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Quadratic infectious diseases mathematical models: Chronic states, sanity levels, and treatment
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8ClENOE~DIREGTe
ELSEVIER
MATHEMATICAL AND COMPUTER MODELLING
Mathematical and Computer Modelling 42 (2005) 1315-1324 www.elsevier.com/locate/mcm
Quadratic Infectious D i s e a s e s M a t h e m a t i c a l Models: Chronic States Sanity Levels and T r e a t m e n t F. J . SOLIS AND B. TAPIA CIMAT Guanajuato Gto. M6xico 36000©cimat .mx
J . V . ROMERO AND J. MORENO I n s t i t u t o de M a t e m S t i c a M u l t i d i s c i p l i n a r U n i v e r s i d a d Polit4cnica de Valencia Valencia, Spain 46071©mat .upv. es
(Received November 2004; accepted December 2004) A b s t r a c t - - S p e c i f i c quadratic continuous models are examined to study the dynamics of interacting cells of an organism that is affected by a disease or virus. We focus only on models that exhibit a chronic state and we study conditions on the parameters involved in the models to guarantee that the organism can obtain a level of sanity. Aggressive and nonaggressive treatments are applied when the organism can not reach such level of recovery. @ 2005 Elsevier Ltd. All rights reserved. Keywords--Infectious
disease, Ctlronic states, Treatment.
1. I N T R O D U C T I O N Viruses are parasites with diverse ways of interacting with their hosts. Some viruses initiate lytic infections, which result in high levels of virus replication followed by clearance of the virus. On the other hand, other viruses establish life-long persistent infections in which the virus has established a quasi-homeostasis with the organism. These behaviors have been studied in contemporary epidemiological theory and they can be traced back to the early part of the 20th century. Before this period, the mechanisms by which infectious disease agents spread within populations or within the human body had been revealed by microbiological research. The theoretical framework most commonly used to mimic the dynamics of viral and bacterial infections is one based on the division of the human population into categories containing susceptibles, infected who are not yet infectious (latent), infectious individuals, and those who are recovered and immune. However, most models that use these categories reflect the dynamics of the number of infected people without reference to the abundance of organisms within each individual. In the literature there exist models whose main interest is the description of the interaction between viruses and the cells of the human body. Those models consider different populations of 0895-7177/05/$ - see front matter (~) 2005 Elsevier Ltd. All rights reserved. doi: 10.1016/j.mcm.2004.12.004
Typeset by AMS-]~X
F.J. SOLIS et
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al.
cells and virus (infected ceils, healthy cells, infectious virus, non infectious virus, etc.) and are represented by equations of quadratic type, displaying most of t h e m one carrying capacity for the growth of the healthy ceils. The specific situations t h a t are modelled in some works are mainly the introduction of intracellular delay (see [1,2]), the application of drugs t h a t can be inhibiting or a genetically engineered defective interfering virus (see [3,4]) and the behaviors of healthy ceils and the h u m a n immunodeficiency virus (HIV) and how they affect each other. One of the aims of this work is to elaborate a parametric study of families of quadratic models describing the behavior of the human ceils when they are affected by a disease or virus under the assumption that the dynanfics of the cells are similar regardless of the region of the organism that is under consideration. We seek conditions on the parameters to guarantee t h a t the organism reaches a level of sanity or to achieve at least that the growth of the infected cells is almost null even though that they m a y be present in the organism. These conditions are mathematically equivalent to one of the following two conditions. First, to have all the orbits of the dynamical systems to approach to an equilibrium point that is considered as the level of total sanity, that is, solutions of the m a t h e m a t i c a l models approach to a state with the total number of cells being healthy. Second, t h a t the mathematical models have a positive equilibrium point, considered as a chronic state, located close to a level of sanity and that the orbits near this point tend to stay in a vicinity of this point. We consider a chronic state as a particular state where a organism has a disease or disorder t h a t is present over long periods of time, often the remainder of his lifetime. A second goal is to study the existence of a complex behavior in the models by means of a strange chaotic attractor. The presence of a chaotic attractor in a model does not allow long t e r m predictions on the behavior of the cells and there is no support by experiments or field studies to validate the model, see [5]. We consider two cases in the models under consideration. First case, when the fraction of healthy cells t h a t are infected can be neglected and second case when the above fraction is important. For b o t h cases we will obtain conditions for the parameters to obtain a unique positive equilibrium point, a chronic state. We will analyze the level of recovery of the organism by a stability analysis of such point and we will study the behavior of the orbits considering levels of sanity and level of quasisanity obtaining in this way a region of optimal initial conditions for recovery. For all those cases when the organism can not recover we will study the application of two types of treatments. One nonaggressive, that is, one which does not affect the healthy cells and an aggressive one, which may affect some healthy cells. We study in both scenarios the effect of the treatment. Finally, we apply our results to some models t h a t have been used to describe the behavior of the HIV dynamics. This paper is organized as follows. Section 2 describes components and parameters of the families of models. The dynamic behavior of the models is analyzed in Sections 3 and 4 with or without treatment. The analogy and difference between different models of infections diseases are compared in Section 5.
2. Q U A D R A T I C M O D E L S A N D D Y N A M I C P R E L I M I N A R I E S Quadratic models arise in a variety of applications, see for example, [6-8] and so on. Two cases (one with the plus and the other with the minus sign) of a general quadratic model are the only cases that fit in our framework, namely,
dX (t) - dt dY (t) - dt
- X (t) (a -F b X (t) 4- ~ Y (t)) nL ( k l Y (t) - k 2 X (t)) Z ( t ) ,
- ± Y (t) (d +
/ d~-F~X(t)
dZ (t)
'
(t) + k 3 Y (t)) + 3 X (t) - k 4 Y (t) Z (t),
Y(t),Z(t)
(1)
dX (t_____~)dY_!t) ~ '
dt
'
dt
J"
Where X(t), Y(t), and Z(t) represent the densities of the healthy cells, infected cells and treatment respectively at time t. Coefficients a and d are the intrinsic growth rates of the healthy
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and infected cells respectively, b and k3 are intraspecific interaction coefficients, ~ and ~ are interspecific interaction coefficients, ~ is the coefficient of healthy cells t h a t become infected, k2 is the coefficient t h a t represents the aggressiveness of the treatment, and finally kl and k4 are the percentages of infected cells t h a t heal and die with the application of the t r e a t m e n t respectively. If we denote by x0 the number of healthy cells and by Y0 the number of infected cells then the total number of cells of the organism at some initial time will be an element of the set {(x0,y0) I x0 _> 0, y0 >_ 0, x0 + Y0 < 1}, which we call the section of initial conditions. We introduce two levels of sanity by dividing this region in three parts. The first division is called the region of sanity, which is the region t h a t the organism is healthy regardless of having some infected ceils, Region II as the zone of quasisanity and Region III as the zone of sickness as shown in Figure la, where the levels of sanity are considered as straight lines, labelled as NC and NCS. Since every organism needs a minimum number of healthy cell to survive and a minimum number of healthy cells to be healthy, we subdivided Region I, taking in consideration those cases where the organism is totally healthy, grouping these cases in a region t h a t we called S. Then, we subdivide Region I I I by considering those cases where the organism can not survive and denoting t h a t region as Region M, as is shown in Figure lb.
(a)
(b)
Figure 1. Set of initial conditions of healthy and infected cells.
There are restrictions over the chronic state and the parameters t h a t we shall impose to ensure realistic h u m a n dynamics. Since a chronic state is a disease or disorder t h a t is present over long periods of time, we identify it as an equilibrium point of system (1). An equilibrium point is a particular solution of (1) that does not change with time, and is determined by setting dX(t) dY(t) dZ(t) _ O. An equilibrium point may be stable or unstable and it can be classified into one of several classes using a linear stability analysis (see [9]). This analysis is restricted to the region immediate to the equilibrium point. Thus, an equilibrium point can be classified as stable node or attractor (unstable node) if nearby orbits converge (diverge) to the equilibrium point, as a saddle point if there are nearby orbits converging and diverging from the equilibrium point and finally, as a center if all nearby solutions are bounded periodic solutions. We remark that the linear analysis is local in nature and thus a global stability of nonlinear systems m a y be a function of the operating conditions for the system, meaning t h a t minor variations in the initial conditions resulted in large differences in their long time behavior. This irregular behavior of a class of nonlinear systems is generated by the existence of a strange chaotic attractor (see [5]). There are basically two necessary criteria for a system to have a chaotic behavior. The system must be nonlinear and must have at least three independent dynamic variables. Of course, there are systems t h a t satisfied these criteria and do no exhibit chaotic behavior. For our models, in order to t h a t the equilibrium point, which we denote by ( x f , y f ) is in the region of interest and it is close to the level of sanity, it is necessary t h a t x f > y f and x I + y f _< 1. dt
--
~
--
dt
--
F.J. SOLIS et aI.
1318
Therefore, we have to impose the following restrictions on the p a r a m e t e r s of the model, a > 0, b < 0, and dk3 < 0. To get rid of some parameters, we nmke the following t r a n s f o r m a t i o n X - ( - a / b ) x , Y = ( - d / k 3 ) y , and Z = z in model (1) to obtain two families of systems of the form -
x (a -
=
±y
(d -
ax
-
ey) +
e~ -
dy)
(py
-
+ ~x -
qx)
z,
syz,
(2)
= f ( x , y , z , Sc,~).
3. Q U A D R A T I C
MODELS WITHOUT
TREATMENT
A first objective is to find conditions on the parameters to make sure the organism reaches a level of sanity or total recovery. To obtain this goal we begin to s t u d y how the organism behaves w i t h o u t a treatment. T h a t is, we will s t u d y a particular case of the families of system (2) namely, J: = x ( a -
az
-
cy) ,
(3) ~1 = -4-y ( d -
ex -
dy)
+ 5x.
We consider separately the cases. First, when d = 0 (fraction of h e a l t h y cells t h a t are infected is null) and later when 5 > 0, since b o t h cases lead us to different analytic conditions. 3.1. C a s e ~ = 0 W i t h 5 = 0 in (3), we obtain two families of models t h a t have been used to described the interaction of two species in population dynamics. Our first step in this case is to guarantee the existence of a chronic state in the region of interest. Conditions for its existence are given in the following proposition. PROPOSITION 1. The system given by (2) with 5 - 0 has a unique positive equilibrium point ( x f , y f ) with x¢ > y f and x f + YI < 1, if and only ifc2e > adc > a2e and d < e. PROOF. See the Appendix. T h e conditions m e a n t h a t the n u m b e r of encounters between the cells is bigger t h a n the rates of growth of b o t h t y p e of cells and t h a t the ratio between the rate of g r o w t h of the healthy ceils and infected cells is less t h a n the ratio of the factors of encounters of the ceils. In the above proposition, we discarded the case w h e n x f + y f = 1, since this condition will have the effect to produce a c o n t i n u u m of equilibrium points at the b o u n d a r y of the region of interest and none at the interior. Having the conditions for a chronic state our next step is to find its behavior when the values of the parameters are changed, its stability properties and finally the behavior of n e a r b y orbits. W h e n we vary the values of the p a r a m e t e r s in b o t h models we observe t h a t when we decrease (increase) the values of a and e the chronic state reduces (increases) its n u m b e r of infected ceils and increases (reduces) its n u m b e r of healthy cells, thus improving the health of the organism. W h e n we vary the values of c and d we obtain the same results as varying a -1 and e -1, respectively. For the first model (with the plus sign) with the infected cells having a negative rate of growth, the stability of the equilibrium point found in Proposition 1 depends on the value of 1 / 7 = aid. If 1 / 7 _> 1, then the point m a y be an a t t r a c t o r of stable spiral. If 1 / 7 < 1 and (a) if c - a = e - d, then the point is a center (nearby orbits are periodic); (b) if c - a > e - d, t h e n is an a t t r a c t o r (attracts all n e a r b y orbits); (e) if c - a < e - d, then the point is an unstable node (some orbits get away for it). For centers, we have t h a t when the chronic state is in Regions I, S, or II, the region of optimal initial conditions, defined as the set of initial conditions t h a t lead the organism to a total recovery, is completely in the same region t h a n the equilibrium point (surround the point), so a t r e a t m e n t
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is not required. We have a similar behavior when the point is an attractor and it is located in Region I or S. In all of the above cases, the region of optimal conditions can be considered in average as a circle with center in the chronic state and a small radius r; which is given by r = m i n { l N S - x o l , I N C S - x o l , Yo, d(P, C), d(P, L)} where P denotes the chronic state, L is the straight line, x + y = 1, C is the boundary of the Region S and d ( A , B ) is the distance from A to B. If the chronic state is in Region II, the organism gets worse in time without any possibility to recover totally but perhaps a t r e a t m e n t could avoid it. For an unstable node the number of infected cells increases with time and therefore, a t r e a t m e n t is strongly required. For the second model (with the minus sign) where the infected cells have a positive rate of growth the fixed point is always a saddle point. Only the points in Regions I and S have regions of optimal initial conditions. These regions are small, thus a t r e a t m e n t is required to increased the size of these regions.
3.2. Case 6 ¢ 0 In this case, we will consider separately the models that we have been studying. The reason is that the algebraic systems that define the equilibrium points are different. For the model given by (2) with the plus sign we have two cases: (1) First Case: ad - ce = 0. Which means that the product of the rates of growth is equal to the product of the factors of the encounters of the cells. PROPOSITION 2. S y s t e m (3) has a positive equilibrium p o i n t ( z / , y I ) of the form
c6
x f=l
ad + c6 - ae' a6 ad+c6-ae'
Yfwith x I > YI, x f + yf < 1, if and only if ad + c 6 - ae > O,
and
a < c,
6 < d - e.
(4)
PROOF. See the Appendix. Notice that this equilibrium point approaches to the point of total sanity as 6 approaches to zero. (2) Second Case: The product of the rates of growth is different to the product of the factors of the encounters of the cells, t h a t is ad - ee 7~ O. PROPOSITION 3. S y s t e m (3) has a positive equilibrium p o i n t (x f , y f ) o f the form C
x I = i - -Yl, a
-
(ae
-
ad
-
e6)
Yl =
-
-
-
e6)
2 (ad - ce)
2 -
-
ce)
'
with x I > Yl, x f + Yl < 1, if and only i£ ae < c6 + ad,
ad < ce,
a < c,
and
a (6 + e) < c (d - 6).
(5)
PROOF. See the Appendix. The equilibrium point found is the &generalization of the equilibrium point given in Proposition 1. For the model given by (2) with the minus sign, where the infected cells have a positive rate of growth, it is not possible to find a &generalization of the equilibrium point given in Proposition 1. However, it is possible to find an equilibrium point if we require t h a t ad - ce ~ O.
1320
F . . J . SOLIS
et al.
PROPOSITION 4. S y s t e m (2) has a p o s i t i v e e q u i l i b r i u m p o i n t (x f , y f ) o f t h e f o r m xf =1-
e(ad-cS-ae) 2a (ad - ce)
'
ad - c5 - ae 2 ( a d - ce) '
Y f -w i t h x f > y f , x f + y f < 1, if and o n l y i f
(ad - c 5 -
ae) 2 = 4aS (ce - ad) ,
ad < c6 + ae,
a < c,
ad (2a - c) < ace - c25,
(6)
and a 2(d+e)
PROOF. See the Appendix. When we make 5 approach to zero the chronic state approaches to a state of total health. For the case 5 ¢ 0, the equilibrium point for the first model can be of three types: center, stable node or unstable node and for the second model only of one type, namely unstable node. In this last case, the organism approaches to a state with the total number of cells been infected. In both models the region of optimal initial conditions is smaller than the one for the case of 5 = 0 and when we vary the value of 5 the chronic state increases (reduces) its number of healthy cells and reduces (increases) its number of infected cells when we decrease (increase) the value of 5. 4.
TREATMENT
Let us include in our systems the equation that models the treatment, which we had left out before. We will consider that the treatment can be non aggressive or aggressive. In the first one, the treatment will affect only to the infected cells making that a proportion of these get healthy and another proportion to die. The second one acts as the first one but now it affects also the healthy cells, that is why it is known as an aggressive treatment. We will consider three cases for the equation for the treatment: In the first, the equation for the treatment is of the form i = 0, meaning that the dose is constant and for the second one i = kl~ ) with kl > 0, meaning that the treatment is applied proportional to the speed of growth of the infected cells. For the third case, we generalize the equation for the treatment to a quadratic equation in ~) to analyze the possibility of a existence of a chaotic attractor, thus the third equation become = ko + k15 + k2($) 2 For the first two cases, we solve the third equation and substitute it in our previous systems giving us the following system 2 = x ( a - a x - cx) + p y n _ q y ' ~ x ,
(7) ~1 = ± y ( - d + e x + dy) + 5x - s y ~.
Where m = 0 and n = 1 corresponds to the constant dose and m = 1 and n = 2 corresponds to the nonconstant dose. The parameter q indicates the aggressiveness of the treatment, p and s are respectively the percentages of infected cells that heal and die with the application of the treatment. For these two cases, the treatment that has better results in the sense of increasing the region of optimal conditions is the one when the dose is constant and nonaggressive treatment, except
Quadratic Infectious Diseases
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for the case when 2a > c and 5 ~ 0. For this last case, it is b e t t e r to have the n o n c o n s t a n t dose and an aggressive treatment. Now let us consider the third case. In order to investigate the key p a r a m e t e r s for which the system m a y have a chaotic behavior we make use of a conjecture t h a t establishes t h a t systems t h a t do not possess the Painleve p r o p e r t y are nonintegrable, and therefore there exists a strong possibility of existence of chaos in t h e m (see [10]). Conditions for the integrability of the system are given in the following proposition. PROPOSITION 5. W i t h t h e p r e v i o u s n o t a t i o n s y s t e m ic = x (a - a x - cy) + (py - qx) z, 9 = +Y (d-
ex-
dy) + 6 x -
(8)
syz,
= ko + k l ~ + k2 @ ) 2 ,
is integrab]e i f k2 7~ 0 and n o n i n t e g r a b l e otherwise.
PROOF. See the Appendix. From this result we discover t h a t k2 is a critical p a r a m e t e r controlling tile possible turbulent d y n a m i c s of our models. Therefore, we can validate our models with d a t a using only the previous two cases of the treatment.
5. M O D E L S
OF H I V
We will like to use our models to c o m p a r e t h e m with other models t h a t have been used in the literature in the s t u d y of some infectious diseases. We concentrate in some models of H I V and we use their d a t a to fit our models, (see [1,2-4,11-13]). We find t h a t the values of the parameters for the model ic = x (a - a x - c y ) , 9 = -y
(d - ex - dy) + ~,
(9)
vary in the intervals given by a C [0.01, 0.02], c • [0.01,0.02], d e [0.26, 0.5], and e E (0.26, 0.5). T h e values of the parameters for the model Jc =
x (a -
= y (d-
ax
-
cy)
,
(1o)
e x - dy) + 5z,
are in the intervals a E [0.01, 0.02], c e [0.01, 0.02], d and e are fl'ee.
ud
LJ.4
u.:.
L~.~
(a) Nonagressive.
U.t
U.U
U.U
1
0
0.!
0.2
03
014
0.5
06
(b) Aggressive.
Figure 2 Numerical simulations with different treatments
0.7
08
0.9
1
1322
F . J . SOLIS et al.
T h e values of 5 for b o t h m o d e l s were chosen to g u a r a n t e e t h e e x i s t e n c e of a chronic s t a t e . F o r t h e a p p l i c a t i o n of a k n o w n t r e a t m e n t (see [3]) t h e p r o p o r t i o n of i n f e c t e d cells t h a t are affected b y t h e t r e a t m e n t is a p p r o x i m a t e l y s = 1.2 a n d this is t h e value t h a t we will consider in our simulations. We v a r i e d all p a r a m e t e r a n d for b o t h m o d e l s t h e b e s t r e s u l t s were o b t a i n e d b y a c o n s t a n t dose a n d non aggressive. In F i g u r e 2 we show some n u m e r i c a l s i m u l a t i o n s using a specific set of p a r a m e t e r s w i t h an a n o n a g g r e s s i v e and an aggressive t r e a t m e n t using a c o n s t a n t dose. 6.
CONCLUSIONS
S t u d y i n g families of simple q u a d r a t i c m o d e l s t h a t d e s c r i b e t h e d y n a m i c s of infected diseases on o r g a n i s m s , which have a clear division b e t w e e n infected cells a n d h e a l t h y ceils, has led us to c o n c l u d e t h a t is possible to o b t a i n c o n d i t i o n s over t h e p a r a m e t e r s t o have a chronic s t a t e of t h e o r g a n i s m . In t h o s e cases, where t h e chronic s t a t e is classified as an a t t r a c t o r , t h e r e is a full r e c o v e r y of t h e o r g a n i s m for a o p t i m a l region of initial c o n d i t i o n s . I n t h e cases where t h e chronic s t a t e is a center, u n s t a b l e n o d e or a saddle, it is n o t possible t o o b t a i n a recovery. W e can r e i n s t a t e t h e level of s a n i t y in these last cases w h e n we i n t r o d u c e a t r e a t m e n t t h a t m a y or m a y not affect t h a t h e a l t h y cells. Since we only a d d a specific t y p e of t r e a t m e n t , a n i n t e r e s t i n g t o p i c to s t u d y in t h e n e a r f u t u r e is to a n a l y z e different p o s s i b l e t y p e s of t r e a t m e n t s t h a t lead to a full recovery.
APPENDIX PROOFS
OF
PROPOSITIONS
PROOF OF PROPOSITION 1. O b s e r v e first t h a t if a d - c e ~ O, t h e n t h e u n i q u e s o l u t i o n to t h e a l g e b r a i c s y s t e m a - a x - c y = 0 a n d d - e x - d y = 0 is given b y ad - cd xf -
ad - ce' ad - ae
Yf -
ad - ce "
L e t us first a s s u m e t h a t s y s t e m (2), w i t h 5 = 0, has a unique p o s i t i v e e q u i l i b r i u m p o i n t ( x f , y f ) w i t h x y > y y a n d x / + y i < 1. Since t h e s y s t e m has a u n i q u e p o s i t i v e e q u i l i b r i u m point, t h e n t h e d e t e r m i n a n t a d - c e can be positive or negative. T h e d e t e r m i n a n t can n o t b e positive, since if it were, t h e n a - c > 0 a n d d - c > 0 in o r d e r to have a p o s i t i v e solution. Since x i + y f < 1, we get t h a t (d - e ) ( a - c) < 0, which is a c o n t r a d i c t i o n . Therefore, t h e d e t e r m i n a n t is negative, t h a t is a d - e e < 0, a n d since t h e solution is positive, we get t h a t a - c < 0 a n d d - e < 0. Now, since x I > Y I , we o b t a i n e d > a e . C o m b i n i n g these two results, we finally g e t e2e > a d e > a 2 e a n d d < e. Now, let us a s s u m e t h a t c 2 e > a d c > a 2 e a n d d < e. F r o m c2e > a d e , we o b t a i n t h a t ce > a d , which m e a n s t h a t t h e d e t e r m i n a n t of t h e a l g e b r a i c s y s t e m is negative. T h u s , we o b t a i n a unique solution. F r o m a d c > a 2 e we get a d - c d < a d - a e a n d since t h e d e t e r m i n a n t is negative, t h e n x z > Y I . T h e fact t h a t t h e s o l u t i o n is positive comes from c 2 e > a 2 e a n d d - e < 0. Finally, since d - e < 0 a n d a - c < 0, t h e n (d - e ) ( a - c) > 0, t h a t is x y + y f < 1. PROOF OF PROPOSITION 2. N o t i c e first t h a t t h e e q u i l i b r i u m p o i n t s ( x f , y y ) satisfy t h e q u a d r a t i c e q u a t i o n ( a d - c e ) y~ + ( a e - a d - e S ) y y + a 5 = 0.
with xy ¢ 0 must
(11)
L e t us a s s u m e t h a t t h e s y s t e m (2) has a positive e q u i l i b r i u m p o i n t ( x f , y f ) , with x I > YI, x f + Y I < 1. Since a d - c e = 0 e q u a t i o n (11) b e c o m e s a linear e q u a t i o n w i t h a unique s o l u t i o n given b y y f = a S / ( a d + c5 - a e ) > 0, which implies a d + c5 - a e > 0. Since x I is positive, t h e n
Quadratic Infectious Diseases
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d-e
> 0. Now, since x f > yf, we o b t a i n t h a t a d - a e > a S , t h a t is 5 < d - e . Finally, from = l - (e/a)yf andyf>0wegetthata-c<0. Let us assume now the following inequalities, a d + c5 ae > 0, d - e > 0, a - c < 0, a n d 5 < d - e. First, since a d + e5 - a e > 0, t h e n y f > 0. Second, d - e > 0 implies t h a t a ( d - e) > 0 a n d since a d + e S - a e > 0, t h e n x I > 0. T h i r d a - c < 0implies, (a-c)yf/a < 0, t h a t is x f + yf < 1. Finally, 5 < d - e implies ( a ( d - e ) ) / ( a d + e5 - a e ) > a S / ( a d + c5 - a e ) , t h a t is, xf + yf < l, zf
xf
> yr.
PROOF OF PROPOSITION 3. Let us assume that system (3) has a positive equilibrium xf + yf < I. By uniqueness, the equilibrium point satisfies
c xf = 1-- -yf a
point (xf,yf) with xf > yf, and
- ( a e - a d - eS) - v / ( a e - a d - c5) 2 - 4 a 5 ( a d - ee) >0,
yf =
2 ( a d - ce)
>0,
which implies t h a t a d - ce < O, a e - c5 - a d < 0 a n d a - c < 0. Since z f > yf, t h e n a ( e + 5) < c ( d - ~). Now, from x f + y f < 1, yf > 0, a n d x / = 1 - ( c / a ) y f , we o b t a i n t h a t a - e < 0. Let us assume now t h a t a e - c5 - a d < O, a d - ce < 0, a - c < 0, a n d a ( e + ~) < c ( d - 5). F r o m the first two inequalities we get y / > 0 a n d from a - c < 0 we get t h a t x f > 0. Also from a-e < 0 a n d 9/ > 0, we get x f + y f < 1. Finally, from a ( e + 5 ) < e(d-5) we o b t a i n t h a t xf
> yr.
PROOF OF PROPOSITION 4. T h e proof is similar to the one for P r o p o s i t i o n 3. PROOF OF PROPOSITION 5. We seek solutions to the s y s t e m (9) of the form c~
x=(t-toF
-
aj(t-to) j ,
0 < It-t01 < R ,
(12)
j=0 where to is a n a r b i t r a r y c o n s t a n t . T h e d o m i n a n t algebraic b e h a v i o r is given b y x ~ a ( t - to) - a , y ~ c ~ ( t - t 0 ) -1, and z ~ c ~ ( t - t o ) -a. Substituting x ~ c ~ ( t - t o ) -3 + ~ l ( t - t o ) r-3, y a ( t - to) -1 + / 3 2 ( t - to) ~'-a, a n d z ~ a ( t - to) - a + ~a(t - to) ~-a into the simplified system leads to r = - 1 , which suggests t h a t (12) misses a n essential p a r t of the solution. U n d e r the change of variables
X ~
• (ln (t - to)) (t-
Y =
to)
'
x2 0n (t - to)) (t to) ' -
x3 (in (t - to)) Z~
(t - to)
'
s y s t e m (9) becomes .~1 = x l + X l ( a t - a x t - c x 2 ) + ( p x 2 - q x l ) x 3 , i2 = x2 + +x2 (dt - exl - dx2) + 5txlsx2x3,
(13)
~d3 = x 3 + k o t 2 + k l (22 - x 2 ) + k 2 t - 2 (22 - x 2 ) 2 •
S y s t e m (13) has a variety of regular solutions, i n c l u d i n g those for which x2(0) = 22(0) # 0. Therefore, every n o n e x p o n e n t i a l solution of (13) corresponds to a movable l o g a r i t h m i c b r a n c h p o i n t of (9), t h u s m a k i n g the s y s t e m n o n i n t e g r a b l e .
1324
F . J . SoLIs et al.
REFERENCES 1. A.S. Perelson, J.D. Murray and P.'vV. Nelson, A model of HIV-1 pathogenesis t h a t includes an intracellular delay, Mathematical Biosciences 163, 201-215, (2000). 2. R.V. Culshaw and S. Ruan, A delay-ditferential equation model of HIV infection of CD4 + T-cells., Mathematical Biosciences 165, 27-39, (2000). 3. A.A. Ding and H. Wu, Relationships between antiviral treatment effects and biphasic viral decay rates in modeling HIV dynamics, Mathematical Biosciences 160~ 63-82, (1999). 4. A.S. Perelson, D.E. Kirsehner and R. De Boer, Dynamics of HIV infection of CD4 + T-Cells, Mathematical Biosciences 114, 81 125, (1993). 5. G.L. Baker and J.B. Gollub, Chaotic Dynamics: An Introduction, Cambridge University Press, Cambridge, (1996). 6. CI.F. Gause, The Strugle for Existence, Williams and Wilkins, Baltimore, MD, (1934). 7. A. Mayer, On the theory of coupled vibrations of two self-excited generators, Tech. Phys. USSR 3, 1056-1071, (1936). 8. W.A. Coppel, On a differential equation of boundary-layer theory, Phil. Trans. Roy. Soc. Ser. A 253, 101136, (1961). 9. M. Braun, Differential Equations and Their Applications, Springer-Verlag, New York, (1983). 10. M.J. Ablowitz, A. Ramani and H. Segur, A connection between nonlinear evolution equations and ordinary differential equations of P-type, I. J. Math. Phys. 21, 715 721, (1980). 11. A.S. Perelson and G.W. Nelson, Modeling defective interfering virus therapy for AIDS: Conditions for DIV survival, Mathematical Biosciences 125, 127-153, (1995). 12. J.L. Spouge, R.I. Shrager and D.S. Dimitrov, HIV-1 infection kinetics in tissue cultures, Mathematical Biosciences 138, 1 22, (1996). 13. N.I. Stilianakis, K. Dietz and D. Schenzle, Analysis of a model for the pathogenesis of AIDS, Mathematical Biosciences 145, 27-46, (1997).
8ClENOE~DIREGTe
ELSEVIER
MATHEMATICAL AND COMPUTER MODELLING
Mathematical and Computer Modelling 42 (2005) 1315-1324 www.elsevier.com/locate/mcm
Quadratic Infectious D i s e a s e s M a t h e m a t i c a l Models: Chronic States Sanity Levels and T r e a t m e n t F. J . SOLIS AND B. TAPIA CIMAT Guanajuato Gto. M6xico 36000
J . V . ROMERO AND J. MORENO I n s t i t u t o de M a t e m S t i c a M u l t i d i s c i p l i n a r U n i v e r s i d a d Polit4cnica de Valencia Valencia, Spain 46071
(Received November 2004; accepted December 2004) A b s t r a c t - - S p e c i f i c quadratic continuous models are examined to study the dynamics of interacting cells of an organism that is affected by a disease or virus. We focus only on models that exhibit a chronic state and we study conditions on the parameters involved in the models to guarantee that the organism can obtain a level of sanity. Aggressive and nonaggressive treatments are applied when the organism can not reach such level of recovery. @ 2005 Elsevier Ltd. All rights reserved. Keywords--Infectious
disease, Ctlronic states, Treatment.
1. I N T R O D U C T I O N Viruses are parasites with diverse ways of interacting with their hosts. Some viruses initiate lytic infections, which result in high levels of virus replication followed by clearance of the virus. On the other hand, other viruses establish life-long persistent infections in which the virus has established a quasi-homeostasis with the organism. These behaviors have been studied in contemporary epidemiological theory and they can be traced back to the early part of the 20th century. Before this period, the mechanisms by which infectious disease agents spread within populations or within the human body had been revealed by microbiological research. The theoretical framework most commonly used to mimic the dynamics of viral and bacterial infections is one based on the division of the human population into categories containing susceptibles, infected who are not yet infectious (latent), infectious individuals, and those who are recovered and immune. However, most models that use these categories reflect the dynamics of the number of infected people without reference to the abundance of organisms within each individual. In the literature there exist models whose main interest is the description of the interaction between viruses and the cells of the human body. Those models consider different populations of 0895-7177/05/$ - see front matter (~) 2005 Elsevier Ltd. All rights reserved. doi: 10.1016/j.mcm.2004.12.004
Typeset by AMS-]~X
F.J. SOLIS et
1316
al.
cells and virus (infected ceils, healthy cells, infectious virus, non infectious virus, etc.) and are represented by equations of quadratic type, displaying most of t h e m one carrying capacity for the growth of the healthy ceils. The specific situations t h a t are modelled in some works are mainly the introduction of intracellular delay (see [1,2]), the application of drugs t h a t can be inhibiting or a genetically engineered defective interfering virus (see [3,4]) and the behaviors of healthy ceils and the h u m a n immunodeficiency virus (HIV) and how they affect each other. One of the aims of this work is to elaborate a parametric study of families of quadratic models describing the behavior of the human ceils when they are affected by a disease or virus under the assumption that the dynanfics of the cells are similar regardless of the region of the organism that is under consideration. We seek conditions on the parameters to guarantee t h a t the organism reaches a level of sanity or to achieve at least that the growth of the infected cells is almost null even though that they m a y be present in the organism. These conditions are mathematically equivalent to one of the following two conditions. First, to have all the orbits of the dynamical systems to approach to an equilibrium point that is considered as the level of total sanity, that is, solutions of the m a t h e m a t i c a l models approach to a state with the total number of cells being healthy. Second, t h a t the mathematical models have a positive equilibrium point, considered as a chronic state, located close to a level of sanity and that the orbits near this point tend to stay in a vicinity of this point. We consider a chronic state as a particular state where a organism has a disease or disorder t h a t is present over long periods of time, often the remainder of his lifetime. A second goal is to study the existence of a complex behavior in the models by means of a strange chaotic attractor. The presence of a chaotic attractor in a model does not allow long t e r m predictions on the behavior of the cells and there is no support by experiments or field studies to validate the model, see [5]. We consider two cases in the models under consideration. First case, when the fraction of healthy cells t h a t are infected can be neglected and second case when the above fraction is important. For b o t h cases we will obtain conditions for the parameters to obtain a unique positive equilibrium point, a chronic state. We will analyze the level of recovery of the organism by a stability analysis of such point and we will study the behavior of the orbits considering levels of sanity and level of quasisanity obtaining in this way a region of optimal initial conditions for recovery. For all those cases when the organism can not recover we will study the application of two types of treatments. One nonaggressive, that is, one which does not affect the healthy cells and an aggressive one, which may affect some healthy cells. We study in both scenarios the effect of the treatment. Finally, we apply our results to some models t h a t have been used to describe the behavior of the HIV dynamics. This paper is organized as follows. Section 2 describes components and parameters of the families of models. The dynamic behavior of the models is analyzed in Sections 3 and 4 with or without treatment. The analogy and difference between different models of infections diseases are compared in Section 5.
2. Q U A D R A T I C M O D E L S A N D D Y N A M I C P R E L I M I N A R I E S Quadratic models arise in a variety of applications, see for example, [6-8] and so on. Two cases (one with the plus and the other with the minus sign) of a general quadratic model are the only cases that fit in our framework, namely,
dX (t) - dt dY (t) - dt
- X (t) (a -F b X (t) 4- ~ Y (t)) nL ( k l Y (t) - k 2 X (t)) Z ( t ) ,
- ± Y (t) (d +
/ d~-F~X(t)
dZ (t)
'
(t) + k 3 Y (t)) + 3 X (t) - k 4 Y (t) Z (t),
Y(t),Z(t)
(1)
dX (t_____~)dY_!t) ~ '
dt
'
dt
J"
Where X(t), Y(t), and Z(t) represent the densities of the healthy cells, infected cells and treatment respectively at time t. Coefficients a and d are the intrinsic growth rates of the healthy
Quadratic Infectious Diseases
1317
and infected cells respectively, b and k3 are intraspecific interaction coefficients, ~ and ~ are interspecific interaction coefficients, ~ is the coefficient of healthy cells t h a t become infected, k2 is the coefficient t h a t represents the aggressiveness of the treatment, and finally kl and k4 are the percentages of infected cells t h a t heal and die with the application of the t r e a t m e n t respectively. If we denote by x0 the number of healthy cells and by Y0 the number of infected cells then the total number of cells of the organism at some initial time will be an element of the set {(x0,y0) I x0 _> 0, y0 >_ 0, x0 + Y0 < 1}, which we call the section of initial conditions. We introduce two levels of sanity by dividing this region in three parts. The first division is called the region of sanity, which is the region t h a t the organism is healthy regardless of having some infected ceils, Region II as the zone of quasisanity and Region III as the zone of sickness as shown in Figure la, where the levels of sanity are considered as straight lines, labelled as NC and NCS. Since every organism needs a minimum number of healthy cell to survive and a minimum number of healthy cells to be healthy, we subdivided Region I, taking in consideration those cases where the organism is totally healthy, grouping these cases in a region t h a t we called S. Then, we subdivide Region I I I by considering those cases where the organism can not survive and denoting t h a t region as Region M, as is shown in Figure lb.
(a)
(b)
Figure 1. Set of initial conditions of healthy and infected cells.
There are restrictions over the chronic state and the parameters t h a t we shall impose to ensure realistic h u m a n dynamics. Since a chronic state is a disease or disorder t h a t is present over long periods of time, we identify it as an equilibrium point of system (1). An equilibrium point is a particular solution of (1) that does not change with time, and is determined by setting dX(t) dY(t) dZ(t) _ O. An equilibrium point may be stable or unstable and it can be classified into one of several classes using a linear stability analysis (see [9]). This analysis is restricted to the region immediate to the equilibrium point. Thus, an equilibrium point can be classified as stable node or attractor (unstable node) if nearby orbits converge (diverge) to the equilibrium point, as a saddle point if there are nearby orbits converging and diverging from the equilibrium point and finally, as a center if all nearby solutions are bounded periodic solutions. We remark that the linear analysis is local in nature and thus a global stability of nonlinear systems m a y be a function of the operating conditions for the system, meaning t h a t minor variations in the initial conditions resulted in large differences in their long time behavior. This irregular behavior of a class of nonlinear systems is generated by the existence of a strange chaotic attractor (see [5]). There are basically two necessary criteria for a system to have a chaotic behavior. The system must be nonlinear and must have at least three independent dynamic variables. Of course, there are systems t h a t satisfied these criteria and do no exhibit chaotic behavior. For our models, in order to t h a t the equilibrium point, which we denote by ( x f , y f ) is in the region of interest and it is close to the level of sanity, it is necessary t h a t x f > y f and x I + y f _< 1. dt
--
~
--
dt
--
F.J. SOLIS et aI.
1318
Therefore, we have to impose the following restrictions on the p a r a m e t e r s of the model, a > 0, b < 0, and dk3 < 0. To get rid of some parameters, we nmke the following t r a n s f o r m a t i o n X - ( - a / b ) x , Y = ( - d / k 3 ) y , and Z = z in model (1) to obtain two families of systems of the form -
x (a -
=
±y
(d -
ax
-
ey) +
e~ -
dy)
(py
-
+ ~x -
qx)
z,
syz,
(2)
= f ( x , y , z , Sc,~).
3. Q U A D R A T I C
MODELS WITHOUT
TREATMENT
A first objective is to find conditions on the parameters to make sure the organism reaches a level of sanity or total recovery. To obtain this goal we begin to s t u d y how the organism behaves w i t h o u t a treatment. T h a t is, we will s t u d y a particular case of the families of system (2) namely, J: = x ( a -
az
-
cy) ,
(3) ~1 = -4-y ( d -
ex -
dy)
+ 5x.
We consider separately the cases. First, when d = 0 (fraction of h e a l t h y cells t h a t are infected is null) and later when 5 > 0, since b o t h cases lead us to different analytic conditions. 3.1. C a s e ~ = 0 W i t h 5 = 0 in (3), we obtain two families of models t h a t have been used to described the interaction of two species in population dynamics. Our first step in this case is to guarantee the existence of a chronic state in the region of interest. Conditions for its existence are given in the following proposition. PROPOSITION 1. The system given by (2) with 5 - 0 has a unique positive equilibrium point ( x f , y f ) with x¢ > y f and x f + YI < 1, if and only ifc2e > adc > a2e and d < e. PROOF. See the Appendix. T h e conditions m e a n t h a t the n u m b e r of encounters between the cells is bigger t h a n the rates of growth of b o t h t y p e of cells and t h a t the ratio between the rate of g r o w t h of the healthy ceils and infected cells is less t h a n the ratio of the factors of encounters of the ceils. In the above proposition, we discarded the case w h e n x f + y f = 1, since this condition will have the effect to produce a c o n t i n u u m of equilibrium points at the b o u n d a r y of the region of interest and none at the interior. Having the conditions for a chronic state our next step is to find its behavior when the values of the parameters are changed, its stability properties and finally the behavior of n e a r b y orbits. W h e n we vary the values of the p a r a m e t e r s in b o t h models we observe t h a t when we decrease (increase) the values of a and e the chronic state reduces (increases) its n u m b e r of infected ceils and increases (reduces) its n u m b e r of healthy cells, thus improving the health of the organism. W h e n we vary the values of c and d we obtain the same results as varying a -1 and e -1, respectively. For the first model (with the plus sign) with the infected cells having a negative rate of growth, the stability of the equilibrium point found in Proposition 1 depends on the value of 1 / 7 = aid. If 1 / 7 _> 1, then the point m a y be an a t t r a c t o r of stable spiral. If 1 / 7 < 1 and (a) if c - a = e - d, then the point is a center (nearby orbits are periodic); (b) if c - a > e - d, t h e n is an a t t r a c t o r (attracts all n e a r b y orbits); (e) if c - a < e - d, then the point is an unstable node (some orbits get away for it). For centers, we have t h a t when the chronic state is in Regions I, S, or II, the region of optimal initial conditions, defined as the set of initial conditions t h a t lead the organism to a total recovery, is completely in the same region t h a n the equilibrium point (surround the point), so a t r e a t m e n t
Quadratic Infectious Diseases
1319
is not required. We have a similar behavior when the point is an attractor and it is located in Region I or S. In all of the above cases, the region of optimal conditions can be considered in average as a circle with center in the chronic state and a small radius r; which is given by r = m i n { l N S - x o l , I N C S - x o l , Yo, d(P, C), d(P, L)} where P denotes the chronic state, L is the straight line, x + y = 1, C is the boundary of the Region S and d ( A , B ) is the distance from A to B. If the chronic state is in Region II, the organism gets worse in time without any possibility to recover totally but perhaps a t r e a t m e n t could avoid it. For an unstable node the number of infected cells increases with time and therefore, a t r e a t m e n t is strongly required. For the second model (with the minus sign) where the infected cells have a positive rate of growth the fixed point is always a saddle point. Only the points in Regions I and S have regions of optimal initial conditions. These regions are small, thus a t r e a t m e n t is required to increased the size of these regions.
3.2. Case 6 ¢ 0 In this case, we will consider separately the models that we have been studying. The reason is that the algebraic systems that define the equilibrium points are different. For the model given by (2) with the plus sign we have two cases: (1) First Case: ad - ce = 0. Which means that the product of the rates of growth is equal to the product of the factors of the encounters of the cells. PROPOSITION 2. S y s t e m (3) has a positive equilibrium p o i n t ( z / , y I ) of the form
c6
x f=l
ad + c6 - ae' a6 ad+c6-ae'
Yfwith x I > YI, x f + yf < 1, if and only if ad + c 6 - ae > O,
and
a < c,
6 < d - e.
(4)
PROOF. See the Appendix. Notice that this equilibrium point approaches to the point of total sanity as 6 approaches to zero. (2) Second Case: The product of the rates of growth is different to the product of the factors of the encounters of the cells, t h a t is ad - ee 7~ O. PROPOSITION 3. S y s t e m (3) has a positive equilibrium p o i n t (x f , y f ) o f the form C
x I = i - -Yl, a
-
(ae
-
ad
-
e6)
Yl =
-
-
-
e6)
2 (ad - ce)
2 -
-
ce)
'
with x I > Yl, x f + Yl < 1, if and only i£ ae < c6 + ad,
ad < ce,
a < c,
and
a (6 + e) < c (d - 6).
(5)
PROOF. See the Appendix. The equilibrium point found is the &generalization of the equilibrium point given in Proposition 1. For the model given by (2) with the minus sign, where the infected cells have a positive rate of growth, it is not possible to find a &generalization of the equilibrium point given in Proposition 1. However, it is possible to find an equilibrium point if we require t h a t ad - ce ~ O.
1320
F . . J . SOLIS
et al.
PROPOSITION 4. S y s t e m (2) has a p o s i t i v e e q u i l i b r i u m p o i n t (x f , y f ) o f t h e f o r m xf =1-
e(ad-cS-ae) 2a (ad - ce)
'
ad - c5 - ae 2 ( a d - ce) '
Y f -w i t h x f > y f , x f + y f < 1, if and o n l y i f
(ad - c 5 -
ae) 2 = 4aS (ce - ad) ,
ad < c6 + ae,
a < c,
ad (2a - c) < ace - c25,
(6)
and a 2(d+e)
PROOF. See the Appendix. When we make 5 approach to zero the chronic state approaches to a state of total health. For the case 5 ¢ 0, the equilibrium point for the first model can be of three types: center, stable node or unstable node and for the second model only of one type, namely unstable node. In this last case, the organism approaches to a state with the total number of cells been infected. In both models the region of optimal initial conditions is smaller than the one for the case of 5 = 0 and when we vary the value of 5 the chronic state increases (reduces) its number of healthy cells and reduces (increases) its number of infected cells when we decrease (increase) the value of 5. 4.
TREATMENT
Let us include in our systems the equation that models the treatment, which we had left out before. We will consider that the treatment can be non aggressive or aggressive. In the first one, the treatment will affect only to the infected cells making that a proportion of these get healthy and another proportion to die. The second one acts as the first one but now it affects also the healthy cells, that is why it is known as an aggressive treatment. We will consider three cases for the equation for the treatment: In the first, the equation for the treatment is of the form i = 0, meaning that the dose is constant and for the second one i = kl~ ) with kl > 0, meaning that the treatment is applied proportional to the speed of growth of the infected cells. For the third case, we generalize the equation for the treatment to a quadratic equation in ~) to analyze the possibility of a existence of a chaotic attractor, thus the third equation become = ko + k15 + k2($) 2 For the first two cases, we solve the third equation and substitute it in our previous systems giving us the following system 2 = x ( a - a x - cx) + p y n _ q y ' ~ x ,
(7) ~1 = ± y ( - d + e x + dy) + 5x - s y ~.
Where m = 0 and n = 1 corresponds to the constant dose and m = 1 and n = 2 corresponds to the nonconstant dose. The parameter q indicates the aggressiveness of the treatment, p and s are respectively the percentages of infected cells that heal and die with the application of the treatment. For these two cases, the treatment that has better results in the sense of increasing the region of optimal conditions is the one when the dose is constant and nonaggressive treatment, except
Quadratic Infectious Diseases
1321
for the case when 2a > c and 5 ~ 0. For this last case, it is b e t t e r to have the n o n c o n s t a n t dose and an aggressive treatment. Now let us consider the third case. In order to investigate the key p a r a m e t e r s for which the system m a y have a chaotic behavior we make use of a conjecture t h a t establishes t h a t systems t h a t do not possess the Painleve p r o p e r t y are nonintegrable, and therefore there exists a strong possibility of existence of chaos in t h e m (see [10]). Conditions for the integrability of the system are given in the following proposition. PROPOSITION 5. W i t h t h e p r e v i o u s n o t a t i o n s y s t e m ic = x (a - a x - cy) + (py - qx) z, 9 = +Y (d-
ex-
dy) + 6 x -
(8)
syz,
= ko + k l ~ + k2 @ ) 2 ,
is integrab]e i f k2 7~ 0 and n o n i n t e g r a b l e otherwise.
PROOF. See the Appendix. From this result we discover t h a t k2 is a critical p a r a m e t e r controlling tile possible turbulent d y n a m i c s of our models. Therefore, we can validate our models with d a t a using only the previous two cases of the treatment.
5. M O D E L S
OF H I V
We will like to use our models to c o m p a r e t h e m with other models t h a t have been used in the literature in the s t u d y of some infectious diseases. We concentrate in some models of H I V and we use their d a t a to fit our models, (see [1,2-4,11-13]). We find t h a t the values of the parameters for the model ic = x (a - a x - c y ) , 9 = -y
(d - ex - dy) + ~,
(9)
vary in the intervals given by a C [0.01, 0.02], c • [0.01,0.02], d e [0.26, 0.5], and e E (0.26, 0.5). T h e values of the parameters for the model Jc =
x (a -
= y (d-
ax
-
cy)
,
(1o)
e x - dy) + 5z,
are in the intervals a E [0.01, 0.02], c e [0.01, 0.02], d and e are fl'ee.
ud
LJ.4
u.:.
L~.~
(a) Nonagressive.
U.t
U.U
U.U
1
0
0.!
0.2
03
014
0.5
06
(b) Aggressive.
Figure 2 Numerical simulations with different treatments
0.7
08
0.9
1
1322
F . J . SOLIS et al.
T h e values of 5 for b o t h m o d e l s were chosen to g u a r a n t e e t h e e x i s t e n c e of a chronic s t a t e . F o r t h e a p p l i c a t i o n of a k n o w n t r e a t m e n t (see [3]) t h e p r o p o r t i o n of i n f e c t e d cells t h a t are affected b y t h e t r e a t m e n t is a p p r o x i m a t e l y s = 1.2 a n d this is t h e value t h a t we will consider in our simulations. We v a r i e d all p a r a m e t e r a n d for b o t h m o d e l s t h e b e s t r e s u l t s were o b t a i n e d b y a c o n s t a n t dose a n d non aggressive. In F i g u r e 2 we show some n u m e r i c a l s i m u l a t i o n s using a specific set of p a r a m e t e r s w i t h an a n o n a g g r e s s i v e and an aggressive t r e a t m e n t using a c o n s t a n t dose. 6.
CONCLUSIONS
S t u d y i n g families of simple q u a d r a t i c m o d e l s t h a t d e s c r i b e t h e d y n a m i c s of infected diseases on o r g a n i s m s , which have a clear division b e t w e e n infected cells a n d h e a l t h y ceils, has led us to c o n c l u d e t h a t is possible to o b t a i n c o n d i t i o n s over t h e p a r a m e t e r s t o have a chronic s t a t e of t h e o r g a n i s m . In t h o s e cases, where t h e chronic s t a t e is classified as an a t t r a c t o r , t h e r e is a full r e c o v e r y of t h e o r g a n i s m for a o p t i m a l region of initial c o n d i t i o n s . I n t h e cases where t h e chronic s t a t e is a center, u n s t a b l e n o d e or a saddle, it is n o t possible t o o b t a i n a recovery. W e can r e i n s t a t e t h e level of s a n i t y in these last cases w h e n we i n t r o d u c e a t r e a t m e n t t h a t m a y or m a y not affect t h a t h e a l t h y cells. Since we only a d d a specific t y p e of t r e a t m e n t , a n i n t e r e s t i n g t o p i c to s t u d y in t h e n e a r f u t u r e is to a n a l y z e different p o s s i b l e t y p e s of t r e a t m e n t s t h a t lead to a full recovery.
APPENDIX PROOFS
OF
PROPOSITIONS
PROOF OF PROPOSITION 1. O b s e r v e first t h a t if a d - c e ~ O, t h e n t h e u n i q u e s o l u t i o n to t h e a l g e b r a i c s y s t e m a - a x - c y = 0 a n d d - e x - d y = 0 is given b y ad - cd xf -
ad - ce' ad - ae
Yf -
ad - ce "
L e t us first a s s u m e t h a t s y s t e m (2), w i t h 5 = 0, has a unique p o s i t i v e e q u i l i b r i u m p o i n t ( x f , y f ) w i t h x y > y y a n d x / + y i < 1. Since t h e s y s t e m has a u n i q u e p o s i t i v e e q u i l i b r i u m point, t h e n t h e d e t e r m i n a n t a d - c e can be positive or negative. T h e d e t e r m i n a n t can n o t b e positive, since if it were, t h e n a - c > 0 a n d d - c > 0 in o r d e r to have a p o s i t i v e solution. Since x i + y f < 1, we get t h a t (d - e ) ( a - c) < 0, which is a c o n t r a d i c t i o n . Therefore, t h e d e t e r m i n a n t is negative, t h a t is a d - e e < 0, a n d since t h e solution is positive, we get t h a t a - c < 0 a n d d - e < 0. Now, since x I > Y I , we o b t a i n e d > a e . C o m b i n i n g these two results, we finally g e t e2e > a d e > a 2 e a n d d < e. Now, let us a s s u m e t h a t c 2 e > a d c > a 2 e a n d d < e. F r o m c2e > a d e , we o b t a i n t h a t ce > a d , which m e a n s t h a t t h e d e t e r m i n a n t of t h e a l g e b r a i c s y s t e m is negative. T h u s , we o b t a i n a unique solution. F r o m a d c > a 2 e we get a d - c d < a d - a e a n d since t h e d e t e r m i n a n t is negative, t h e n x z > Y I . T h e fact t h a t t h e s o l u t i o n is positive comes from c 2 e > a 2 e a n d d - e < 0. Finally, since d - e < 0 a n d a - c < 0, t h e n (d - e ) ( a - c) > 0, t h a t is x y + y f < 1. PROOF OF PROPOSITION 2. N o t i c e first t h a t t h e e q u i l i b r i u m p o i n t s ( x f , y y ) satisfy t h e q u a d r a t i c e q u a t i o n ( a d - c e ) y~ + ( a e - a d - e S ) y y + a 5 = 0.
with xy ¢ 0 must
(11)
L e t us a s s u m e t h a t t h e s y s t e m (2) has a positive e q u i l i b r i u m p o i n t ( x f , y f ) , with x I > YI, x f + Y I < 1. Since a d - c e = 0 e q u a t i o n (11) b e c o m e s a linear e q u a t i o n w i t h a unique s o l u t i o n given b y y f = a S / ( a d + c5 - a e ) > 0, which implies a d + c5 - a e > 0. Since x I is positive, t h e n
Quadratic Infectious Diseases
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d-e
> 0. Now, since x f > yf, we o b t a i n t h a t a d - a e > a S , t h a t is 5 < d - e . Finally, from = l - (e/a)yf andyf>0wegetthata-c<0. Let us assume now the following inequalities, a d + c5 ae > 0, d - e > 0, a - c < 0, a n d 5 < d - e. First, since a d + e5 - a e > 0, t h e n y f > 0. Second, d - e > 0 implies t h a t a ( d - e) > 0 a n d since a d + e S - a e > 0, t h e n x I > 0. T h i r d a - c < 0implies, (a-c)yf/a < 0, t h a t is x f + yf < 1. Finally, 5 < d - e implies ( a ( d - e ) ) / ( a d + e5 - a e ) > a S / ( a d + c5 - a e ) , t h a t is, xf + yf < l, zf
xf
> yr.
PROOF OF PROPOSITION 3. Let us assume that system (3) has a positive equilibrium xf + yf < I. By uniqueness, the equilibrium point satisfies
c xf = 1-- -yf a
point (xf,yf) with xf > yf, and
- ( a e - a d - eS) - v / ( a e - a d - c5) 2 - 4 a 5 ( a d - ee) >0,
yf =
2 ( a d - ce)
>0,
which implies t h a t a d - ce < O, a e - c5 - a d < 0 a n d a - c < 0. Since z f > yf, t h e n a ( e + 5) < c ( d - ~). Now, from x f + y f < 1, yf > 0, a n d x / = 1 - ( c / a ) y f , we o b t a i n t h a t a - e < 0. Let us assume now t h a t a e - c5 - a d < O, a d - ce < 0, a - c < 0, a n d a ( e + ~) < c ( d - 5). F r o m the first two inequalities we get y / > 0 a n d from a - c < 0 we get t h a t x f > 0. Also from a-e < 0 a n d 9/ > 0, we get x f + y f < 1. Finally, from a ( e + 5 ) < e(d-5) we o b t a i n t h a t xf
> yr.
PROOF OF PROPOSITION 4. T h e proof is similar to the one for P r o p o s i t i o n 3. PROOF OF PROPOSITION 5. We seek solutions to the s y s t e m (9) of the form c~
x=(t-toF
-
aj(t-to) j ,
0 < It-t01 < R ,
(12)
j=0 where to is a n a r b i t r a r y c o n s t a n t . T h e d o m i n a n t algebraic b e h a v i o r is given b y x ~ a ( t - to) - a , y ~ c ~ ( t - t 0 ) -1, and z ~ c ~ ( t - t o ) -a. Substituting x ~ c ~ ( t - t o ) -3 + ~ l ( t - t o ) r-3, y a ( t - to) -1 + / 3 2 ( t - to) ~'-a, a n d z ~ a ( t - to) - a + ~a(t - to) ~-a into the simplified system leads to r = - 1 , which suggests t h a t (12) misses a n essential p a r t of the solution. U n d e r the change of variables
X ~
• (ln (t - to)) (t-
Y =
to)
'
x2 0n (t - to)) (t to) ' -
x3 (in (t - to)) Z~
(t - to)
'
s y s t e m (9) becomes .~1 = x l + X l ( a t - a x t - c x 2 ) + ( p x 2 - q x l ) x 3 , i2 = x2 + +x2 (dt - exl - dx2) + 5txlsx2x3,
(13)
~d3 = x 3 + k o t 2 + k l (22 - x 2 ) + k 2 t - 2 (22 - x 2 ) 2 •
S y s t e m (13) has a variety of regular solutions, i n c l u d i n g those for which x2(0) = 22(0) # 0. Therefore, every n o n e x p o n e n t i a l solution of (13) corresponds to a movable l o g a r i t h m i c b r a n c h p o i n t of (9), t h u s m a k i n g the s y s t e m n o n i n t e g r a b l e .
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