SYMBIOTIC MODELS WITH AN SIR DISEASE

Transcription

1 SYMBIOTIC MODELS WITH AN SIR DISEASE A Thesis Presented to the Faculty of California State Polytechnic University, Pomona In Partial Fulfillment Of the Requirements for the Degree Master of Science In Mathematics By Michael Bonilla 2016

2 SIGNATURE PAGE THESIS: SYMBIOTIC MODELS WITH AN SIR DISEASE AUTHOR: Michael Bonilla DATE SUBMITTED: Spring 2016 Department of Mathematics and Statistics Dr. Jennifer Switkes Thesis Committee Chair Mathematics & Statistics Dr. Hubertus Von Bremen Mathematics & Statistics Dr. Ryan Szypowski Mathematics & Statistics ii

3 ACKNOWLEDGMENTS First I would like to deeply thank Dr. Switkes for her guidance, advice, and support in completing this thesis and in my coursework. I would also like to thank Dr. Von Bremen and Dr. Szypowski for there teachings and advice through my years of course work at Cal Poly. Lastly I would like to thank my family for their support during the writing of this thesis. iii

4 ABSTRACT In this thesis we will investigate how a species that is being afflicted with an SIR disease reacts when another species is introduced into the system that has a symbiotic relationship with the diseased species. First we will introduce the classical SIR model and results we can observe later on in our model. Next we will start with a basic symbiotic model of two species and explore its behavior. Later from the simple symbiotic model we will build in more complex behavior into the model such as births, deaths, and overcrowding parameters and analyze these models. We will notice in analyzing these models some interesting bifurcations occur when some parameters are varied. Lastly we introduce our own model that combines the two ideas so that we have a model of two symbiotic species with disease in one of the species. We compare and contrast the models previously explored and explore the varying changes in behavior as the disease affects the symbiotic relationship and how the symbiotic relationship affects the disease dynamics in the infected species. iv

5 Contents 1 Introduction 1 2 Classical SIR Model Introduction SIR Model Analysis of SIR Model Classical Symbiotic Species Interaction Model Basic Symbiotic Model Symbiotic Model with Birth and Deaths Symbiotic Model with Overcrowding Symbiotic Model with Overcrowding, Births, Deaths Symbiotic Model with Disease Basic Symbiotic Model With Disease Symbiotic Model with Disease and Birth and Death Rates Symbiotic Model with Disease and Overcrowding General Symbiotic Model with Disease Equilibrium Analysis of Symbiotic Model with Disease 50 v

6 5.1 Equilibrium Analysis of Symbiotic Model with Disease Equilibrium Analysis of Symbiotic Model with Disease, Births, and Deaths Equilibrium Analysis of Symbiotic Model with Disease and Overcrowding Equilibrium Analysis of General Symbiotic Model with Disease Conclusion 71 Bibiliography 73 vi

7 Chapter 1 Introduction Much work has been done in the analysis and research of mathematical models of interactions between species. Since the early species models like the Lotka-Volterra predator prey models, many models have been created in this fashion to model other types of behavior between species. Some of these behaviors between species include competitive, parasitic, and symbiotic relationships. Models have become more sophisticated over time in other ways that even incorporate other factors that affect species and their interactions [3][6] such as births, deaths, and overcrowding behaviors. Here we will focus on the interaction between two species in a symbiotic interaction and start off with a basic model that is easy to understand. We later introduce more complicated dynamics in each species such as births, deaths, and overcrowding in order to explore how the populations of both species change over time due to these dynamics. Along with population dynamics we will also look into the dynamics of disease within a single species. For our purposes we will focus on the classical Kermack-McKendrick SIR model [7] which attempts to model the dynamics of disease in a single species that allows for the species to recover from the disease and attain an immunity. There have also been developments 1

8 in species interaction modeling that allows us to incorporate a disease model in one or both species and analyze the interaction between the species [1][4]. In our model we will focus on a symbiotic model between two species and introduce an SIR disease into one of the species and observe how the two species interact. As we are looking at a symbiotic we hope to see how the interaction of the diseased species is affected by the symbiotic non-diseased species. Also we will investigate how introducing a species with a symbiotic relationship with another species afflicted with an SIR disease affects the population dynamics when different symbiotic models are used. For the most part in the analysis we will attempt to solve each system analytically until the models become too complicated to solve analytically and we turn to technology to carry out further analysis in solving the system of differential equations numerically. 2

9 Chapter 2 Classical SIR Model 2.1 Introduction Before we introduce our developed model of disease present in a symbiotic species model we will look into the classical SIR (Susceptible, Infected, Recovered) model and some results from the model. Disease models like the SIR model vary in structure and uses. More complicated models involve more groups of a single population in different stages of a disease like the SEIR and MSEIR models. Some models are simpler in structure like the SI and SIS models. These types of models are used depending on the disease in question, the population being affected, and other factors that can be observed from interactions within the population. The following is a classical SIR model proposed by Kermack and McKendrick in 1927 [5] that takes into account three groups of a certain population. Among the three groups are the susceptible class, the infective class, and the recovered class. As the classes can suggest the SIR model attempts to model a disease affecting a population susceptible to a disease transmitted by an infected group of the population. In this scenario we also include the possibility of recovering from the disease 3

10 and having permanent immunity once recovered. 2.2 SIR Model We have the following set of differential equations modeling the SIR scenario we have presented: ds/dt = β IS di/dt = β IS γi. (2.1) dr/dt = γi The following table contains a description of the variables used in the model above and it is important to note that in this model the population is constant and contained in a geographical area. Variables Description S I R β γ Susceptibles Infectives Recovered Contact rate Recovery rate. The S variable represents the susceptible population of the species that are vulnerable to infection and have no immunity. I represents the infected population in the model and these individuals can spread the disease with contact with susceptible individuals from the S class of the population. Lastly the R variable represents the population of individuals that have attained permanent immunity from the disease that can no longer carry the disease. Here we have β being the contact rate of the disease between the 4

11 infected population and the susceptible population. For example if it takes the disease a certain amount of time to infect a susceptible individual it is reflected in β. Similarly the recovery rate γ reflects the amount of time required for an infected individual to recover and move into the recovered group of the population. Therefore when we look at the terms of the system of differential equations we have the terms β IS and γi occurring. For β IS this term tells us the rate at which susceptible individuals are being infected and moving into the infected class depending on the sign of the term since all of our variables are non-negative. This β IS term also tells us that the rate of infection is proportional to the product of the infected and susceptible populations which is the contact rate between the two populations or in other words how often they interact with one another. The γi term tells us that the rate at which infected individuals are recovering and moving into the recovered class of the population is proportional to the population of the infected population. 2.3 Analysis of SIR Model Analyzing the SIR model we find the equilibrium points to be (n,0,m) for n,m R. This means that the SIR model will be at equilibrium when the infection dies out and we have only susceptible individuals, recovered individuals, or both groups being present after the infection has died out. The equilibrium point with all populations zero is not relevant in this scenario as the model assumes we will have constant population throughout. Now we look into the stability of the solutions and the system overall by looking at the Jacobian of the system and its eigenvalues. We get for the Jacobian to be 5

12 β I β S 0 β I β S γ 0. 0 γ 0 (2.2) The eigenvalues of this system are evaluated at (n,0,m) and are λ 1 = λ 2 = 0,λ 3 = β n γ. Since we have some zero eigenvalues the standard results from linearization do not hold here and therefore we must analyze the model further. For this model we have initial conditions S 0,I 0 > 0 and R 0 = 0 since we must have a positive number of susceptible and infected individuals in order for the model to make sense. Also note that we must have the recovered population being zero since the disease has not had time to spread to more individuals or the infected individuals to recover. Now that we have this let us solve the S and I differential equations since they do not include the R variable and we can divide the two equations to get the following: di β IS γi =. (2.3) ds β IS We can solve (2.3) by separating variables: γ di = 1 ds (2.4) β S γ di = 1 ds (2.5) β S γ I = lns S +C 1 β (2.6) γ I + S ω lns = C 1 for ω =. β (2.7) We have an equation in terms of S and I with only the unknown constant of integration we get in the process of solving the system but we can find what C 1 is by using the initial 6

13 conditions we set before. Using our initial conditions we have C 1 = I 0 + S 0 ω lns 0. Therefore we have from (2.7) our final equation for S and I being I + S ω lns = I 0 + S 0 ω lns 0. (2.8) Using this we now have an idea of how the S,I phase plane will look like but before we get to this there are some things to point out. Notice that the entire population of the species in the SIR model is constant since if we add the three differential equations of S,I,R we have dn = dt ds di dr dt + dt + dt = 0. Therefore N = S + I + R = N 0 at any time, so at t = 0 we have N(0) = S(0) + I(0) + R(0) = S 0 + I 0 + R 0 = S 0 + I 0 = N 0. Also over time we expect the S population to decrease and the R population to increase since the susceptible population differential equation rate function is negative in sign and the recovered population differential equation rate function is positive in sign. Now notice the differential equation for the infected population can be either positive or negative depending on the S,β,γ values. Suppose for some initial conditions S 0 > 0,I 0 > 0 we have di dt t=0 = I 0 (β S 0 γ). (2.9) Here in (2.9) di/dt is positive for S 0 > β γ so I(t) > I 0 on some interval [0,t] if S 0 > β γ. Also di/dt is negative for S 0 < γ so I(t) < I 0 on some interval [0,t] if S 0 < β γ. Note in β the case for S 0 > γ the I differential equation rate function is positive and then will β become negative since we said the S population will decrease over time. Therefore when S 0 > γ the infected population will initially increase to a peak and from there I will β decrease from there on. This is known as an epidemic where the infection spreads and the infected population is able to grow more than the initial amount but we still end up having the infected population fully recover eventually. In the other case where there is 7

14 no epidemic the disease in the entire population does not spread fast enough to cause the infected group to survive or the recovery rate for the population is more than enough to cause the disease from spreading. The following figure is a phase portrait in the S I plane of various initial conditions showing when we have an epidemic and when we do not. Figure 2.1: Phase portrait of S and I for β = 1 and γ = 1 In the plot we can see the direction field showing how the populations S and I increase or decrease. The black line in the plot is the line I = S + N 0 and this line bounds all solutions of the system and all solutions start here given the initial conditions represented by the green dots. We can confirm in the plot when S 0 > γ β = 1 the infected population grows until S = 1 and then the infected population decreases towards zero. It is easy to 8

15 see that the R population, when I goes to zero, will increase towards N 0 S(t f ) where t f > 0 is the time when I = 0. Therefore the SIR model is stable in all populations and the equilibria are of the form (n,0,m) for n,m R depending on the parameters and initial conditions. The following set of plots gives us an example of solution curves to the SIR model and shows for S 0 > γ β we have an epidemic since the initial infected population γ grows then shrinks. We can also confirm when we have S 0 < β the infected population does not increase and just decreases towards zero. Figure 2.2: SIR Model plots with β,γ = 1 (a) Model with no epidemic, S 0 = 0.5,I 0 = 1.5 (b) Model with an epidemic, S 0 = 1.8,I 0 = 0.2 9

16 Chapter 3 Classical Symbiotic Species Interaction Model Now that we have covered some of the basics of disease modeling in a SIR setting we are moving on to introducing some of the basic symbiotic models of species interactions. We will look at three variations of the symbiotic species model and for all three of these we will consider the amount of species in a specific geographical area opposed to proportions of species in the entire population. Before introducing the models we will list the variables used in the models which are all non-negative. 10

17 Variable P Q c e a α d β b f Description Species P Species Q Symbiosis parameter for Q affecting P Symbiosis parameter for P affecting Q Birth parameter for species P Death parameter for P Birth parameter for Q Death parameter for Q Overcrowding parameter for P Overcrowding parameter for Q 3.1 Basic Symbiotic Model The first model we introduce is dp/dt = cpq dq/dt = eqp (3.1) We can see here that the terms cpq and eqp are the same except for the different constants which in this case are the different symbiosis parameters and we will see how these constants affect the solutions to the system. Both of these terms represent rate of the growth of the populations being proportional to the product of the population of both species P and Q. This model is simple and it is easy to see since every term is positive we will have unstable growth in both terms for all positive initial conditions and the equilibrium points are (n,0),(0,n) for n R. Not only can we find the equilibria of this system we can even solve explicitly for the solutions to the differential equations by first 11

18 dividing differential equations: dp c = (3.2) dq e c dp = dq (3.3) e c dp = dq (3.4) e c P = Q + k 1. (3.5) e Now that we have an equation for P in terms of Q, we can solve for the constant k c 1 giving us k 1 = P 0 e Q 0 but in the following equations in solving the Q differential equation we will leave the k 1 constant for simplicity. Substituting P into the Q differential equation we get dq c = eq( Q + k 1 ) (3.6) dt e dq Q c = edt (3.7) ( e Q + k 1 ) dq = edt. (3.8) Q( c e Q + k 1 ) that To proceed the left integral in (3.8) is computed using partial fractions and we have and our integral in (3.8) is now 1 1 c c = c (3.9) Q( Q + k 1 ) Qk 1 ek 1 ( Q + k 1 ) e e 12

19 1 c c dq = Qk 1 ek1( e Q + k 1 ) edt (3.10) lnq ln( c e Q + k 1 ) = et + k 2 k 1 (3.11) 1/k Q 1 c = exp[et]k 3 for k 3 = exp[k 2 ] ( e Q + k 1 ) (3.12) Q k 1 c = exp[etk1]k 4 for k 4 = k Q k 3 (3.13) e + 1 c e Q + k 1 = 1 exp[ etk1 ]k 5 for k5 = k Q 4 (3.14) k1 Q = c. exp[ etk 1 ]k 5 e (3.15) 0 From (3.15) we can use Q to solve for P using (3.5) and k5 which is k5 = Q. There fore our final solutions for both P and Q are the following: P 0 c e ( c 0 Q0) P exp[ et( P c Q 0 0 e 0 )] Q c e P0 e Q 0 P0 e c Q0 c P exp 0 c [ et(p 0 e Q 0 )] Q 0 e e P(t) = + c Q(t) = for P(0) = P 0 > 0 and Q(0) = Q 0 > 0. 0 (3.16) We will look into analyzing the equilibrium points by looking at the Jacobian of the system around each point in order to determine the stability. The Jacobian in this case is cq cp J(P,Q) =. (3.17) eq ep For our equilibrium point at the origin the eigenvalues are zero so the linearization about this equilibrium does not tell us much about the stability of the origin, although we expect any equilibrium to be unstable since the differential equations have positive rate functions here for P > 0,Q > 0. As for the infinite number of nonzero equilibria (n,0) and (0,n) we have one eigenvalue being zero and another being en and cn respectively 13

20 for each equilibrium. Since we have one zero eigenvalue and one positive eigenvalue, for n > 0, we have non-isolated fixed points which makes sense since we have an infinite number of fixed points on the P and Q axis. We can still see the behavior of the system if we look at the phase portrait of the system given here: Figure 3.1: Phase Portrait with c,e = 1 We can see in the phase portrait we have straight lines which means the P and Q c populations grow proportionally by e. This is confirmed by our result of (3.5) which gives us the equations of the curves in the phase portraits being lines with the same slope but different P intercept. Also from the phase portrait we can confirm that both species P and Q will grow without bound which can be seen in the following plot: 14

21 Figure 3.2: Plot of P(t) and Q(t) for c,e = Symbiotic Model with Birth and Deaths The next model adds onto the basic symbiotic model by including birth and death terms into the model in order to control the population growth in both species: dp/dt = P(cQ + a α) dq/dt = Q(eP + d β ) (3.18) This model is not as simple as the one before and therefore we would need to analyze it further in order to determine how stable or unstable the system is. Here we still have the same terms as in the previous model but we add the terms ap,αp,dq,β Q. These 15

22 terms depending on the constant being multiplied by tells us that the rate of births and deaths is proportional to their respective population sizes. Therefore when we have a minus sign in front of a term then this term tells us that the population is decreasing at a rate proportional to the population size. In this model we have two equilibrium points, (0,0) and ( B, A = a α and B = d β to make derivations easier. Also the Jacobian here is e A c ). Here we let J(P,Q) = cq + A cp. (3.19) eq ep + B The eigenvalues from the Jacobian for the origin equilibrium point are λ 1,2 = A,B and for the second equilibrium are λ 1,2 = ± AB. Note that A and B are not always positive therefore the signs of both these constants determine the stability and type of equilibrium we have here. Let us first focus on the equilibrium at the origin which has eigenvalues that are non zero because we feel the case with one or both species having equal birth and death rates is unnecessary to examine here and eliminates some of the dynamics we want to see in the model. Therefore the origin can be a stable, unstable, or saddle node depending on the signs of A,B. We will focus on the case where the equilibrium at the origin is the only one that exists in the first quadrant and on the nonnegative P and Q axes. In this scenario we can not have both A,B being negative or else the second equilibrium will appear in the first quadrant. Therefore letting both A and B being positive we will get the origin being an unstable node which we can confirm by the following phase portrait: 16

23 Figure 3.3: Phase portrait with A,B,c,e = 1 From this phase portrait we can see that both populations will grow without bound reaffirming what we have observed in the differential equations representing the growth of both populations, which are positive rate in this case. Now we can allow one of the variables A or B to become negative and we still have only the first equilibrium existing in the first quadrant. For our purposes we will allow B = 1 and the rest of the parameters are one as before and in this case we expect the origin to be a saddle node which is confirmed in the following phase portrait: 17

24 Figure 3.4: Phase portrait with A,c,e = 1 and B = 1 From the phase plot and the parameter values in the differential equations for the population growth rate of species P all terms are positive and therefore P will always grow. For the Q species the differential equation representing the growth rate is only negative in one parameter which represents a net decrease in the population since the death parameter is greater than the birth parameter here. Since the P population is growing without bound P and this affects the Q species growth rate differential equation since the symbiotic term eqp will grow as P grows. In fact the symbiotic term eqp will grow faster over time than the birth-death parameter term BQ and therefore the Q species will eventually grow without bound since the unbounded growth of P increases the symbiotic impact on species Q. Now we allowed B to be negative in this scenario but 18

25 if we allow A to be negative instead of B then we will have the same situation except Q will have the noticeable unbounded growth in its population. Similarly like before letting Q have the positive rate function for population growth will allow the symbiotic impact Q has on P to increase over time and eventually the symbiotic impact will outpace the AP birth-death term, which is negative here, and the P population will then grow without bound as we have seen before. Figure 3.5: Phase portrait with B,c,e = 1 and A = 1 The second equilibrium exists in the first quadrant if and only if A,B < 0 meaning the death rates of both species must be greater than their respective birth rates. When we do have A,B < 0 the eigenvalues given by the Jacobian earlier are always a pair of real eigenvalues, one being positive and the other negative. Therefore the second equilibrium 19

26 is always a saddle node when it exists in the first quadrant. Now if we want to have both equilibria coexist in the first quadrant then we have the origin being a stable node and the second equilibrium is a saddle node. This means for certain initial conditions we will either have both species heading towards the origin meaning both species die out or both species populations will grow without bound. The following phase plot shows the two equilibria existing at the same time: Figure 3.6: Phase portrait with A,B = 1 and c,e = 1 We can see the phase portrait does show the origin being a stable node and the second equilibrium, here it is (1,1), is a saddle node. Therefore depending on the initial conditions of the populations both species populations will either tends towards zero or both will thrive and grow without bound. 20

27 3.3 Symbiotic Model with Overcrowding We move onto another variant of the symbiotic model. Previously we introduced death and birth terms as a way to temper the growth of both populations so we do not have unstable growth of both species. This time we introduce the idea of overcrowding of one species in a certain geographical area. Overcrowding can represent the effect of having a large enough population leading to many of one species occupying an area with limited resources. We have the following system of differential equations to represent this: dp/dt = P( bp + cq) dq/dt = Q( f Q + ep) (3.20) Here we have the same symbiotic terms cpq and eqp as before in the previous models representing our symbiotic interactions between the species. Now we have the square terms in both species variables, 2 bp and f Q 2, meaning the rate the population decreases as a result of overcrowding is proportional to the population size squared. In nature this overcrowding leads to limitations on resources such as space and food in the environment and therefore the rate the population decreases is related to the population squared as we have explained this term in the model. Analyzing the equilibrium point for this system is not as fruitful here as the only equilibrium point is the origin which gives us two zero eigenvalues in the Jacobian given here: 2bP + cq cp J(P,Q) = (3.21) eq 2 f Q + ep There is an interesting case in this model when b f = ce. When we have b f = ce we obtain an infinite number of equilibria of the form (n, b c n) which is a line of equilibria. For these equilibria the eigenvalues are λ 1 = 0,λ 2 = en bn and we still have the case 21

28 of one zero eigenvalue. Therefore we will look at the phase portraits of the system for various parameter values. Also note that the green dots are initial conditions for the phase plane trajectories: Figure 3.7: Phase portrait b, f = 0.5 and c,e = 1 for b f < ce In this plot above we can see that the product of the parameters related to the symbiotic relationship between species P and Q is greater than the overcrowding parameters product which leads to the behavior above. When the symbiotic relationship between the two species is greater than the hindering effects of overcrowding in both individual species both species will thrive and grow without bound. 22

29 Figure 3.8: Phase portrait b, f = 1 and c,e = 1 for b f = ce Here we have a special scenario when the product of the overcrowding parameters is equal to the product of the symbiotic parameters. Instead of both species thriving and growing without bound both species will head to an equilibrium point where both species coexist at a fixed level depending on the initial populations of both species. It seems to be the case depending on which species population is greater that species will decrease in size while the other increases to meet at an equilibrium somewhere on the line defined by the points (P, b P). c 23

30 Figure 3.9: Phase portrait b, f = 1.5 and c,e = 1 for b f > ce In the last scenario when the product of the overcrowding parameters is greater than the product of the symbiotic parameters we can see both species are heading towards extinction. So the overcrowding effect on one or potentially both species is more significant here than the symbiotic relationships between the two species causing both species populations to decrease towards the origin here. 3.4 Symbiotic Model with Overcrowding, Births, Deaths The final model combines both of the population growth inhibitors, birth and death terms and the overcrowding terms. Therefore we have the following model incorporating all of 24

31 these terms: dp/dt = P( bp + cq + a α) dq/dt = Q( f Q + ep + d β ) (3.22) Simply put this model is the basis for the symbiotic part of our model to be presented later and it is included here to see how this system behaves compared to the previous three symbiotic models and models to be presented later. Again this model is more complicated than the last and we need to analyze it further in order to determine the stability of the system. For simplicity we have A = a α and B = d β as before. B A Bc+A f Bb+ Ae This model give us four equilibrium points (0,0),(0, f ),( b, 0),( b f ce, b f ce ). The Jacobian of the system that will help us classify these equilibria is J(P,Q) = 2bP + cq + A cp. (3.23) eq 2 f Q + ep + B Here we have the eigenvalues for the origin equilibrium being λ 1 = A,λ 2 = B. For the cb f second equilibrium we have λ1 = + A,λ 2 = B and similarly for the third equilibrium we have eigenvalues λ 1 = A,λ 2 = ea + B. Lastly for the fourth equilibrium point the b Y eigenvalues are λ = ± Y 2 4XZ 1,2 2X for X = b f ce, Y = f ( Ae + Bb) + b( A f + Bc ), and Z = (Bc + A f )(Bb + Ae). We will analyze each equilibrium separately to see what kind of behavior each equilibrium can exhibit and also note A,B = 0 or else the model will be reduced to previous models we have already seen before and we feel eliminating this population dynamic from one species or both would be less revealing for the model. We first start with the origin which is associated with eigenvalues λ 1 = A,λ 2 = B. This means that depending on the signs of A,B we can have a stable, unstable, or saddle node. If we only want the origin to be the only equilibrium point in the first quadrant then we must have that A,B < 0 and b f ce and when this occurs then the origin is a stable node 25

32 which is confirmed in the following plot: Figure 3.10: Phase Portrait for A,B = 1,c,e = 1,b, f = 1.5 and (0,0) being the only equilibrium The second equilibrium is associated with eigenvalues λ 1 = cb f + A, λ 2 = B and for the equilibrium to exist in the first quadrant we need only B > 0. Because of this restriction we have one eigenvalue always being negative and the second eigenvalue λ = cb f + A can vary in sign or even become zero. Therefore this eigenvalue changes sign at A = cb cb cb f, so if A > f we will have a saddle node, if A < f then we have a stable cb node, and if A = f then our linearization analysis is inconclusive. Again we would like to see the behavior of the system when the second equilibrium is the only biologically meaningful equilibrium point other than the origin. To achieve this we must have B > 0, 26

33 A < 0, and for simplicity we will assume b f = ce which eliminates the fourth equilibrium from existing in the first quadrant; since the denominator will be zeroed the equilibrium cb will not exist. So the stability of this equilibrium changes stability at A = f as we will see in the following plots: Figure 3.11: Phase portrait B = 1,A = 1.5 for b, f,c,e = 1 with equilibria at (0,0) and (0,1) In this case, shown by figure 3.11, we have A < cb f which gives us two negative eigenvalues meaning the equilibrium point on the Q axis is stable and the Q species will survive while the P species will decrease towards zero. This outcome tells us that the P species death parameter, α, has a larger impact on P than the symbiotic effect by Q, c, on species P. Also the population dynamic B, positive here, is not large enough or the 27

34 overcrowding effect by f is more influential here to keep the Q species from growing in order to aid the P species in its symbiotic effect in order to keep species P from being eliminated. Figure 3.12: Phase portrait B = 1,A = 1 for b, f,c,e = 1 cb Here in figure 3.12 we have the case when A = f and we can see in the phase portrait we have infinite number of equilibria along some curve like in section 3.3. Since we have this equality one eigenvalue is now zero so our linearization analysis is not viable here. Biologically here we can see that the parameters A and cb f being equal gives us a scenario where depending on the initial populations we have both species heading towards a stable equilibrium in such a way that one species will reduce in size while the other grows towards this new equilibrium. 28

35 Figure 3.13: Phase portrait B = 1,A = 0.5 for b, f,c,e = 1 and equilibria are at (0,0) and (0,1) If A > cb the point becomes a saddle node and since we have a saddle node at f the origin we can see in the phase portrait both populations of species thrive and grow without bound. This is shown in the plot above in figure This time the P species death parameter does not hinder the population as much as before while the benefits of the symbiotic effect and the negative affects of the overcrowding in the Q populations allows both species to thrive and grow. For the third equilibrium the story is similar to the second equilibrium analysis. We need A > 0 in order for the equilibrium to exist in the first quadrant meaning one of ea the eigenvalues will always be negative. Again the other eigenvalue λ = b + B can 29

36 change sign depending on the values of the associated parameters. If B > b ea then we ea will have a saddle, if B < b then we have a stable node, and if B = b then our analysis is inconclusive and must pursue a different approach. Therefore we will use phase portraits to confirm that the behavior of the equilibrium points are similar to the previous equilibrium point: Figure 3.14: Phase portrait B = 1.5,A = 1 for b, f,c,e = 1 with equilibria at (0,0) and (1,0) ea Here in figure 3.14 we have ea b < B giving us a negative and positive eigenvalue so the equilibrium on the P axis is a saddle node and since the origin is a saddle as well then both populations will always thrive and grow without bound. Similarly like before the Q species death parameter along with the symbiotic parameter, e, and overcrowding term 30

37 b of species P is favorable here in that it allows for both species to overcome hindering effects in order to grow and thrive without bound. Figure 3.15: Phase portrait B = 1,A = 1 for b, f,c,e = 1 Like in the previous equilibrium point when we get the equality case of ea = B we b get an infinite number of equilibria along some curve where both species head towards a stable population depending on the initial populations shown above in figure Again like before hindering effects of the death parameters in both species, the overcrowding parameter in P, and the symbiotic term e allows both species to survive and head towards a stable population depending on initial populations. Also we have the behavior that in order to reach these stable population levels one species must grow while the the other is reduced. 31

38 Figure 3.16: Phase portrait B = 0.5,A = 1 for b, f,c,e = 1 Lastly in the case where ea > B we get two negative eigenvalues and the equilibb rium at the P axis is stable meaning the P population will head towards a stable level while the Q species is eliminated. We have here the Q species death parameter having a larger influence than the death parameter resulting in the Q species dying out. Another contribution to this is that the symbiotic term of the P species affecting Q, e, is not helpful enough and/or the P species overcrowding is great enough along with possibly the birth parameter of P not being great enough to allow the P species to grow in order to aid the Q species from dying out. Finally the fourth equilibrium has some rather complicated looking eigenvalues but we can look at some restrictions we can impose in order to determine the stability of the 32

39 equilibrium. First off we are only concerned about the stability of this equilibrium when it exists in the first quadrant and with this restriction we can prove that the radical in the eigenvalues Y 2 4XZ > 0 when the equilibrium exists in the first quadrant. ( ) Bc+A f Bb+Ae Proposition 1. If the equilibrium, exists in the first quadrant then this b f ce b f ce equilibrium can only be a stable or saddle node. (Bb+Ae) b f ce Proof. Suppose the equilibrium exists in the first quadrant meaning that (Bc+A f ), > b f ce 0. So the product of (Bc + A f )(Bb + Ae) > 0 as well since the product of the two components of the equilibrium must be positive. We have the quantity Y 2 4XZ which can be shown to be equal to Y 2 4XZ = (Ab f + Ae f + Bbc + Bb f ) 2 4(b f ce)(bc + A f )(Bb + Ae) (3.24) = (Ab f + Ae f + Bbc + Bb f ) 2 4b f (Bc + A f )(Bb + Ae) + 4ce(Bc + A f )(Bb + Ae) (3.25) = (Ab f Ae f Bb f + Bbc) 2 + 4ce(Bc + A f )(Bb + Ae). (3.26) This quantity is always positive as long as our equilibrium is in the first quadrant. Now that we have the radical of the eigenvalues of the fourth equilibrium always being positive when it exists in the first quadrant we can further analyze the stability of the equilibrium. We have eliminated the possibility of this equilibrium being a center or a spiral but depending on the parameter values of the model we can have either a stable, unstable, or saddle node for the fourth equilibrium. We can see from earlier that the quantity Y 2 4XZ determines if we get a saddle node or a node that is stable or unstable depending on if 4XZ is positive or negative. If 4XZ is negative then the quantity Y 2 4XZ > Y and we have a pair of real valued eigenvalues, one being positive and the other being negative. If 4XZ is positive then 33

40 Y 2 4XZ < Y and the eigenvalues will be a pair of positive or negative real eigenvalues depending on the sign of Y. We can see 4XZ only changes sign when X changes sign and if X > 0 then the numerators of the equilibrium point, also the components of Z, have to be positive as well. Therefore Y must be positive as well meaning the fourth equilibrium point can only be a saddle or stable node. There are several cases to explore with the fourth equilibrium point so we will start with the case where the fourth equilibrium is the only biologically relevant point along with the origin. In order to achieve this we must have A,B < 0 and b f < ce and in terms of stability we should have the origin being a stable node while the fourth equilibrium is a saddle. We can see this in the following phase portrait: Figure 3.17: Phase portrait with A,B = 1, b, f = 1, and c,e =

41 Biologically we have two scenarios happening depending on the initial populations where both species will die out or both end up thriving and growing without bound. Having the requirement that b f < ce means the product of the overcrowding parameters effect is less than the product of the symbiotic parameters effect so the symbiotic effect between both species can be seen as greater than the respective overcrowding effects in each individual species. Furthermore since both species have greater death parameters than birth parameters then depending on the initial populations we can see in figure 3.17 if one or both species has a large enough initial population the negative effects of overcrowding and deaths can be overtaken by the symbiotic relationship. If the initial population of one or both species is not great enough to overcome the negative effects of overcrowding and deaths then both species will die out since the symbiotic effect will not be as influential due to low populations. Now we will look at the case when we include one of the equilibria on either the P or Q axis as the third equilibrium in the first quadrant. Let us focus on the equilibrium on the Q, (0, B f ), axis case and we have seen before the behavior of this equilibrium and the P axis equilibrium, ( A b,0) are very similar in terms of stability. When we have the three equilibria existing in the first quadrant the origin is always a saddle while (0, B f ) and the fourth equilibrium vary on stability depending on the values of the parameters. In order for all three to exist in the first quadrant we must have A < 0 and B > 0, also b f = ce. We know that the equilibrium on the Q axis changes stability depending on the relation of A f + Bc being negative or positive or zero and also notice that this quantity is also the numerator for the P component of the fourth equilibrium. Let us look at the case when A f < Bc where before we would have the second equilibrium being a stable node. In this case we would require that X < 0 and Bb + Ae < 0. Therefore with the restriction of X < 0 then the fourth equilibrium will have one positive and one negative eigenvalues 35

42 meaning it is a saddle. We can see this behavior in the following phase portrait: Figure 3.18: Phase portrait for A = 1.5,B = 1 and b,c, f = 1 and e = 1.5 and equilibria at (0,0), (0,1), and (1,2.5) We can see for certain initial populations the model has either the Q species heading towards a stable level while the P species dies out or both species end up thriving and growing without bound. This is a result of the symbiotic parameter of P affecting Q, e, being large enough such that a large enough P population gives us the two populations thriving and growing without bound. Due to the higher death rate in P than Q we also have the P population dying out if the initial population of P is low enough since the symbiotic relationship of Q affecting P, c, is not large enough to curtail the larger negative effects of the P species death parameter and overcrowding. 36

43 We move onto the case where A f > Bc where the second equilibrium is a saddle node and for the fourth equilibrium we have X > 0 meaning it will be a stable node since A f + Bc > 0 requiring Ae + Bb > 0. We can see this here in the phase portrait: Figure 3.19: Phase portrait for A = 0.5,B = 1 and c,e = 1 and b, f = 1.5 with equilibria at (0,0), (0,2/3), (0.2,0.8) Looking at the phase portrait in figure 3.19 since the origin and second equilibrium are saddle the trajectories are heading towards the fourth equilibrium since it is stable. This means that for all initial populations we will have both populations heading towards a stable level where they can coexist. Biologically we have a balance of the overcrowding parameters negative effects, the symbiotic parameters, and the death rate of P not being as influential here to have for all initial populations to reach a stable level where both 37

44 populations survive. Now we are left with the case when A f + Bc = 0 but if we have this case then the P component of the fourth equilibrium is zeroed out and the equilibrium point lies on the Q axis as well alongside the second equilibrium. In fact we can show when A f + Bc = 0 the fourth equilibrium is equal to the second equilibrium and these two equilibria actually collide as the sign of the quantity A f + Bc changes. Therefore the behavior here is reduced to the case we have seen before when we had the origin and second equilibria in the first quadrant. If we look at the case if we had the origin, fourth equilibrium and the third equilibrium on the P axis we get similar results as before with the second equilibrium on the Q axis. This time we will require that A > 0 and B < 0 with again b f = ce and we will look at the stability as Ae + Bb changes sign. Like before we will first look at Bb < Ae which will give us a stable node for the third equilibrium and we need Ae + Bb < 0 and X < 0 so the fourth equilibrium will be a saddle node. 38

45 Figure 3.20: Phase portrait for A = 1,B = 1.5 and b, f,e = 1 and c = 1.5 with equilibria at (0,0), (2.5,1), and (1,0) Here we can see that for certain initial populations we either have the P species surviving at a stable level while the Q population dies out or both species end up growing without bound. Just like before the symbiotic term c is large enough so that if the Q population and P population is large enough then both populations grow without bound. If both populations are too small then the symbiotic parameter c does not help enough and the death parameter of Q is large enough for Q to die out while P survives. Next the case for when Bb > Ae requires that Ae + Bb > 0 and X > 0 so the fourth equilibrium will be stable while the origin and third equilibria are saddle nodes. 39

46 Figure 3.21: Phase portrait for A = 1,B = 0.5 and b, f = 1.5 and c,e = 1 with equilibria at (0,0), (1,0), and (0.8,0.2) In the above phase portrait all trajectories head towards the fourth equilibrium meaning biologically the two populations will reach a stable population size where both can coexist. Again we have a balance between the overcrowding effects of both species and the death parameter of species Q in order to insure that both species are able to survive. Like earlier we are left with the scenario when Ae + Bb = 0 and like before we have the fourth equilibrium point collides with the third equilibrium point on the P axis. Therefore we get the same behavior as before when we had only the origin and second equilibrium point as the only equilibria in the first quadrant. The last case we want to investigate is the case when all four equilibria are present in 40

47 the first quadrant. For this we require that A,B > 0 for which then we must have X > 0 so the origin, second, and third equilibria are all saddle nodes while the fourth equilibrium is a stable node shown below: Figure 3.22: Phase portrait for A,B = 1 and b, f = 1.5 and c,e = 1 with equilibria at (0,0), (0,2/3), (2/3,0), and (2,2) Therefore all initial population sizes will tend towards the fourth equilibrium and both species will coexist. Since both populations have a greater birth rate than death rate then the only parameters here hindering the growth of both species are the overcrowding parameters. Here we have achieved a balance where no matter the initial population size there is sufficient growth from both species given the beneficial symbiotic parameters and birth parameters while not allowing one or both species to grow without bound by 41

48 having great enough overcrowding in both species. 42

49 Chapter 4 Symbiotic Model with Disease We have introduced so far a basic SIR model and a variety of symbiotic species models and from these models we are intrigued at how two populations of different species interact with one another while one of these species is being affected by a disease. Our general model in an attempt to model this interaction is similar to our previous symbiotic model with all the inhibiting terms to the population growth of both species. Also included in the model are two more differential equations representing the population groups of the infected and recovered populations of the infected species. While these two equations model the disease progressing in one of the species populations we also have interactions with the symbiotic species population not affected by the disease. There are restrictions to this model including both species inhabiting one geographical area and there is no immigration or emigration to other areas. The population of species being infected with the disease can only infect the susceptible population of its own species and not cross over to the other species in the area. As with the SIR model our model will have three classes of the disease stages in one population of species, susceptible, infected, and recovered. For the following table we have all the variables used in our models and note all variables are 43

50 non-negative unless specified otherwise. Variable P Q I R a b c d e f k h δ l g r α β γ ε Description Species P Species Q Infected species Q Recovered species Q Birth term for species P Overcrowding parameter for P Symbiosis parameter for Q,R affecting P Birth parameter for Q,R Symbiosis parameter for P affecting Q,R Overcrowding parameter for Q,I,R Interaction parameter of P affecting I Birth parameter of I into Q Contact rate between Q and I Birth rate of I into I Interaction parameter of I affecting P Recovery parameter for I Death parameter for P Death parameter for Q Death parameter for I Death parameter for R 44

51 4.1 Basic Symbiotic Model With Disease Here we have our proposed model in its simplest form with no terms attempting to reduce all populations of species: dp/dt = P(c (Q + R) + ki) dq/dt = Q(eP δ I) di/dt = I (δ Q + gp r) (4.1) dr/dt = R(eP) + ri As we can see from these equations there are similarities to the symbiotic and SIR models we have covered so far. Like in the symbiotic model in section 3.1 we have an interaction term in Pc(Q + R) that represents the symbiotic relationship between species P and Q. However we have an extra R term and since Q and R are not infected and are of the same species we will assume these two populations experience the same symbiotic relationship with species P. Furthermore the kpi term represents the symbiotic relationship the P species has on the infected population of Q, I. Note here k does not have to be non-negative, in other words the symbiotic constant k can represent a beneficial relationship between P and I or a harmful relationship on P. In the second equation we have the same eqp term seen before in the symbiotic model and now we introduce from the SIR model the δ QI representing the susceptible population of Q being infected and moving in to the infected population. Next the third equation has again terms familiar to us from the SIR model, δ IQ and ri, representing the introduction of new infected individuals from the susceptible class and the recovery of infected individuals into the recovered population respectively. Again we have a new term in gip representing the interaction of species P with the infected population. Like we had for k earlier, g can be negative or positive representing a beneficial symbiotic relationship or a harmful re 45

52 lationship for I. Lastly the final equation has the same recovery term in the SIR model, ri, representing the rate infected individuals recover from disease and attain permanent immunity. Additionally we have this epr term representing the symbiotic relationship between P and R. We are assuming here the constant e can be different from what we had in the first equation since e gives the rate of how P affects R in this relationship. 4.2 Symbiotic Model with Disease and Birth and Death Rates From the model in section 4.1 we now add births and deaths to the model seen before in the symbiotic model in section 3.2 but we have some notable differences in the different constants for the rates of births and rates for the different disease classes: dp/dt = P(a + c(q + R) + ki α) dq/dt = Q(d + ep δ I β ) + hi + dr di/dt = I (l + δ Q + gp γ r) (4.2) dr/dt = R(eP ε) + ri We can see in this model the same birth constants from section 3.2 for P and Q being a,α and d,β respectively. Their roles here remain the same as in section 3.2 giving us the births and deaths of P and Q being proportional to the population sizes. Now some noticeable additions to the Q differential equation include hi and dr. We will allow the possibility of infected individuals to be able to reproduce and these offspring are born as susceptible individuals or infected as we shall later see. The hi terms allows this and the birth rate of susceptible individuals from infected parents is proportional to the infected population I. Of course we can remove the possibility of infected individuals from hav 46