - x O ( x - x z ) , (4.1)

dt

where the roots xl and x2 are

1 xl( t ) = ~ [~7 - /~(t)(1 - z(t)) - n(t)],

1 x2(t) = ~ [~5 - / / ( t ) (1 - z(t)) + n(t)], (4.2)

where

n(t) = ~/(~ - /? ( t ) (1 - z(t))) 2 + 4/~(t)g(1 - p).

The solution of (4.1) with the initial condit ion x(0) = Xo is

Xl(t) - x2(t)Cexp(fon(s)ds ) x(t) =

(4.3)

(4.4)


where the integration constant C is given by

X 0 - - Xl(O ) c - . ( 4 . 5 )

X o - x 2 ( O )

The functions fl(t) and z(t) are given by the formulas (2.7), (2.8), and (3.7). The equat ion for y(t) is the Bernoulli equat ion

dy + fl(t)Y z _ y[fl(t)(1 - z(t)) + 5x(t) - ~ - 7] = 0. (4.6) dt

The solution o f (4.6) with the initial condit ion y(0) = Y0 is

yoF(t) y ( t ) - 1 + y o H ( t ) ' (4.7)

where

E J l F(t) = exp - (~ + 7)t + (fl(s)(1 - z(s)) + Ox(s))ds , 0

H ( t ) = f ' o f l ( s ) F ( s ) d s . (4.8)

In Fig. 3 the funct ion y( t ) is represented for k = 0.5, 6 = 1/7, p = 0.7, A = 50, 7 = 0.005, Yo = 0.1, f = 0.07 and B~ = 49. In Fig. 3 are also represented the cases where only the contact rate is reduced (~ = 0), where only drugs are administered (~ = 0) and where no control is applied (~ = 15 = 0).

In Fig. 4 the funct iony( t ) is represented for k = 0.5, 6 = 1/7, A = 50, 7 = 0.005, Yo = 0.1, ~ = 0.07, B9 = 49 and several values o f p . We have assumed that the

1.o-

0.5

0.1

0

5=~ =0

B5 = 0 , v = 0.07

Bu = 4 9 , v = 0

~ _ _ a 6 = ~ 9 . v = 0.07

I I I I I ' I 10 30 60

Fig. 3

1 i

J

I '1 I I m

100 [ Doys] t

60

Y

J. Gonzalez-Guzman

0.1 P = 0.1

P= 0.3

P= 0 5

0.05' P = 0.7

P = 0 . 9

P= 1.0

I I I 0 10 20 30 [ Days ] t

Fig. 4

initial value Xo is equal to 1 - p . That means that the disease is homogeneously distributed in the territory, or that the non-accessible part of the population has the same proportion of affected ones as the total population at the beginning of the control program.

The solutions have been computed with a Runge-Kutta method from equation (1.4).

For a given intensity of the controls zi and ~, the contact rate fl(t) and the protected proportion of the population tends to

pg - _ , ( 4 . 9 ) z(oo) fi + v

respectively, when t tends to infinity. If we introduce these asymptotic values in the equation for y(t), we obtain an approximate solution which has the limit for t ~ ov

y(oo) = max(0 , 1 - z(oo) 7 + g(1 - x(ov))~ (4.10) - 7

where

t 1 x(oo) = ~ [ ~ - k(1 - z(oo)) + ~ ( f - k(1 - z(oo))) 2 + 4kg(1 - p ) ] , i fg ~ O,

l l - p , if g=O. (4.11)

We can therefore ask: what reduction of the affected population (with respect to the state of the epidemic without control) can we expect for a given intensity of the control program for large values of t?

We define that reduction as

R Y(~)I . . . . o - y(~)l.,o - ( 4 . 1 2 )

100 y(oo)l . . . . o

Mixed Program for Parasitic Disease Control

Table 1

61

p ~ k Bt2 ~ y(oo)[o, 0 y(oo)[a, e R%

0.5 1/7 0.5 0 0.026 0.99 0.89 10 0.5 1/7 0.5 45.4 0 0.99 0.89 10 0.5 1/7 0.5 45.4 0.026 0.99 0.56 43.1 0.9 1/300 0.01 0 0.00014 0.50 0.45 10 0.9 1/300 0.01 4.5 0 0.50 0.45 10 0.9 1/300 0.01 4,5 0.00014 0.50 0.40 19.9 1 1/7 0.5 0 0.012 0.99 0.89 I0 1 1/7 0.5 0 0.076 0.99 0.50 50 1 1/7 0.5 49 0 0.99 0.50 50 1 1/7 0,5 32,2 0.1 0.99 0 100 0.7 1/7 0.5 0 0.07 0.99 0.680 31.2 0.7 1/7 0.5 49 0 0.99 0.6 50 0.7 1/7 0.5 49 0.07 0.99 0 100 0.3 1/7 0.5 49 0.07 0.99 0 100 0.1 1/7 0.5 49 0.07 0.99 0.04 96

In Table 1, R is calculated for several values of the constants and several control intensities. The values of ~ and A are taken equal to 0.005 and 50, respectively.

We can see that the joint effect of both controls may be greater than the sum of the effects of each control separately. For example, i f p = 0.5, ~ = 1/7, k = 0.5, Bti = 45.4, and g = 0.026, the reduction for each control is 10% while the mixed reduction is 43.1%.

5. Optimal Control for the Maintenance of the Disease Below a Given Level

Let us consider the following situation: the proport ion y(t) of the affected population has been reduced by a certain control program to a desired low level Yl and one wants to maintain the disease below this level for the future with a minimal cost. By using the control variables u(t) and v(t) we can achieve and maintain an adequate level for the contact rate fl* and an adequate proport ion of protected population z*. We shall state an optimal strategy for the maintenance of the disease by the calculation of optimal values of fi* and z* and the corresponding values of the controls u* and v*. We assume that the cost of the maintenance program is

c=f;,[Du(t)+Lv(t)]e-~(~-T')dt, (5.1)

where D and L are certain constants and r is a discount rate. T' is the first time at which both fl and z attain the values fl* and z*, respectively. (They attain these values not necessarily at the same time, the first must "wait" for the other.) I f the controls assume constant values u and v, then the cost is

D L C = - - u + - - v. (5.2)

r r

For simplicity we setp -- 1. The equation for y(t) with constant values offi and z is the non-homogeneous Riccati differential equation


dy - - = f l y ( 1 - 2 - y ) - (7 + v ) y . (5.3) dt

The solution of (5.3) with initial condi t ion y ( T ' ) = Y2 is

y( t ) =

1 Y + v z ) y 2

z)(t-7")) (5.4) Y2-- y 2 - - l + - ~ - - + Z exp - - f l 1 y + v

t~

if 1 - (7 + v)/fl - z > 0. Otherwise

Y2 y ( t ) = 1 + y E f l ( t - T ' ) ' (5.5)

I f the initial condi t ion satisfies: Y2 < 1 - (~ + v)/fl - z the funct ion (5.4) increases, monotonica l ly , to the value 1 - (y + v)/[3 - z. Thus, if we want to main ta in the disease level below y, the following relat ion must be satisfied:

7 + v 1 z <~ Yl. (5.6)

We have seen that, in order to main ta in constant the contac t rate fl and the p ropor t ion of protected popula t ion z, it is enough to set u = (1 - f l / k ) ( A / B ) and v = 6 z / ( 1 - z), respectively. Thus the condi t ion (5.6) means for the cont ro l parameters :

~ + v v 1 - - - - ~< Yl. (5.7)

k(1 - B u / A ) 6 + v

I f 1 - B u / A <~ O,y( t ) tends to zero when t tends to infinity, i.e., the disease should be eradicated wi thout drug adminis t ra t ion. In this case the cost is

D A C = - - - - . (5.8)

r B

I f 1 - u B / A > 0, we obta in f rom (5.7) the limit condi t ion

1 il t~ u B 1 - Yl - ~~~m ,,

I t is easy to see by derivat ion that (5.9) represents a strictly decreasing funct ion of v. I f the condi t ion

7 < ~ (5.10)

holds, the second derivative d2u/dv 2 is negative. In this case, the min imal value of the cost (5.2) subject to the condi t ion (5.9) must be at tained at the bounda ry of the rectangle [0, ~] • [0, ~], due to the linearity of the cost with respect to u and v (see Fig. 5).


C=Lv C

!"'-,

~ - - ~ A u

Fig. 5

In order to obtain the optimal controls u* and v* it is enough to compare the costs hA and h8 at the extrema A and B of the graphic of the function (5.9) and the special cost (5.8).

Through the relations

v* z* = - - (5.11)

6+v*'

we can determine the optimal levels/Y* and z*.

Conclusions

Optimal control theory can be applied to an epidemic control problem to find concrete strategies in order to minimize the costs. The analysis of the corresponding model in our case shows that the drug application program and the use of insecticide must be as intensive as possible. The mixed control program can have greater long-time effects than the sum of the effects of each control separately. We have constructed a procedure to find the optimal balance between both control methods in order to maintain the proportion of affected population below a given level. No such optimal balance has been elaborated for the problem of reduction of the affected population. For this case use of Pontryagin's maximum principle alone leads to very difficult mathematical problems. The control model does not consider long-time effects of the insecticide nor the spatial spread of the disease.

Acknowledgments. I would like to thank Prof. Klaus Dietz for his valuable observations and suggestions.

References

l. Dietz, K. : Models for parasitic disease control. Bull. of the I.S.I, 46, 531 -544 (1975) 2. Sethi, S.: Quantitative guidelines for communicable disease control program. Biometrics 30,

681-691 (1974)


3. Sethi, S. : Optimal control of some simple deterministic epidemic models. J. Op. Res. Soc. 29, 129-136 (1978)

4. Pontryagin, L. S., Boltyanski, V. G., Gamkrelidze, R. V., Mischenko, E. F. : The mathematical theory of optimal processes. New York: Wiley 1962

Received July 23, 1979

a mixed program for parasitic disease control

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