Dynamics of SI models with both horizontal and vertical transmissions as well as Allee effects

Mathematical Biosciences 248 (2014) 97–116 Contents lists available at ScienceDirect Mathematical Biosciences journal homepage: www.elsevier.com/loc...

Download PDF

2MB Sizes 2 Downloads 73 Views

Report

PDF Reader
Full Text

Mathematical Biosciences 248 (2014) 97–116

Contents lists available at ScienceDirect

Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs

Dynamics of SI models with both horizontal and vertical transmissions as well as Allee effects q Yun Kang a,⇑, Carlos Castillo-Chavez b,c,d,e,f a

Science and Mathematics Faculty, School of Letters and Sciences, Arizona State University, Mesa, AZ 85212, USA Mathematical, Computational and Modeling Sciences Center, Arizona State University, Tempe, AZ 85287-1904, USA c School of Human Evolution and Social Changes, Santa Fe Institute, Santa Fe, NM 87501, USA d School of Sustainability, Santa Fe Institute, Santa Fe, NM 87501, USA e Cornell University, Biological Statistics and Computational Biology, Ithaca, NY 14853-2601, USA f Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, 77 MASS Ave. 33-404, Cambridge, MA 02139-4307, USA b

a r t i c l e

i n f o

Article history: Received 5 September 2013 Received in revised form 11 December 2013 Accepted 13 December 2013 Available online 31 December 2013 Keywords: Allee effects Horizontal transmission Vertical transmission Disease-driven extinction Disease-free dynamics Diffusive instability

a b s t r a c t A general SI (Susceptible-Infected) epidemic system of host–parasite interactions operating under Allee effects, horizontal and/or vertical transmission, and where infected individuals experience pathogeninduced reductions in reproductive ability, is introduced. The initial focus of this study is on the analyses of the dynamics of density-dependent and frequency-dependent effects on SI models (SI-DD and SI-FD). The analyses identify conditions involving horizontal and vertical transmitted reproductive numbers, namely those used to characterize and contrast SI-FD and SI-DD dynamics. Conditions that lead to disease-driven extinction, or disease-free dynamics, or susceptible-free dynamics, or endemic disease patterns are identiﬁed. The SI-DD system supports richer dynamics including limit cycles while the SI-FD model only supports equilibrium dynamics. SI models under ‘‘small’’ horizontal transmission rates may result in disease-free dynamics. SI models under with and inefﬁcient reproductive infectious class may lead to disease-driven extinction scenarios. The SI-DD model supports stable periodic solutions that emerge from an unstable equilibrium provided that either the Allee threshold and/or the disease transmission rate is large; or when the disease has limited inﬂuence on the infectives growth rate; and/or when disease-induced mortality is low. Host-parasite systems where diffusion or migration of local populations manage to destabilize them are examples of what is known as diffusive instability. The exploration of SI-dynamics in the presence of dispersal brings up the question of whether or not diffusive instability is a possible outcome. Here, we brieﬂy look at such possibility within two-patch coupled SIDD and SI-FD systems. It is shown that relative high levels of asymmetry, two modes of transmission, frequency dependence, and Allee effects are capable of supporting diffusive instability. Published by Elsevier Inc.

1. Introduction Parasitism contributes to the selection of future generations of hosts through their impact on factors that lead to reductions in ﬁtness [31] and as a result, wildlife managers must account for emerging and/or re-emerging diseases. Competition for space and resources (ﬁnding mates or food) also impact the reproductive q This research of CCC is partially supported by the grant number 1R01GM100471-01 from the National Institute of General Medical Sciences (NIGMS) at the National Institutes of Health. The research of Y.K. is partially supported by NSF DMS (1313312), Simons Collaboration Grants for Mathematicians (208902) and the research scholarship from School of Letters and Sciences. The authors would also like to thank Simon Levin for suggesting that we connect with the concepts of diffusive instability.

⇑ Corresponding author. Tel.: +1 4807275004. E-mail addresses: [email protected] (Y. Kang), [email protected], [email protected] (C. Castillo-Chavez). 0025-5564/$ - see front matter Published by Elsevier Inc. http://dx.doi.org/10.1016/j.mbs.2013.12.006

ability and likelihood of survival of individuals, particularly those housing pathogens or parasites. Hosts’ dynamics (survival in particular) often depends on the ability of a population to maintain a critical mass [36]. The impact of heterogenous transmission factors including multiple transmission modes by altering a population’s dynamics may lessen the plausibility of conservation goals or the economic viability of selected management policies [44]. Hence, it is not surprising that the pressure which parasites or pathogens place on their hosts and the relation of such interactions to community and/or ecosystem structure has been the subject of continuous empirical and theoretical studies. Some of the theoretical consequences associated to host-pathogen dynamics with factors like: (i) multiple modes of disease transmission; (ii) host population density; and (iii) the presence or absence of critical host population thresholds, are addressed in this manuscript. Modes of disease transmission, like horizontal and vertical, differentially facilitate the colonization of host populations by

98

Y. Kang, C. Castillo-Chavez / Mathematical Biosciences 248 (2014) 97–116

bacteria, fungi, or viruses. Colonization (horizontal transmission) is sometimes seen as the result of close interactions (contacts) between disease-free host and infected individuals. A contact process that implicitly assumes the sharing of a common, local habitat. The passage of a disease-causing agent from a mother to offspring during the ‘‘ birth’’ process is also sometimes possible (vertical transmission). Feline leukemia (FeLV) and feline immunodeﬁciency (FIV) viruses are transmitted horizontally and vertically. Leishmaniasis, a disease caused by the protozoan parasite Leishmania infantum, is transmitted horizontally and vertically. Domesticated dog populations are presumed to be a reservoir for Leishmania infantum; a reservoir maintained by the differential contributions of multiple modes of transmission [46]. The deadly septicaemia, which manages to kill 80% of septicaemia-infected birds, gets lodged in the ovary of surviving birds; passed later to the birds eggs (vertical transmission); spreading horizontally within the hatcher and brooder. Traditionally, density-dependent transmission (DDT) and frequency-dependent transmission (FDT) are two extreme forms of disease transmission. DDT refers to parasite transmission in which the rate of contact between susceptible hosts and the source of new infections increases with host density while FDT refers to the rate of contact between susceptible hosts and the source of new infections is independent of host density. Teasing out the roles of density- and frequency-dependent transmission (DDT versus FDT) on the dynamics of host-parasite systems is carried out for theoretical and policy reasons. FDT is the result of densityindependent contact rates between susceptible and infected individuals. DDT assumes that infection risks increase with host density. Density-dependent transmission (DDT) may require a minimal number of available susceptible hosts, that is, a threshold density, for transmission to occur. Density-dependent parasitic disease transmission plays a role in regulating host population size [3] while frequency-dependent parasitic transmission does not require host density thresholds or regulatory host population constraints on the birth or death rates to ‘‘work’’. In population biology we often lack absolutes. And so, vectorand sexually-transmitted diseases have been seen to thrive in frequency-dependent transmission settings while density-dependent infections that lead to pathogens being shed by infected hosts into common environments may sometimes need a critical mass of susceptible individuals to thrive [4,5]. Pathogens can be spread via ‘‘direct’’ contacts (kissing can spread herpes viruses), aerosol (sneezing can spread inﬂuenza viruses), or via indirect contacts (ingesting water contaminated with fecal material can cause result in cholera infections), or through vectors (ticks and mosquitoes often spread viruses and bacteria to their hosts), or via some combination of direct and indirect modes, sometimes mediated by a vector. Empirical work on mice, voles, lady bird beetles, frogs, and plants has shown that pathogen transmission often involve DD and FD transmission modes, with one predominant mode [31]. The negative impact of deliberate releases of pathogens via aerosol or in water systems tends to increase with host density. On the other hand, sexually transmitted pathogens seem to thrive equally well or bad in small or large population settings while some vector-borne diseases have been shown to support frequency-dependent transmission patterns [4,5,21]. Antonovics and Alexander [5] manipulated the density and frequency of infected hosts Silene latifolia and in the process they found out that deposition of the anther smut fungus Microbotryum violaceum by pollinating insects managed to increase with the frequency of infection. A pathogen may or may not be deleterious enough to regulate the dynamics of host populations and so it is not surprising that the impact of pathogens on hosts is tied in to virulence. Pathogen’s levels of virulence differentially impacts host’s ﬁtness. Often, increases in virulence result in a reduced probability survival or a

diminished ability of a host to reproduce successfully, or both [3,31,27]. Pathogens whose transmission successes increases with host density seem to have managed to select for variants capable of regulating a host population. [19] studied a host-pathogen system where a detailed account of virus titer on infected hosts could be estimated. Their study focused on studying the ability of the Myxoma virus to control an exploding rabbit populations over a long window in time. Empirical evidence from systems involving conjunctivitis in house ﬁnches or parasitic nematodes in red grouse and feral Soay sheep provide an example of a system where disease regulates population size [24,30,29]. Pathogen infections are contributors to the decline or the extinction of some species [19,14,49,52]. The deleterious role of chytridiomycosis in amphibians, chestnut blight in American chestnuts, avian malaria in Hawaiian birds, devil facial tumour disease in Tasmanian devils, or sudden oak death in Californian trees provide classical examples of the role of disease in regulating a population. Theory suggests that density-dependent specialist pathogens (i.e., those infecting a single host) alone rarely drive their hosts’ extinction but can lead to extinction of the pathogen while frequency-dependent transmission may be capable of supporting signiﬁcant decreases, including the potential extinction of host and parasite populations in the presence of moderately lethal pathogens [10,21,39]. The impact of disease outbreaks can be devastating and their dynamics must be particularly monitored within populations near extinction; that is, those with population levels near established Allee effects thresholds [1,12,35,50,51]. The relevance of threshold effects has been identiﬁed within a wide array of taxa [12,40]. Populations under Allee effects or facing extinction or both must be effectively managed [16,27]. The fragility of these populations means that limiting the transmission of highly deleterious diseases is critical [11,27]. Recurrent infectious disease outbreaks tend to enhance the deleterious role of Allee effects within diseases capable of inducing reductions in host ﬁtness [25,11,54,27,52,28,22,37,38]. The results of this manuscript seem to be in sync with the overall conclusions reached the study of predator–prey systems (e.g., [13,20,18,33,8,34,36]). Parasites and hosts co-evolve in response to environmental clues and/or selective pressures [39]. Mammals, birds, ﬁsh, and insects generate mobility patterns as they track resources and as it is well known movement and/or dispersal can impact disease dynamics [2]. In short, mobility has been a key player in the evolution of host-parasite systems. Studies that in addition to disease and mobility (dispersal) also include the impact of Allee effects are not well understood [45,26,37,38]. Hilker et al. [26] used a reaction–diffusion SI model within a frequency-dependent transmission framework in their explorations of the impact of disease and mobility on the spatiotemporal patterns of disease transmission. SI models that incorporate disease-reduced fertility have been explored by a number of researchers (see [17,7]). In [38] a two-patch SI model with density-dependent transmission is used to show that the differential movement of susceptible and infected individuals can enhance or suppress the spread of a disease. A SI model that incorporates a horizontally and vertically transmitted disease; infectives giving birth to infectives; susceptibles giving birth to susceptibles; Allee effects within the net reproduction term; disease-induced death rate; and disease reduced reproductive ability, is used in this manuscript to begin to address questions that include: What is the role of multiple modes of transmission? Will density-dependent and frequencydependent vertical transmission affect host-parasite dynamics differentially? Under what conditions would Allee effects alter disease-free dynamics or facilitate disease-driven extinction? Would Allee thresholds on reproductive ﬁtness become altered (reduced) by disease? What is the role of DDT or FDT in support of diffusive instability?

99

Y. Kang, C. Castillo-Chavez / Mathematical Biosciences 248 (2014) 97–116

In Section 2, a general SI (Susceptible-Infected) model with Allee effects built in the reproduction that incorporates horizontal and vertical transmission modes, is formulated. The basic dynamic properties of the model are characterized, in particular, sufﬁcient conditions in support of disease-free and persistence of species results are identiﬁed (Theorem 2.1 and its Corollary 2.2). In Section 3, the dynamics of SI models under frequencydependent or density-dependent horizontal transmission are contrasted. Boundary dynamics are characterized (Proposition 3.1) and and sufﬁcient conditions for disease- and susceptiblepopulation persistence are provided in Theorem 3.2. A classiﬁcation of interior dynamics comes in Theorem 3.3. In Section 4, disease-driven extinction, disease-free or susceptible-free dynamics, and endemic persistent dynamics are characterized. The nature of bifurcations supported by SI models is studied with the aid of the reproduction numbers linked to horizontal and vertical transmission modes. In Section 5, sufﬁcient conditions leading to diffusive instability (Theorem 5.1) are identiﬁed. The nature of mechanisms potentially capable of supporting diffusive instability in SI-models and prey–predator models, is brieﬂy discussed. The implications of the results in this manuscript are discussed in the Conclusion.

Thus, the population model described by Eq. (2.2) converges to 0 if initial conditions are below K or converges to its carrying capacity K þ whenever the initial conditions are above K . Note: The formulation of this SI model (2.1) is similar in approach to that found in [6,11,12,27,52,36], particularly in the way we model the effects of Allee effects and disease. Our models allow for infectives to give birth to infectives with the caveat that their reproductive ability may be reduced; a feat that being captured with the parameter q. The need for biological consistency (well posed model) is addressed with the help of the state-space naturally associated with Model (2.1), namely, X ¼ fðS; IÞ 2 R2þ g with its interior deﬁned as X ¼ fðS; IÞ 2 R2þ : S > 0 and I > 0g. The state space when /ðNÞ ¼ b is X ¼ fðS; IÞ 2 R2þ : S þ I > 0g. The assumption that f ðNÞ is differentiable leads to the following theorem for Model (2.1): Theorem 2.1 (Basic dynamical features of (2.1)). Assume that r > 0; d > 0; q 2 ½0; 1 and both f ; / are continuous in X with f satisfying Condition (2.3), then the system (2.1) is positively invariant and bounded in X with the following property

lim sup NðtÞ ¼ lim sup ðSðtÞ þ IðtÞÞ 6 K þ : t!1

2. An SI model with Allee effects and vertical transmission The model outlined in this section deals with a population facing a disease that can be effectively captured within a SI framework under assumptions that include the possibility of multiple modes of transmission, that is, horizontal and vertical. It is therefore assumed that infected individuals can give birth to infected hosts; that Allee effects alter the net reproduction term (possibly due to mating limitations or predator saturation); the presence of increases in mortality due to disease-induced deaths; and the fact that infected individuals may experience reductions in reproductive ability. We let S and I denote the susceptible and infective populations, respectively, with N ¼ S þ I denoting the total host population. The approach followed from [11] leads to the following set of nonlinear equations after the incorporation of the above assumptions:

dS ¼ dt

rSf ðNÞ |ﬄﬄﬄ{zﬄﬄﬄ}

Growth with Allee effects

dI I ¼ /ðNÞ S þ dt N

I /ðNÞ S N |ﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄ}

qrIf ðNÞ |ﬄﬄﬄﬄ{zﬄﬄﬄﬄ}

ð2:1Þ dI |{z}

1. If Nð0Þ 2 ð0; K Þ, then limt!1 NðtÞ ¼ 0. 2. If there exists a positive number a > K such that r qf ðaÞ > d, then

lim inf NðtÞ ¼ lim inf ðSðtÞ þ IðtÞÞ P a for any Nð0Þ > a: t!1

3. If

t!1

þ maxK 6N6K þ fK /ðNÞ N

þ r qf ðNÞg < d, then lim sup IðtÞ ¼ 0. t!1

Proof. Since both S ¼ 0 and I ¼ 0 are invariant manifold for the system (2.1), then according to the continuity of the system, we can easily show that (2.1) is positively invariant in X. In addition, the system (2.1) gives the following equation:

dN ¼ rðS þ qIÞf ðNÞ dI dt

ð2:4Þ

1. If K 6 N 6 K þ , we have the following inequalities

dN dt ¼ rðS þ qIÞf ðNÞ dI 6 rNf ðNÞ

r qNf ðNÞ dN ¼ Nðr qf ðNÞ dÞ 6

Additional death due to infections

where r > 0; q 2 ½0; 1; d > 0 are respectively the intrinsic growth rate, the reduction of growth rate due to disease, and the excess death rate from the disease. The horizontal transmission term /ðNÞ includes density-dependent transmission, /ðNÞ ¼ bN (the law of mass action) or frequency-dependent transmission, /ðNÞ ¼ b (proportionate mixing or standard incidence). In the absence of disease, the SI Model (2.1) reduces to the following single species growth model:

ð2:2Þ

where the per capita growth function rf ðNÞ is subject to strong Allee effects, i.e., there exists an Allee threshold K and a carrying capacity K þ such that

f ðNÞ < 0 if 0 < N < K or N > K þ ;

which indicates that limt!1 NðtÞ ¼ 0 if Nð0Þ < K since f ðNÞ satisﬁes the condition (2.3), i.e., f ðNÞ > 0 when K 6 N 6 K þ . Similarly, if N < K or N > K þ , then we have the following inequalities

dN dt ¼ rðS þ qIÞf ðNÞ dI 6 r qNf ðNÞ

rNf ðNÞ dN ¼ Nðrf ðNÞ dÞ 6

f ðK Þ ¼ f ðK þ Þ ¼ 0:

ð2:3Þ

ð2:6Þ

which indicates that lim supt!1 NðtÞ 6 K if Nð0Þ > K since f ðNÞ satisﬁes the condition (2.3), i.e., f ðNÞ < 0 when N < K or N > K þ . The discussion above also implies that lim supt!1 NðtÞ 6 K þ . 2. If there exists a positive number a > K such that r qf ðaÞ > d, then according to (2.5), we have

dN j P r qf ðNÞ djN¼a > 0: Ndt N¼a

f ðNÞ > 0 if K < N

ð2:5Þ

þ

dN ¼ rNf ðNÞ dt

< Kþ;

In addition, we have the following:

where N ¼ S þ I. We have the following three cases

Horizontal transmission

Vertical transmission

t!1

Therefore, lim inf t!1 NðtÞ P a for any Nð0Þ > a.

100

Y. Kang, C. Castillo-Chavez / Mathematical Biosciences 248 (2014) 97–116 þ

3. If maxK
dI I S/ðNÞ ¼ /ðNÞ S þ qrIf ðNÞ dI ¼ I þ rqf ðNÞ d dt N N þ K /ðNÞ þ rqf ðNÞ d < 0
In addition, we have the following: 1. If Nð0Þ 2 ð0; hÞ, then lim NðtÞ ¼ 0. t!1 2. If there exists a positive number a > h such that qf ðaÞ > d, then

lim inf NðtÞ P a for any Nð0Þ > a: t!1

2

3. If b þ qð1hÞ < d, we have lim sup IðtÞ ¼ 0. 4 t!1

which indicates that lim supt!1 IðtÞ ¼ 0. h Notes: Some of the consequences that follow from Theorem 2.1 are:

Proof. The application of Theorem 2.1 is direct by letting r ¼ 1; f ðNÞ ¼ ðN hÞð1 NÞ and /ðNÞ ¼ b or bN. We only show the item 3. Since we have

1. The size of the initial population is extremely important for persistence regardless of the disease due to Allee effects. 2. The total population will not be above its carrying capacity K þ in the long run. 3. Species persistence requires that the initial population a is above the Allee threshold K , and excess deaths rdq are small enough, smaller than the per capita growth function evaluated at the total population a, i.e., f ðaÞ > rdq. 4. In the absence of vertical transmission, disease free dynamics requires that

N ¼ S þ I P S and

max þ

K 6N6K

/ðNÞ d < þ: N K

In the presence of vertical transmission, disease free dynamics requires that

/ðNÞ r qf ðNÞ d < þ: max þ N K 6N6K þ Kþ K

lim sup NðtÞ 6 1; t!1

therefore, the system (2.7), (2.8) has the following property

S/ðNÞ bS þ rqf ðNÞ ¼ þ qðN hÞð1 hÞ 6 b þ qðN hÞð1 hÞ N N qð1 hÞ2 6bþ : 4 For the system (2.9), (2.10), we also have

S/ðNÞ þ rqf ðNÞ ¼ bS þ qðN hÞð1 hÞ 6 b þ qðN hÞð1 hÞ N qð1 hÞ2 : 6bþ 4 2

Therefore, if b þ qð1hÞ < d, we have lim sup IðtÞ ¼ 0 for the system 4 t!1 (2.7), (2.8) and the system (2.9), (2.10). h

leads to the following two SI models with both horizontal and vertical transmission and Allee effects:

Notes: The traditional basic reproduction number for SI- Allee effects free- and vertical transmission free-models, namely R0 ¼ bd is naturally no longer relevant. The item 3 of Corollary 2.2 indicates that the additional condition is needed for disease-free dynamics. Since our systems (2.7), (2.8) and (2.9), (2.10) involve both horizontal and vertical disease transmissions, their dynamics can be governed by the vertical transmission reproduction number, i.e., Rv0 ¼ qd , and the horizontal transmission reproduction number, i.e., Rh0 ¼ bd. The remainder of this article focuses on the dynamics of the systems (2.7), (2.8) and (2.9), (2.10).

dS bSI ¼ SðN hÞð1 NÞ dt N

ð2:7Þ

3. Mathematical analysis

dI bSI ¼ dI þ qIðN hÞð1 NÞ dt N

ð2:8Þ

For convenience, we can consider that f ðNÞ has a generic form of ðN K ÞðK þ NÞ and /ðNÞ ¼ b (i.e., frequency-dependent) or bN (i.e., the law of mass action) then scaling and setting

S!

S I K q b d ; b! ; d! þ ; I ! þ ; K ! þ ; t ! rt; q ! r r r K K K

and

Notice that the system (2.7), (2.8) is not deﬁned at ðS; IÞ ¼ ð0; 0Þ but from Corollary 2.2, we know that

limðSðtÞ; IðtÞÞ ¼ ð0; 0Þ whenever Sð0Þ þ Ið0Þ < h:

t!1

dS ¼ SðN hÞð1 NÞ bSI dt dI ¼ bSI dI þ qIðN hÞð1 NÞ dt

ð2:9Þ ð2:10Þ

where f ðNÞ ¼ ðN hÞð1 NÞ is the per capita growth in the absence of disease; the parameter h ¼ KK þ represents the Allee threshold; q 2 ½0; 1 represents the reduce reproductive ability due to the disease; b represents the disease transmission rate while d denotes the additional death rate coming from infections. The direct application of Theorem 2.1 to the systems (2.7), (2.8) and (2.9), (2.10) gives the following corollary:

Thus, we artiﬁcially deﬁne ð0; 0Þ as the extinction equilibrium. Hence, the systems (2.7), (2.8) and (2.9), (2.10) have the same boundary dynamics since both of them can be reduced to the system given by

dS ¼ SðS hÞð1 SÞ if I ¼ 0 dt and

dI ¼ qIðI hÞð1 IÞ dI dt

if S ¼ 0:

Therefore, the systems (2.7), (2.8) and (2.9), (2.10) have the following three boundary equilibria

Corollary 2.2 (Basic dynamic features of (2.7), (2.8) and (2.9), (2.10)). The systems (2.7), (2.8) and (2.9), (2.10) are positively invariant and bounded in their state space X with the following property

E0;0 ¼ ð0; 0Þ;

lim sup NðtÞ 6 1:

(2.9), (2.10) support the following additional boundary equilibria on the I-axis:

t!1

Eh;0 ¼ ðh; 0Þ; v

q

If, in addition, R0 ¼ d >

4 ð1hÞ2

E1;0 ¼ ð1; 0Þ: holds, then systems (2.7), (2.8) and

101

Y. Kang, C. Castillo-Chavez / Mathematical Biosciences 248 (2014) 97–116

E0;h

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 0 ð1 hÞ2 4=Rv0 1 þ h A; ¼ @0; 2 2 0

E0;1

1þh ¼ @0; þ 2

Thus, we have

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 hÞ2 4=Rv0 1 þ h 1þh h < h2 ¼ < < K2 2 2 2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 hÞ2 4=Rv0 1þh ¼ þ < 1: 2 2

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 ð1 hÞ2 4=Rv0 A: 2

We have arrived at the following proposition regarding the boundary equilibria of the systems (2.7), (2.8) and (2.9), (2.10): Proposition 3.1 (Boundary equilibria of the systems (2.7), (2.8) and (2.9), (2.10)). The systems (2.7), (2.8) and (2.9), (2.10) always have boundary equilibria E0;0 ¼ ð0; 0Þ; Eh;0 ¼ ðh; 0Þ; E1;0 ¼ ð1; 0Þ. If in addition Condition Rv0 ¼ qd > 4 2 holds then both systems will ð1hÞ support two additional boundary equilibria E0;h ¼ ð0; h2 Þ and E0;1 ¼ ð0; K 2 Þ where

1þh h < h2 ¼ 2

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 hÞ2 4=Rv0 2

1þh < K2 ¼ þ 2

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 hÞ2 4=Rv0 2

Proof. If S ¼ 0, the systems (2.7), (2.8) and (2.9), (2.10) are reduced to the following equation:

Therefore, if ð1 hÞ2 > 4d=q ¼ 4=Rv0 , we have

E0;h

E0;1

0

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 ð1 hÞ2 4=Rv0 A; 2

0

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 ð1 hÞ2 4=Rv0 A: 2

1þh ¼ ð0; h2 Þ ¼ @0; 2

1þh ¼ ð0; K 2 Þ ¼ @0; þ 2

Notice that

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ , therefore, ð1 hÞ2 4=Rv0 =2 < 1h 2

1þh h2 ¼ 2

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 hÞ2 4=Rv0 2

>

K2 ¼

1þh þ 2

2

<

Theorem 3.2 (Persistence of disease or susceptibles). Assume that Rv0 > 4 2 and the initial condition Nð0Þ 2 ðh2 ; K 2 Þ then the following ð1hÞ statements follow: 1. For the systems (2.7), (2.8) and (2.9), (2.10), a sufﬁcient condition for the persistence of disease is Rh0 > 1. 2. For the system (2.7), (2.8), a sufﬁcient condition for the persistence of susceptibles is Rh0 < q1 while for the system (2.9), (2.10) is Rh0 < q1K 2 . The persistence of disease (or susceptibles) means that there exists a positive number such that

lim inf IðtÞ½or SðtÞ P for any Nð0Þ 2 ðh2 ; K 2 Þ: t!1

Proof. Deﬁne hðNÞ ¼ qðN hÞð1 NÞ d ¼ Rv0 ðN hÞð1 NÞ 1=Rv0 . Notice that h2 and K 2 are roots of the equation ðN hÞð1 NÞ ¼ 1=Rv0 if 2 the condition ð1hÞ > 1=Rv0 holds, thus we have 4

hðNÞ ¼ qðN hÞð1 NÞ d ¼

q d

ðN h2 ÞðK 2 NÞ:

This indicates that hðaÞ > 0 for any a 2 ðh2 ; K 2 Þ. Since hðNÞ ¼ qf ðNÞ d then Theorem 2.1 and (its) Corollary 2.2 imply that

1þh 1h >h 2 2

lim inf NðtÞ ¼ lim inf ðSðtÞ þ IðtÞÞ P a for any Nð0Þ ¼ a 2 ðh2 ; K 2 Þ:

1þh 1h ¼ 1: 2 2

The use of Theorem 2.1 and (its) Corollary 2.2 again allows to conclude that the systems (2.7), (2.8) and (2.9), (2.10) attract to the compact set 0 6 N 6 1 and are positively invariant within a 6 N 6 1 for any a 2 ðh2 ; K 2 Þ.

and

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 hÞ2 4=Rv0

Notes: The results in Proposition 3.1 are used to determine the global dynamics in the absence of interior equilibrium (see the proof of Theorem 3.3 for details).

< 1:

The nature of the stability of these boundary equilibria is listed in Table 3.1.

dI ¼ IðqðI hÞð1 IÞ dÞ ¼ 0 ) qðI hÞð1 IÞ d ¼ 0: dt

The stability of the boundary equilibria is obtained from the signs of eigenvalues of the corresponding Jacobian matrices. We omit the details but collect the results in Table 3.1. h

t!1

t!1

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Table 3.1 1þh ð1hÞ2 4=Rv0 1þhþ ð1hÞ2 4=Rv0 The local stability of boundary equilibria for the systems (2.7), (2.8) and (2.9), (2.10) where h2 ¼ ; K2 ¼ ; Rv0 ¼ qd , and Rh0 ¼ bd. 2 2 Boundary equilibria

Stability condition

E0;0 Eh;0

Always locally asymptotically stable

E1;0

Saddle if Rh0 > 1; Locally asymptotically stable if Rh0 1

E0;h

For Model (2.9), (2.10)-Saddle if Rh0 > 1=q; Source if Rh0 < 1=q pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þh ð1hÞ2 4=Rv0 For Model (2.9), (2.10)-Saddle if > qdb (i.e., Rh0 > q1h2 ); 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 1þh ð1hÞ 4=Rv0 Source if < qdb (i.e., Rh0 < q1h2 ) 2

E0;1

For Model (2.9), (2.10)-Saddle if Rh0 < 1=q; Locally asymptotically stable if Rh0 > 1=q; pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þhþ ð1hÞ2 4=Rv0 < qdb (i.e., Rh0 < q1K 2 ); For Model (2.9), (2.10)-Saddle if 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þhþ ð1hÞ2 4=Rv0 Locally asymptotically stable if > qdb (i.e., Rh0 > q1K 2 ) 2

For Model (2.7), (2.8)-Saddle if Rh0 < 1; Source if Rh0 > 1. For Model (2.9), (2.10)-Saddle if Rh0 < 1=h; Source if Rh0 > 1=h

102

Y. Kang, C. Castillo-Chavez / Mathematical Biosciences 248 (2014) 97–116

Letting BS ¼ fðS; IÞ 2 X : a 6 S þ I 6 1g \ fI ¼ 0g and BI ¼ fðS; IÞ 2 X : a 6 S þ I 6 1g \ fS ¼ 0g leads to the facts that (i) BS and BI are positively invariant and that (ii) the omega limit set of BS is E1;0 while the omega limit set of BI is E0;1 . The results (Theorem 2.5 and its corollary) in [32] guarantee that the persistence of disease is determined by the sign of dI dI Idt jBS ¼ Idt jE1;0 while the persistence of susceptibles is determined dS dI by the sign of Sdt jBI ¼ Idt jE0;1 . Letting /ðNÞ ¼ b or bN means that the dynamics of I-class are governed by

dI I S/ðNÞ ¼ /ðNÞ S þ qIf ðNÞ dI ¼ I þ qf ðNÞ d ; dt N N

1. A sufﬁcient condition for the permanence of the system (2.7), (2.8) is that the initial condition Nð0Þ 2 ðh2 ; K 2 Þ and

( () max 1;

if /ðNÞ ¼ b and

d ¼ 1=Rh0 < 1 b

and

dI

¼ ðbS þ qf ðNÞ dÞjBS ¼ ðbS þ qf ðSÞ dÞjE1;0 ¼ b d > 0 Idt BS if /ðNÞ ¼ bN

and

d ¼ 1=Rh0 < 1: b

The results (Theorem 2.5 and its corollary) in [32] guarantee that the persistence of disease for the systems (2.7), (2.8) and (2.9), Rv ð1hÞ2 (2.10) as long as 1 < minf 0 4 ; Rh0 g and the initial condition Nð0Þ 2 ðh2 ; K 2 Þ. The dynamics of the S-class are governed by

dS I I/ðNÞ ; ¼ Sf ðNÞ /ðNÞ S ¼ S f ðNÞ dt N N which gives

dS

I/ðNÞ

d ¼ f ðNÞ ¼ f ðIÞ bjE0;1 ¼ f ðK 2 Þ b ¼ b > 0 Sdt BI N BI q if /ðNÞ ¼ b and

d >1 bq

qð1 hÞ2

)

4b 4b

qð1 hÞ2

) < Rh0 <

1

q

:

2. A sufﬁcient condition for the permanence of the system (2.9), (2.10) is that the initial condition Nð0Þ 2 ðh2 ; K 2 Þ and

( ) d qð1 hÞ2 h qK 2 < ¼ 1=R0 < min 1; b 4b ( ) 4b 1 : () max 1; < Rh0 < qK 2 qð1 hÞ2

which gives

dI

¼ ðb þ qf ðNÞ dÞjBS ¼ ðb þ qf ðSÞ dÞjE1;0 ¼ b d > 0 Idt BS

(

d b

q < ¼ 1=Rh0 < min 1;

We postulate (throughout the rest of this manuscript) that the system (2.7), (2.8) or the system (2.9), (2.10) have disease-free dynamics if its attractor is E0;0 [ E1;0 ; or the system (2.7), (2.8) or (2.9), (2.10) has susceptibles-free dynamics if its attractor is E0;0 [ E0;1 ; or the system (2.7), (2.8) or (2.9), (2.10) has disease-driven extinction if its attractor is E0;0 .

3.1. Interior equilibrium Notice that the equilibria of the system (2.7), (2.8) satisfy the following equations:

bI NðN hÞð1 NÞ ¼ 0 ) S ¼ 0 or I ¼ S0 ¼ S ðN hÞð1 NÞ ; N b bS þ qðN hÞð1 NÞ d ¼ 0 ) I ¼ 0 or I0 ¼ I N N½d qðN hÞð1 NÞ : S¼ b The equilibria of the system (2.9), (2.10) satisfy the following equations

S0 ¼ S½ðN hÞð1 NÞ bI ¼ 0 ) S ¼ 0 or I ¼

ðN hÞð1 NÞ ; b

I0 ¼ I½bS þ qðN hÞð1 NÞ d ¼ 0 ) I ¼ 0 or

and

dS

I/ðNÞ

d ¼ f ðNÞ ¼ f ðIÞ bIjE0;1 ¼ bK 2 > 0 Sdt BI N BI q if /ðNÞ ¼ bN

and

d > K2: bq

Therefore, applications of the results in (Theorem 2.5 and its corollary, [32]) allows us to conclude that: 1. A sufﬁcient condition for the persistence of susceptibles in the 2 system (2.7), (2.8) is that ð1hÞ > q1 > b=d ¼ Rh0 as long initial 4d condition are such that Nð0Þ 2 ðh2 ; K 2 Þ. 2. A sufﬁcient condition for the persistence of susceptibles in the 2 system (2.9), (2.10) is that ð1hÞ > q1 > bK 2 =d ¼ Rh0 K 2 as long 4d initial condition are such that Nð0Þ 2 ðh2 ; K 2 Þ. h Note: The system (2.7), (2.8) or the system (2.9), (2.10) satisﬁes the deﬁnition of permanence provided that there exists a positive number such that for any Nð0Þ 2 ðh2 ; K 2 Þ

lim inf minfIðtÞ; SðtÞg P t!1

An application of Theorem 3.2 leads to the following permanency results:

S

d qðN hÞð1 N Þ ¼ : b b

If we let ðS ; I Þ be an interior equilibrium of the system (2.7), (2.8) or the system (2.9), (2.10) then we have that: 1. The following equation

N ¼ S þ I ¼ )

N ½d þ ð1 qÞðN hÞð1 N Þ b

bd ¼ ðN hÞð1 N Þ 1q

ð3:1Þ

for the system (2.7), (2.8) must be satisﬁed, and so, we see that the system (2.7), (2.8) has no interior equilibrium if d P b (i.e., Rh0 P 1) and qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 hÞ2 4ðbdÞ 1þh ð1 hÞ2 ðb dÞ 1q if > >0 2 2 1q 4 N ðN hÞð1 N Þ I ¼ ; b " # d qðN hÞð1 N Þ ðN h2 ÞðN K 2 Þ ð1 hÞ2 d S ¼ N ¼ qN if > : b b 4 q N ¼

103

Y. Kang, C. Castillo-Chavez / Mathematical Biosciences 248 (2014) 97–116

The system (2.7), (2.8) may have the following two interior equilibria Ni ; i ¼ 1; 2, i.e.,

h < S1 þ I1 ¼ N 1 ¼

1þh 2

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 hÞ2 4ðbdÞ 1q 2

<

and

1þh b < S2 þ I2 ¼ N2 2 2ð1 qÞ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2 1 þ h 1b q þ 4ð1d q hÞ 1þh b ¼ þ 2 2ð1 qÞ 2 b <1þh ð1 qÞ

1þh 2

and

1þh 1þh < S2 þ I2 ¼ N2 ¼ þ 2 2

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 hÞ2 4ðbdÞ 1q 2

< 1:

The Jacobian matrix of Model (2.7), (2.8) evaluated at the interior equilibrium ðS ; I Þ can be represented as follows h i h i 1 0 S 1 þ h 2N þ ðNbI Þ2 S 1 þ h 2N ðNbS Þ2 B C J ðS ;I Þ ¼ @ h i h iA bI bS I qð1 þ hÞ 2qN þ ðN Þ2 I qð1 þ h 2N Þ ðN Þ2 ð3:2Þ

where N ¼ S þ I . Its two eigenvalues ki ; i ¼ 1; 2 satisfy the following equalities:

if

2 1þh b b > ; 1þh 2 2ð1 qÞ 1q d d and h > >4 h : 1q 1q On the other hand if h < 1d q then the system (2.9), (2.10) may have at most one interior equilibrium, namely,

S2 þ I2 ¼ N2

k1 þ k2 ¼ ðS þ qI Þð1 þ h 2N Þ and ¼

bS I ð1 qÞð2N 1 hÞ k1 k2 ¼ : N

ð3:3Þ

Thus, if an interior equilibrium ðS ; I Þ exists, it would be locally asymptotically stable provided that N > 1þh , or a saddle if 2 . We conclude that the system (2.7), (2.8) has either no inteN < 1þh 2 rior or two interior equilibria Ni ; i ¼ 1; 2 and if we happen to have two interior equilibria then we must have that N 1 is always a saddle and N 2 is always a source. 2. The following equation

d ðN hÞð1 N Þ N ¼ S þ I ¼ þ ð1 qÞ b b b d ) ¼ ðN hÞð1 N Þ N 1q b

ð3:4Þ

d

1 þ h 1q þ 4ð1q hÞ 1þh b N ¼ 2 2ð1 qÞ 2

2 1 þ h 1b q d 1þh >h ¼ if 1q 2 4 r ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2

2 1 h 1b q 4 dhb 1 þ h 1b q 1q b d if > h 2 2ð1 qÞ 1q 4

I ¼

:

The Jacobian matrix of the system (2.9), (2.10) evaluated at the interior equilibrium ðS ; I Þ is represented as

J ðS ;I Þ ¼

S ð1 þ h 2N Þ S ð1 þ h b 2N Þ qI ð1 þ h 2N Þ I ðb þ qð1 þ hÞ 2qN Þ

ð3:5Þ

where N ¼ S þ I . Its two eigenvalues ki ; i ¼ 1; 2 satisfy the following equalities:

k1 k2 ¼ bS I ½ð1 qÞð2N 1 hÞ þ b:

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ

2

2

k1 þ k2 ¼ ðS þ qI Þð1 þ h 2N Þ and

for the system (2.9), (2.10). According to Corollary 2.2, we have N < 1, thus (3.4) implies that the system (2.9), (2.10) has no interior equilibrium if Rh0 6 1 and b

1þh b þ 2 2ð1 qÞ

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2 1 þ h 1b q þ 4ð1d q hÞ

ðN hÞð1 N Þ ; b

" # d qðN hÞð1 N Þ ðN h2 ÞðN K 2 Þ ð1 hÞ2 v S ¼ ¼q if > 1=R0 : b b 4

ð3:6Þ

Thus, when the interior ðS ; I Þ exists, it is locally asymptotically stab ble as long as N > 1þh while a saddle whenever N < 1þh 2ð1 qÞ. It is 2 2 a source if

1þh b 1þh < N < : 2 2ð1 qÞ 2 If the system (2.9), (2.10) has two interior equilibria N i ; i ¼ 1; 2 then using their expressions and the criteria for interior stability allow us to conclude that N 1 is always a saddle while N 2 can be a sink or source. If, on the other hand, the system (2.9), (2.10) has only one interior equilibrium, namely N2 , then we see that it can be a sink or source depending on parameter values. The above discussion can be summarized in the following theorem: Theorem 3.3 (Interior equilibria of Models). Let Ei1 ¼ ðS1 ; I1 Þ and Ei2 ¼ ðS2 ; I2 Þ then existence and stability conditions for the interior equilibria of the system (2.7), (2.8) are listed in Table 3.2.

The system (2.9), (2.10) may have the following two interior equilibria:

S1 þ I1 ¼ N1 1þh b ¼ 2 2ð1 qÞ 1þh b < 2 2ð1 qÞ

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2 1 þ h 1b q þ 4ð1d q hÞ 2

Table 3.2 The local stability of interior equilibrium for the system (2.7), (2.8). Interior Equilibrium

Condition for existence

Condition for local asymptotically stable

Ei1

Rv0 <

4 ð1hÞ2

Always a saddle

4 ð1hÞ2

< Rv0 <

Rv0 <

4 ð1hÞ2

4 ð1hÞ2

< Rv0 <

Ei2

q < 1 or bd 1q bd

q < 1 or bd 1q bd

Always locally asymptotically stable.

104

Y. Kang, C. Castillo-Chavez / Mathematical Biosciences 248 (2014) 97–116

Table 3.3 The local stability of interior equilibrium for the system (2.9), (2.10). Interior Equilibrium

Condition for existence

Ei1

max

Ei2

n

o

1q q ; Rh0 ; bþ1 q1 d

Case (1):max Rv0 > Rv0 >

4 ð1hÞ2 4 ð1hÞ2

n

Condition for stability Always a saddle

> 1=h and (a) Rv0 < 4 2 or (b) Rv0 > 4 2 &N 1 < h2 ð1hÞ ð1hÞ o > 1=h and (a) Rv0 < 4 2 or (b) Rv0 > 4 2 &N 2 < h2 or (c)

1q q ; Rh0 ; bþ1q1 d

ð1hÞ

; N 2 > K 2 ;Case (2):1 <

Rh0

< 1=h and (a) Rv0 <

Locally asymptotically stable if N 2 > 1þh 2 while

ð1hÞ

4 ð1hÞ2

or (b) Rv0 >

4 ð1hÞ2

; N 2 < h2 or (c)

it’s a source if

1þh 2

b 1þh 2ð1 qÞ < N 2 < 2

; N2 > K 2

Table 3.4 No interior equilibrium and the related global dynamics for the systems (2.7), (2.8) and (2.9), (2.10). Cases

The system (2.7), (2.8)

No interior equilibrium

Case (1): Case (3):

Rh0

6 1; Case (2):

4 ð1hÞ2

<

1q bd

The system (2.9), (2.10) 1q bd

<

4 ; ð1hÞ2

< Rv0 .

1q 4 < Case (1): Rh0 6 1; Case (2): bhd 2 ; Case (3): ð1h1b qÞ n o 1q h 1q v 4 max d ; R0 ; bþq1 > 1=h; R0 > 2 ; h2 < N i < K 2 ; i ¼ 1; 2; Case (4): ð1hÞ

Rh0 < 1=h; Rv0 > Disease-free dynamics Susceptible-free dynamics Disease-driven extinction Permanence

Rh0 6 1 n

> max

n

Rv0 > max

> q1.

1q ; Rv0 bd

o

.

Nð0Þ 2 ðh2 ; K 2 Þ and Rh0

1

; h2 < N 2 < K 2

Rh0 6 1 o 2

min Rh0 ; ð1hÞ 4d 4 ð1hÞ2

4 ð1hÞ2

Rh0

< q < minf q

2 ; ð1hÞ g. 4d

Rv0 >

4 ð1hÞ2

n

1 bK 2

;

and

4 ð1hÞ2 1q bhd

o

<

and Conditions of Case (2) or Case (3) or Case (4) 4 2. ð1h1b qÞ

n h o 2 R Nð0Þ 2 ðh2 ; K 2 Þ and Rh0 < q1K 2 < min qK02 ; ð1hÞ . 4dK 2

Existence and stability conditions for the interior equilibria of the system (2.9), (2.10) are listed in Table 3.3. Sufﬁcient conditions leading to no interior equilibrium and the related global dynamics of the systems (2.7), (2.8) and (2.9), (2.10) are listed in Table 3.4.

4 Therefore, if Rv0 < ð1hÞ 2 then the systems (2.7), (2.8) and (2.9), (2.10)

have no interior if Rh0 1 or if the following conditions

1q 4 < for the system ð2:7Þ; ð2:8Þ b d ð1 hÞ2 and

Proof. The above discussion shows that a necessary condition for the systems (2.7), (2.8) and (2.9), (2.10) to a have interior equilibRh0

> 1 while the existence of interior equilibrium for rium is that the systems (2.7), (2.8) and (2.9), (2.10) can be classiﬁed with the v 4 4 . conditions Rv0 > ð1hÞ 2 and R0 < ð1hÞ2 If Rv0 < 4 2 then the systems (2.7), (2.8) and (2.9), (2.10) have ð1hÞ no boundary equilibria E0;h and E0;1 on I-axis and therefore

hðNÞ ¼ qðN hÞð1 NÞ d < 0 for all N > 0: Necessary conditions for the existence of interior equilibrium for the systems (2.7), (2.8) and (2.9), (2.10) are tied in to the existence of solutions of the following equations:

b d ð1 hÞ2 0< < 1q 4 ! 4 1q i:e:; Rv0 < < ð1 hÞ2 b d

4 1q for the system ð2:9Þandð2:10Þ:

2 < ðbh dÞ b 1 h 1q 4 If Rv0 > ð1hÞ 2 , then the systems (2.7), (2.8) and (2.9), (2.10) have

boundary equilibria E0;h and E0;1 on the I-axis. Additional conditions are needed to guarantee the existence of interior equilibrium for the systems (2.7), (2.8) and (2.9), (2.10). The systems (2.7), (2.8) and (2.9), (2.10) are discussed separately. 1. For the system (2.7), (2.8), if

d

S ¼ N

q

ð3:7Þ

b d d ¼ ðN hÞð1 NÞ; N > h; or 1q b b b d 1þh b ¼ ðN hÞð1 NÞ; N > ; 1q b 2 2ð1 qÞ 2 b 4ðbh dÞ > > 0: 1h 1q 1q

d qðN hÞð1 N Þ ðN h2 ÞðN K 2 Þ ¼ qN > 0: b b

q 4 The schematic nullclines for the system (2.7), (2.8) when 1 < ð1hÞ 2 bd

are illustrated in Fig. 3.1. Two interior equilibria occur whenever 4 ð1hÞ2

and

4 < ð1hÞ 2 Eq. (3.7) should have

solutions in the interval ð0; h2 Þ or ðK 2 ; 1Þ since

bd ¼ ðN hÞð1 N Þ; 1q <

1q bd

q < Rv0 < 1 with one interior a saddle (i.e., the horizontal line bd

intercepts in the green region of ðN hÞð1 NÞ) and the other a sink (i.e., the horizontal line intercepts in the blue region of ðN hÞð1 NÞ). There is no interior equilibrium when the horizontal line intercepts (crosses) the black region of ðN hÞð1 NÞ, i.e.,

ð3:8Þ

4 ð1hÞ2

q < 1 < Rv0 . bd

2. For the system (2.9), (2.10), whenever

1q bhd

>

4 2

ð1h1b qÞ

Eq. (3.8)

should have solutions in the interval ð0; h2 Þ or ðK 2 ; 1Þ since

105

Y. Kang, C. Castillo-Chavez / Mathematical Biosciences 248 (2014) 97–116 Nullclines for Model (7)−(8) d/ρ<(1−θ)2/4 0.2 (1−θ)2/4

0.18

(β−d)/(1−ρ) No Interior Equilibrium

0.16 d/ρ 0.14 0.12 0.1

(β−d)/(1−ρ) Two Interior Equilibria

0.08 0.06

(N−θ)(1−N) 0.04 0.02 0 0.1

θ

0.2

0.3

0.4 θ

0.5

0.6 (1+θ)/2

0.7

0.8

K2

0.9

1

1.1 N=S+I

2

2

ð1hÞ 1q bd b 4 Fig. 3.1. Schematic nullclines for the system (2.7), (2.8) when 1 q < 4 (i.e., bd < ð1hÞ2 ). Potential (two) interior equilibria are the intercepts of the horizontal line 1q and the

curve ðN hÞð1 NÞ with their stability determined by the location of the intercept. The green region of ðN hÞð1 NÞ is a saddle and the blue region is a sink. Two interior 2

v ð1hÞ 1q bd d 4 equilibria (see the purple horizontal line) occur whenever 1 q < q < 4 (i.e., ð1hÞ2 < R0 < bd ) where one interior is saddle and the other one a sink. (For interpretation of the

references to colour in this ﬁgure caption, the reader is referred to the web version of this article.) Nullclines for Model (9)−(10)

Nullclines for Model (9)−(10)

0.2

0.2 min{d/β,d/(1−ρ),(β+ρ−1)/(1−ρ)}<θ 2 and d/ρ<(1−θ) /4

2

(1−θ) /4 0.18

θ
2

(1−θ) /4 0.18

0.16

0.16

d/ρ

d/ρ β(N−d/β)/(1−ρ) Two Interior Equilibria

0.14 0.12

β(N−d/β)/(1−ρ) One Interior Equilibrium

0.14

0.1

0.1

0.08

0.08 β(N−d/β)/(1−ρ) Two Interior Equilibria

0.06

β(N−d/β)/(1−ρ) No Interior Equilibrium

β(N−d/β)/(1−ρ) One Interior Equilibrium

0.06 (N−θ)(1−N)

(N−θ)(1−N)

0.04

0.04

0.02 0 0

β(N−d/β)/(1−ρ) One Interior Equilibrium

0.12

0.02 d/β θ

0.2

0.4 θ2

0.6

0.8

(1+θ)/2

1 N=S+I

K

2

Fig. 3.2. Schematic nullclines for the system (2.9), (2.10) when

0 0

1.2

θ

0.2 d/β

0.4 θ

0.6 (1+θ)/2

K

0.8

1

2

N=S+I

2

1q bhd

>

4 b

ð1h1qÞ

2

. Potential interior equilibria are intercepts of the line

b ðN 1q

dbÞ and the curve ðN hÞð1 NÞ

with their stability determined by the location of the intercept. The green region of ðN hÞð1 NÞ is a saddle; the red region is a source and the blue region is a sink. The left graph corresponds to the case when Rh0 > 1=h: one (see the black line 1b q ðN dbÞ) or two interior equilibria (see the purple or dark green line 1b q ðN dbÞ) are possible. The right graph corresponds to the case when 1 < Rh0 < 1=h, potentially one interior equilibria (see the dark green or purple line

b ðN 1q

dbÞ). (For interpretation of the references to

colour in this ﬁgure caption, the reader is referred to the web version of this article.)

S ¼

d qðN hÞð1 N Þ ðN h2 ÞðN K 2 Þ ¼q > 0: b b

The schematic nullclines for the system (2.9), (2.10) when 1q bhd

>

4 2

ð1h1b qÞ

are found in Fig. 3.2. There are two cases depending

on the sign of db h: (1) If

d b

(i.e., the line intercepts the blue region of ðN hÞð1 NÞ). If Condition N2 < h2 or N2 < K 2 does not hold, i.e., h2 < N 2 < K 2 then the system (2.9), (2.10) has only one interior equilibrium N 1 , a saddle. (2) If db < h (see Fig. 3.2(b)) then only one interior equilibria occurs whenever

> h () Rh0 < 1=h (see Fig. 3.2(a)), two

1 < Rh0 < 1=h; h <

interior equilibria occur whenever

1þh b > ; 2 2ð1 qÞ

Rh0 > 1=h;

where

N1

< h2 ;

N2

< h2

or

N2

< K2

is always a saddle (i.e., the line

h2 < N2 < K 2

or b ðN 1q

dbÞ ¼ 1b q ðN and N 2 can be

and

N2 < h2 or N2 < K 2

and this interior equilibrium N 2 can be a sink or a source, depending on parameters’ values. There is no interior if the line intercepts the black region of ðN hÞð1 NÞ when

4 1q

2 < ðbh dÞ b 1 h 1q

and

N1

d 1q

1 Þ Rh0

sink intercepts the green region of ðN hÞð1 NÞ) (i.e., the line intercepts the red region of ðN hÞð1 NÞ) or source

4 1h

2 >

b 1q

1q : ðbh dÞ

106

Y. Kang, C. Castillo-Chavez / Mathematical Biosciences 248 (2014) 97–116

In short, sufﬁcient conditions for the existence of interior equilibria and their stability have been identiﬁed and listed in Table 3.2 for the system (2.7), (2.8) and in Table 3.3 for the system (2.9), (2.10). The above analysis has identiﬁed conditions (sufﬁcient) that guarantee the absence of interior equilibria for the systems (2.7), (2.8) and (2.9), (2.10); listed inTable 3.4. In the absence of interior equilibrium, we can conclude thanks to the Poincaré–Bendixson Theorem [23], that a trajectory starting with arbitrary initial conditions in X converge to its locally asymptotically stable boundary equilibria since the systems (2.7), (2.8) and (2.9), (2.10) support each a global compact attractor fðS; IÞ 2 X : 0 6 S þ I 6 1g in X. The fact that E0;0 is always an attractor results according to Proposition 3.1 in the following three cases: 1. Disease-free dynamics that corresponds to the case where E0;0 and E1;0 are the only locally asymptotically stable boundary equilibria while other existing boundary equilibria are unstable. This implies that b 6 d (i.e., Rh0 1) is a sufﬁcient condition in support of disease-free dynamics within the systems (2.7), (2.8) and (2.9), (2.10). 2. Susceptible-free dynamics that corresponds to the case where E0;0 and E0;1 are the only locally asymptotically stable boundary equilibria while other existing boundary equilibria (including

Thus, if the system (2.7), (2.8) has boundary equilibria on the I-axis, 1q 4 4 i.e., Rv0 > ð1hÞ 2 , then the no interior equilibria condition bd < ð1hÞ2 says that the boundary equilibrium E0;h is a source while E0;1 is locally asymptotically stable according to Proposition 3.1 and the fact that

bd d q 1 > ) Rv0 ¼ < : 1q q d b Therefore, the system (2.7), (2.8) have two interior equilibria provided that

( ) b d ð1 hÞ2 d q 1 4 v 0< < < () R0 ¼ < min ; 1q b ð1 hÞ2 4 q d or

0<

b d d ð1 hÞ2 4 q 1 < < () < Rv0 ¼ < : 1q q 4 d b ð1 hÞ2

Corollary 2.2, Proposition 3.1, and Theorem 3.2, 3.3 lead to the study of three cases for the system (2.7), (2.8): 1. The disease-driven extinction occurs in the situation depicted in 2

ð1hÞ bd Fig. 4.1. First, no interior equilibrium, which requires 1 q> 4 .

Within Fig. 4.1, we see that the existence and the stability of

those on the I-axis) are unstable. This implies that Rh0 > q1K 2

h b 4 boundary equilibria requires Rv0 ¼ qd < ð1hÞ 2 and R0 ¼ d > 1. Thus,

4 and Rv0 > ð1hÞ 2 for the systems (2.7), (2.8) and (2.9), (2.10), in

a sufﬁcient condition that makes Fig. 4.1 possible is

addition to the conditions of non-existence of interior equilibrium. Our results imply that large value of Rh0 and Rv0 may lead to the susceptible-free dynamics 3. Disease-driven extinction that corresponds to the case where E0;0 is the only locally asymptotically stable boundary equilibria provided that there is no boundary equilibria on the I-axis a result based on Theorem 3.2. This implies that 1 < Rh0 < q1K 2 4 and Rv0 < ð1hÞ 2 for the systems (2.7), (2.8) and (2.9), (2.10) in conjunction to the conditions that there are no interior equilibrium. Detailed conditions on the three cases discussed above are listed in Table 3.4. h

max

q 1q ;

d bd

<

4 ð1 hÞ2

and Rh0 ¼

b > 1: d

The system (2.7), (2.8) may also support disease-driven extinction whenever it supports an interior equilibrium. In such a case, disease-driven extinction occurs as a result of catastrophic events, that is, when a stable limit cycles merges with the adjacent saddle, leading to the annihilation of the susceptible and infected sub-populations. 2. An endemic situation occurs whenever the system (2.7), (2.8) supports the interior equilibria shown in Fig. 4.2. A necessary condition is that

Notes: Theorem 3.3 implies the following:

1q bd

4 > ð1hÞ 2 and thus we can conclude that the

sufﬁcient condition leading to Fig. 4.2(a) is 1. Small values of q make the systems (2.7), (2.8) and (2.9), (2.10) prone to disease-driven extinction since one necessary condi4 tion for disease-driven extinction requires that Rv0 < ð1hÞ 2 according to Theorem 3.2. This also suggests that vertical transmission may save a species from extinction provided that the reproductive ability of infectives is large enough (some additional conditions must be met). 2. The system (2.7), (2.8) has simpler dynamics than the system (2.9), (2.10). In fact, the system (2.7), (2.8) has no interior equilibria or two interior equilibria (a saddle and a sink) while the system (2.9), (2.10) may have one or two interior equilibria.

q 4 1q d 1 h Rh0 ¼ < < and q < < 1 or 1 < R < 0 b d ð1 hÞ2 b d q while the sufﬁcient condition leading to Fig. 4.2(b) is

1.2 2

β> d and (1−θ) /4
1 0.8

4. Classiﬁcations on dynamics and related bifurcation diagrams I

Disease−driven Extinction

This section focuses on the classiﬁcation of the dynamics and related bifurcations of the systems (2.7), (2.8) and (2.9), (2.10) by using the horizontal transmission reproduction number Rh0 ¼ bd and the vertical transmission reproduction number Rv0 ¼ qd. 4.1. The SI model with frequency-dependent horizontal transmission

0.6 0.4 0.2 0 0

For the system (2.7), (2.8), notice that

bd d q 1 < () Rv0 ¼ > : 1q q d b

0.5

1

1.5

S

Fig. 4.1. Schematic phase plane for the system (2.7), (2.8) when it experiences the possibility of disease-driven extinction.

107

Y. Kang, C. Castillo-Chavez / Mathematical Biosciences 248 (2014) 97–116

1.2 2

β< d and (1−θ) /4<(β−d)/(1−ρ)<β
I

I

1.2 2

β> d and (β−d)/(1−ρ)<(1−θ) /4
1

Edemic

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0 0

0.5

1

S

Edemic

0 0

1.5

0.5

(a) Schematic phase plane

1

S

1.5

(b) Schematic phase plane

Fig. 4.2. Schematic phase plane for the system (2.7), (2.8) when it has the endemic occurs.

1.2

1.2 2

2

β≤ d and d/ρ>(1−θ) /4

β> d and (β−d)/(1−ρ)<(1−θ) /4
1

1 Susceptibles−free Dynamics

0.8

0.8

I

I

Disease−free Dynamics 0.6

0.6

0.4

0.4

0.2

0.2

0 0

0.5

1

1.5

0 0

0.5

1

S

1.5

S

Fig. 4.3. Schematic phase plane for the system (2.7), (2.8) when it has no coexistence of susceptibles and infectives.

1q q d 1 h h : < < R ¼ and q < < 1 or 1 < R < 0 0 b d q ð1 hÞ2 b d 4

3. Disease-free or susceptible-free dynamics occur when the system (2.7), (2.8) has no interior equilibrium with either E1;0 or E0;1 locally asymptotically stable, as shown in Fig. 4.3. Fig. 4.3(a) highlights a disease-free situation for which the condition

Rh0 6 1 and

4 ð1 hÞ2

< Rv0 ¼

q

4.2. The SI model with density-dependent horizontal transmissions

d

is sufﬁcient. Fig. 4.3(b) highlights a susceptible-free state for which sufﬁcient the condition is

Rv0 <

4 ð1 hÞ2

<

1q bd

and

tends lead to the coexistence of susceptibles and infectives; and small values of Rv0 tends to lead to disease-driven extinction. 3. The SI model with frequency-dependent transmissions or the system (2.7), (2.8) supports relatively simple equilibrium dynamics. It can support no interior or two interior equilibria, with one of the interior equilibrium always stable (see Theorem 3.3, Tables 3.3, 3.4 and Fig. 3.1).

d
1 or Rh0 > > 1 :

q

In this subsection, the dynamics and potential bifurcations of the SI model, with density-dependent horizontal transmission, given by the system (2.9), (2.10), are classiﬁed. The classiﬁcation of stability of boundary equilibria for the system (2.9), (2.10) on the 2

I-axis E0;h and E0;1 when ð1hÞ > qd can be determined from the signs 4 of h2 qdb and K 2 qdb. Since

v

q

The vertical transmission reproduction number, R0 ¼ d , and the horizontal transmission reproduction number, Rh0 ¼ bd help, using the above discussions and the analytical results in previous sections, understand the effects of parameters q; d; b; h on the dynamics of the system (2.7), (2.8). The results can brieﬂy summarized as follows: Rh0

1. A horizontal transmission reproduction number less than 1 supports disease-free dynamics for the system (2.7), (2.8) (see Theorem 3.3 when combined with the relevant results in Table 3.4). 2. Both initial condition Nð0Þ ¼ Sð0Þ þ Ið0Þ and the value of the vertical transmission reproduction number, Rv0 , are important in determining global dynamics (see Corollary 2.2 and Theorem 3.3). We can conclude that large values of Rv0 tend to lead to susceptible-free dynamics; while intermediate values of Rv0

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 hÞ2 4d=q 1þh h < h2 ¼ < K2 2 2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼

ð1 hÞ2 4d=q

1þh þ 2

2

< 1:

Hence, the signs can be determined by solving Rv0 ¼ qd from the equations h2 ¼ qdb and K 2 ¼ qdb, i.e.,

1þh 2 Rv0 ¼ and

q d

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 hÞ2 4d=q

¼

d ¼ ) qb qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ h 4b ð4b 1Þ2 þ ð1 hÞ2 1 8bh 2

2bh

108

Y. Kang, C. Castillo-Chavez / Mathematical Biosciences 248 (2014) 97–116

Two dimensional Bifurcation Diagram: d−ρ 1

β=0.1; θ=0.15

0.9 0.8 0.7

ρ

0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.01 0.02 d=0.015

0.03

0.04

0.05 d

0.06

0.07

0.08

0.09

0.1

Fig. 4.4. An example of two dimensional bifurcation diagram (d q) of the system (2.9), (2.10) when b ¼ 0:1 and h ¼ 0:15. The black area indicates that the reproduction v 4 4 ; c1 g < Rv0 < c2 ; the green area number of vertical transmission Rv0 ¼ qd is large, i.e., Rv0 > maxfð1hÞ 2 ; c 2 g; the cyan area indicates that R0 has intermediate values, i.e., maxf ð1hÞ2 v v 4 4 . The green dots indicate that the system (2.9), (2.10) has only one interior indicates that Rv0 has small values, i.e., ð1hÞ 2 < R0 < c 1 and the white area indicates that R0 < ð1hÞ2

equilibrium which can be a source, saddle or sink while the red dots indicate that the system (2.9), (2.10) has two interior equilibria where one is a saddle and the other one can be a sink or source. (For interpretation of the references to colour in this ﬁgure caption, the reader is referred to the web version of this article.)

1þh þ 2

q d

¼

d ¼ ) 2 qb qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ h 4b ð4b 1Þ2 þ ð1 hÞ2 1 8bh 2bh

Letting c2 ¼

(

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 hÞ2 4d=q

1þh4bþ

ð4b1Þ þð1hÞ 18bh 2bh

ð4b1Þ þð1hÞ 18bh 2bh

4 and

leads, making use of Proposition

3.1, to the following (a two dimensional bifurcation diagram example is shown in Fig. 4.4 when b ¼ 0:1 and h ¼ 0:15) results: 1. Black area in Fig. 4.4: E0;h is a saddle and E0;1 is locally asymptotically stable if

Rv0 ¼

q d

(

> max

4

ð1 hÞ2

ð1 hÞ2

; c1

< Rv0 ¼

q d

< c2 :

3. Green area in Fig. 4.4: E0;h is a source and E0;1 is a saddle if

:

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2

1þh4b

c1 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2

max

)

4

)

; c2 :

2. Cyan area in Fig. 4.4: E0;h is a source and E0;1 is locally asymptotically stable if

ð1 hÞ

2

< Rv0 ¼

q d

< c1 :

4. White area in Fig. 4.4: there is no boundary equilibrium on Iaxis, i.e.,

Rv0 ¼

q d

<

4 ð1 hÞ2

:

We therefore identify only four cases for the system (2.9), (2.10): 1. Case one: There is no boundary equilibrium E0;h and E0;1 when the reproduction number of vertical transmission Rv0 ¼ qd is small 4 enough, i.e., Rv0 < ð1hÞ 2 . For a certain range of parameter values, the system (2.9), (2.10) can have a unique interior attractor,

4 Fig. 4.5. Schematic phase plane of the system (2.9), (2.10) when I-class is not able to persist in the absence of S-class, i.e., Rv0 < ð1hÞ 2.

109

Y. Kang, C. Castillo-Chavez / Mathematical Biosciences 248 (2014) 97–116

Fig. 4.6. Schematic phase plane of the system (2.9), (2.10) when I-class is able to persist in the absence of S-class, i.e.,

Fig. 4.7. Schematic phase plane of the system (2.9), (2.10) when I-class is able to persist in the absence of S-class, i.e.,

Fig. 4.8. Schematic phase plane of the system (2.9), (2.10) when I-class is able to persist in the absence of S-class, i.e.,

which can be an interior equilibrium (see Fig. 4.5; where within the sub-ﬁgure (a) corresponds to the white area with blue dots, i.e., Rh0 ¼ bd > 1h, and (b) corresponds to the white area with red dots, i.e., 1 < Rh0 < 1h, below the green area of Fig. 4.4) or a stable limit cycle through Hopf-Bifurcation. This is the case when the system (2.9), (2.10) can support disease-driven extinction as it was the case for the system (2.7), (2.8). 2. Case two: There are boundary equilibria E0;h and E0;1 when the reproduction number of vertical transmission Rv0 ¼ qd is large v 4 enough, i.e., ð1hÞ 2 < R0 and the reproduction number of horizontal

qð1hÞ2 4b

4b

qð1hÞ2

4b

qð1hÞ2

> db.

< Rh0 .

< Rh0 .

transmission has a large value, i.e., Rh0 > 1h. An example of this case is shown in the black area [whose dynamics is corresponding to the sub-ﬁgure (a) of Fig. 4.6] and the green area [whose dynamics is corresponding to the sub-ﬁgure (b) of Fig. 4.6] on the right of the purple vertical line d ¼ 0:015 of Fig. 4.4. 3. Case three: There are boundary equilibria E0;h and E0;1 and the reproduction number of horizontal transmission has intermediate values, i.e., 1 < Rh0 < 1h. An example of this case is shown in the green area [whose dynamics corresponds to the sub-ﬁgure (a) of Fig. 4.7] and the black area [whose dynamics is

110

Y. Kang, C. Castillo-Chavez / Mathematical Biosciences 248 (2014) 97–116 Bifurcation Diagram of β

Bifurcation Diagram of ρ

1

0.64 0.62

d=.45; θ=0.15; ρ=0.35

0.95 β=1; θ=0.15; d=0.45

0.6

0.9

0.58

0.85

0.8

*

N*=S +I*

N*=S*+I*

0.56 0.54

0.75

0.52 0.7 0.5 0.65 0.48 0.6

0.46 0.44 0

0.1

0.2

0.3

0.4

0.5 ρ

0.6

0.7

0.8

0.9

0.55 0.4

1

0.5

0.6

0.7 β

0.8

0.9

1

Bifurcation Diagram of θ 0.62 Bifurcation Diagram of d 1

β=1; d=0.45; ρ=0.35;

0.6 β=1; θ=0.15; ρ=0.35

0.9

0.58

0.8 0.56

N*=S*+I*

N*=S*+I*

0.7

0.6

0.54

0.52

0.5 0.5 0.4 0.48 0.3 0.46

0.2

0.1 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.44 0

0.05

0.1

0.15

d

0.2

θ

0.25

0.3

0.35

0.4

0.45

Fig. 4.9. Bifurcation Diagrams of q; b; d; h for the system (2.9), (2.10) where the blue dots in the left ﬁgure means the interior equilibrium is locally asymptotically stable while the red dots means source. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

corresponding to the sub-ﬁgure (b) of Fig. 4.7] on the left of the purple vertical line d ¼ 0:015 of Fig. 4.4. 4. Case four: There are boundary equilibria E0;h and E0;1 and the reproduction number of vertical transmission has intermediate n o 4 values, i.e., max ð1hÞ < Rh0 < c2 . An example of this case 2 ; c1 is shown in the cyan area of Fig. 4.4 whose dynamics on the right side of the purple vertical line d ¼ 0:015, i.e., the reproduction number of horizontal transmission has large values, i.e.,

The values of Rh0 determines whether the system (2.9), (2.10) can have unique interior equilibrium (Rh0 > 1b) or two interior equilibria (1 < Rh0 < 1b). 2. The large values of q; h; b and the small values of d can destabilize the system (2.9), (2.10) (see Fig. 4.9). 3. The system (2.9), (2.10) can have a stable limit cycle; an example is included in Fig. 4.10.

Rh0 > 1h. Dynamics represented by the sub-ﬁgure (a) of Fig. 4.8 and the dynamics on the left side of the purple vertical line

5. Diffusive instability

d ¼ 0:015, i.e., 1 < Rh0 < 1h, represented by the sub-ﬁgure (b) of Fig. 4.8.

The ecological and evolutionary dynamics of host-pathogen or host-parasite systems are theoretically challenging for factors that include the impact of recurrent disease invasion, the ability of a parasite or pathogen to modify a host’s mobility-tied ﬁtness, or reducing life span, or limiting/eliminating reproductive ability. Dispersal is capable of shaping the boundaries of habitats through increases or reductions on the size of the sphere-of-inﬂuence of infectious hosts, a cumulative process possibly altering infection rates (reductions or increases in effective contact rates), or its ability to generate clusters, or disease-driven selection of particular behavioral types [2,15,41,43]. Diffusive instability arises when diffusion or migration destabilizes stable situations [47,41,48]. It may emerge as a result of the dynamics of meta-population systems when coupled by dispersal or reaction-diffusion [15,41,43]. The possible emergence of diffusive instability from two-patch systems, coupled by dispersal, when the local dynamics are governed by variants of the general SI-FD or SI-DD systems, is brieﬂy addressed in this section. [47] using a simple predator–prey model studied the possibility of diffusive instability in predator–prey systems. Hence, ﬁrst, some

The above discussion and the associated analytical results, including Proposition 3.1, Theorems 2.1, 3.2, 3.3 lead us to conclude that the effects of parameters q; d; b; h on the dynamics of the system (2.9), (2.10) can be summarized as follows: 1. The values of the reproduction number of horizontal transmission Rh0 ¼ bd and the reproduction number of vertical transmission Rv0 ¼ qd determine the dynamics of the system (2.9), (2.10): 4 If 1 < Rh0 < 1b and Rv0 < ð1hÞ 2 , then the system (2.9), (2.10) has no boundary equilibrium on the I-axis and it may have the disease-driven extinction in certain range of parameter values. If the system (2.9), (2.10) has the intermediate values of Rv0 , n o 4 i.e., max ð1hÞ < Rv0 < c2 , then the system tends to have 2 ; c1 susceptible-free dynamics.

111

Y. Kang, C. Castillo-Chavez / Mathematical Biosciences 248 (2014) 97–116 Time Series of S/I−class 0.7 S−class I−class

β=1; θ=0.15; d=0.45; ρ=0.35; S(0)=0.26; I(0)=0.2 0.6

0.5

S/I

0.4

0.3

0.2

0.1

0 0

100

200

300

400

500 Time

600

700

800

900

1000

Fig. 4.10. An example of a stable limit cycle when b ¼ 1; h ¼ 0:15; d ¼ 0:45; q ¼ 0:35; Sð0Þ ¼ 0:26; Ið0Þ ¼ 0:2 where the blue dots in the right ﬁgure means the population of Iclass while the red dots means the population of S-class. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

classical results addressing the emergence of diffusive instability in predator–prey systems, are revisited. Segel and Jackson [47] showed that the addition of random dispersal was enough to generate instability from an otherwise initially stable uniform steadystate distribution. Diffusive instability, as shown by [41], also arises from the effects of dispersal on predator–prey interactions under the pressure of Allee effects. Segel and Levin [48] used approximate methods and a multiple-time scale theoretical approach in their development of a small amplitude nonlinear theory of prey–predator interactions under random dispersal; a process modeled via diffusion-like terms in discrete and continuous settings. Segel and Levin [48] showed that dispersal can destabilize spatially uniform states; diffusive instability moving the system to new nonuniform steady states. Levin and Segel [48,42] noted that the emergence of diffusive instabilities may explain some of the spatial irregularities observed in nature. The possibility of diffusive instability in general SI models is brieﬂy discussed since identifying conditions that lead to diffusive instability on systems where disease and dispersal play a non-independent role are explored. The discussion below, we hope, will instigate further research on the study of diffusive instability in the settings introduced in this manuscript. A general SI-model can be represented abstractly via the following set of equations

dS ¼ f ðS; IÞ dt dI ¼ gðS; IÞ; dt

ð5:1Þ

operating under the assumption that the system (5.1) has a local asymptotically stable interior equilibrium ðS ; I Þ, an assumption formulated using the inequalities

fS þ g I < 0 and f S g I fI g S > 0 where fS ¼

@f ðS ; I Þ; @S

fI ¼

@f @I

ðS ; I Þ; g S ¼

ð5:2Þ @g ðS ; I Þ; @S

gI ¼

@g @I

ðS ; I Þ.

The inclusion of dispersal leads, for example, to the study of symmetric two-patch models. An example of such a system is given by the following set of equations:

dS1 ¼ f ðS1 ; I1 Þ þ lS ðS2 S1 Þ dt dI1 ¼ gðS1 ; I1 Þ þ lI ðI2 I1 Þ dt dS2 ¼ f ðS2 ; I2 Þ þ lS ðS1 S2 Þ dt dI2 ¼ gðS2 ; I2 Þ þ lI ðI1 I2 Þ dt

ð5:3Þ

where lS is the dispersal rate of the S-class and lI is the dispersal rate of I-class. A typical pseudo diffusion model analog, involving constant diffusion coefﬁcients, is given by the following system:

@S ¼ DS DS þ f ðS; IÞ dt @I ¼ DI DI þ gðS; IÞ dt

ð5:4Þ

where D is the Laplacian; DS ; DI are the constant diffusion coefﬁcients for susceptibles and infectives, respectively. We say the SI Model (5.1) supports diffusive instability (or Turing Effects) if ðS ; I Þ is a locally asymptotically stable interior equilibrium of the system (5.1) but ðS ; I ; S ; I Þ becomes unstable when embedded in the symmetric two-patch model given by the system (5.3) for certain values of lI ; lS . We can achieve similar results as long as the ðS ; I Þ equilibrium is unstable for the Diffusion the system (5.4) at least for some values of DS ; DI . The following theorem provides conditions that support the diffusive instability of the systems (2.7), (2.8) and (2.9), (2.10): Theorem 5.1 (Diffusive instability). The general SI model (5.1) can have diffusive instability only if fS g I < 0. In particular, the system (2.7), (2.8) can support diffusive instability provided that the following inequalities hold

112

Y. Kang, C. Castillo-Chavez / Mathematical Biosciences 248 (2014) 97–116

(

)

N ðN hÞð1 N Þ and b qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 hÞ2 4ðbdÞ 1þh 1þh 1q þ > ; N ¼ 2 2 2

bd d ð1 hÞ2 and < min ; 1q q 4 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 hÞ2 4ðbdÞ ð1 þ hÞ þ ð1 þ hÞ2 3h 1þh 1q þ < : 2 2 3

I ¼

we have

The system (2.9), (2.10) does not support diffusive instability.

fS > 0 ) 1 þ h 2N þ Proof. Recall that the general SI model (5.1) has locally asymptotically stable interior equilibrium ðS ; I Þ if

bI 2

ðN Þ

2

J ðS ;I ;S ;I Þ

fS lS 6 g 6 S ¼6 4 lS

fI

lS

0

g I II

0

lI

0

fS lS

fI

lI

gS

g I lI

0

3

which has four eigenvalues ki ; i ¼ 1; 2; 3; 4. These four eigenvalues satisfy the following two conditions: 1. k1 and k2 are eigenvalues of the Jacobian matrix of the SI model (5.1) evaluated at ðS ; I Þ, thus they are both negatives. 2. k3 and k4 have the following properties:

k3 þ k4 ¼ 2ðfS þ g I lS lI Þ < 0 and This implies that ðS ; I ; S ; I Þ is an interior equilibrium of its twopatch model (5.3) with its stability being determined by the sign of K. Thus, diffusive instability can occur only if K < 0 which indicates that fS g I < 0, that is, if either fS or g I , is positive then ðfS lI þ g I lS Þ > 0 can be made positive and large enough with the right combination of lS ; lI ; in other words, we conclude that for these parameter values, we have that K < 0. For example, if fS > 0 then we can select the dispersal rate of the I-class lI large enough and the dispersal rate of S-class lS small so that the condition K < 0 is met. Now, under g I > 0 diffusive instability may be possible as long as lS is large and lI is small. Relying on the discussion in Section 8.9 of [9], we conclude that ðS ; I Þ is a steady state of its reaction-diffusion model, namely Model (5.4), where the necessary and sufﬁcient conditions for diffusive instability are given by

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ fS þ g I < 0; f S g I fI g S > 0 and f S DI þ g I DS > DS DI ðfS g I fI g S Þ

which also implies that fS g I < 0. From Theorem 3.3, we know that if an interior equilibrium ðS ; I Þ is locally asymptotically stable then for the system (2.7), (2.8) or the system (2.9), (2.10).

1þh N ¼S þI > : 2

Thus, for the system (2.7), (2.8), its Jacobian matrix (3.2) evaluated at the interior equilibrium ðS ; I Þ gives

"

fS ¼ S 1 þ h 2N þ

#

bI

"

2

ðN Þ

2

3ðN Þ þ 2ð1 þ hÞN h ¼ 0 ) N ¼

1þh N ¼ þ 2

bd 1q

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 hÞ2 4ðbdÞ 1q 2

<

ð1 þ hÞ

< min

n

d ð1hÞ 4

q;

ð1 þ hÞ þ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 þ hÞ2 3h

o 2

3

;

for the existence

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 þ hÞ2 3h 3

:

For the system (2.9), (2.10), its Jacobian matrix (3.5) evaluated at ðS ; I Þ gives

fS ¼ S ð1 þ h 2N Þ and g I ¼ qI ð1 þ h 2N Þ which implies that fS < 0 and g I < 0 since N > 1þh . Therefore, we 2 conclude that the system (2.9), (2.10) does not support diffusive instability. h Remark: A direct application of the proof for Theorem 5.1 leads to the following statements:

k3 k4 ¼ K ¼ fS g I fI g S 2ðfS lI þ g I lS Þ þ 4lS lI :

3ðN Þ þ 2ð1 þ hÞN h > 0: N

Since

therefore fS > 0 requires that of N based on Theorem 3.3,

7 7 7 5

ðN hÞð1 N Þ N

2

¼

fS þ g I < 0 and f S g I fI g S > 0: Since the two-patch model (5.3) is symmetric, thus, ðS ; I ; S ; I Þ is its interior equilibrium. The Jacobian matrix of two-patch model (5.3) evaluated at ðS ; I ; S ; I Þ can be represented as follows:

¼ 1 þ h 2N þ

and g I ¼ I qð1 þ h 2N Þ

#

bS ðN Þ

2

which implies that g I < 0 since N > 1þh . Therefore the possibility of 2 diffusive instability in the system (2.7), (2.8) requires that fS > 0. Since

1. If fS > 0 and g I < 0, then diffusive instability for the patchy model (5.3) requires that llSI is large enough and lS < f2S while diffusive instability for the reaction–diffusion model (5.4) requires that DI is large enough. DS 2. If fS < 0 and g I > 0, then diffusive instability for the patchy model (5.3) requires llSI to be large enough and lI < g2I while diffusive instability for the reaction-diffusion model (5.4) requires DDSI to be large enough. In addition, Theorem 5.1 indicates that the SI system (2.7), (2.8) with frequency-dependent horizontal transmission can support diffusive instability under certain conditions. For example, when the system (2.7), (2.8) has b ¼ 1; d ¼ 0:85; q ¼ 0:05; h ¼ 0:2, then it has two interior equilibria

Ei1 ¼ ðS1 ; I1 Þ ¼ ð0:4666247185; 0:08749213472Þ and Ei2 ¼ ðS2 ; I2 Þ ¼ ð0:5439015973; 0:1019815495Þ where Ei1 is a saddle and Ei2 is locally asymptotically stable with

fS ¼ 0:0830521552 > 0;

f I ¼ 0:7590531079;

g S ¼ 0:02446282446 and g I ¼ 0:1334319123 < 0: Thus if we choose llSI (or DDSI ) large enough and lS < f2S then diffusive instability occurs. These results agree with the study of predator–prey systems by Timm and Okubo [53], which suggest that the existence of diffusive instability in such systems may require density effects on intra-speciﬁc coefﬁcients and on a predator’s diffusivity that must be sufﬁciently larger when compared to the prey’s. There is a critical value involving the ratio of the prey/predator diffusivities that must be crossed before diffusive instability sets in. An alternative SI model 5.5, 5.6 with Allee effects and horizontal and vertical transmission disease that can also support diffusive instability is given by the system

113

Y. Kang, C. Castillo-Chavez / Mathematical Biosciences 248 (2014) 97–116

dS ¼ SðS hÞð1 S IÞ bSI dt dI ¼ bSI dI þ qIðI hÞð1 S IÞ dt

then we obtain the following SI the system (5.11), (5.12) by letting q ¼ 0; a1 ¼ a2 ¼ 1:

ð5:5Þ ð5:6Þ

dS bSI ¼ f ðS; IÞ ¼ SðN hÞð1 NÞ dt N dI bSI ¼ gðS; IÞ ¼ dI; dt N

with a locally asymptotically stable interior equilibrium ðS ; I Þ given by

fS ¼ S ð1 N S þ hÞ;

f I ¼ S ðS þ b hÞ;

g S ¼ I ðqI þ b þ qhÞ;

For example, when b ¼ :1; h ¼ :2; d ¼ 0:095; q ¼ 0:001, the system (5.5), (5.6) has a unique locally asymptotically stable interior equilibrium ðS ; I Þ ¼ ð0:95; 0:044Þ with

fS ¼ 0:70680 < 0;

ðS ; I Þ ¼

f I ¼ 0:8075;

¼

g S ¼ 0:004406864 and g I ¼ 0:000006864 > 0:

ðS þ qIÞðS þ a1 I hÞð1 S a2 IÞ bSI 0;

if S ¼ 0 and ða1 I hÞð1 a2 IÞ 6 0

;

dI ¼ gðS; IÞ ¼ bSI dI; dt

N

2

! @f bI ðS ; I Þ ¼ S 1 þ h 2N þ 2 ; @S ðN Þ ! @f bS f I ¼ ðS ; I Þ ¼ S 1 þ h 2N 2 < 0 @I ðN Þ fS ¼

2

gS ¼

ð5:8Þ

@g bðI Þ ðS ; I Þ ¼ 2 > 0; @S ðN Þ

gI ¼

@g bS I ðS ; I Þ ¼ 2 < 0: ð5:13Þ @I ðN Þ

Thus, the system (5.11), (5.12) can have diffusive instability if

fS > 0 ) 1 þ h 2N þ

ð5:9Þ

bI 2

ðN Þ

¼ 1 þ h 2N þ

bI ðN Þ

2

2

where the assumptions of the model and the detailed biological meaning of parameters can be found in Kang and Castillo–Chavez [37], cannot support diffusive instability. The model can have a locally asymptotically stable interior equilibrium ðS ; I Þ with

¼

fI ¼ ½ðq þ a1 ÞS þ 2qa1 I hð1 S a2 I Þ

3ðN Þ þ 2ð1 þ hÞN h > 0: N

Therefore, according to the proof of Theorem 5.1 and the discussion on the system (2.7), (2.8), we can conclude that sufﬁcient conditions leading to diffusive instability are that llSI (or DDSI ) is large enough and the following inequalities hold

fS ¼ ½2S þ ða1 þ qÞI hð1 S a2 I Þ ðS þ qI ÞðS þ a1 I hÞ bI ;

fS d ; < 1; 2 b qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ h þ ð1 hÞ2 4ðb dÞ

ð5:10Þ

lS <

a2 ðS þ qI ÞðS þ a1 I hÞ bS ; g S ¼ bI ;

dN N ðN hÞð1 NÞ d where < 1; ; b b b qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ h þ ð1 hÞ2 4ðb dÞ

and

Thus, if we choose llSI (or DDSI ) large enough and lI < g2I , then diffusive instability occurs. This suggests that the existence of diffusive instability in 5.5, 5.6 requires that susceptible’s diffusivity is sufﬁciently larger than that of infectives with a critical value involving the ratio of the susceptible/infectives diffusivities moving beyond a threshold after which diffusive instability sets in. The SI system (5.8), (5.9) with Allee effects and disease modiﬁed ﬁtness studied by Kang and Castillo–Chavez [37] is given by

dS ¼ f ðS; IÞ ¼ dt

ð5:12Þ

who supports the unique locally asymptotically stable interior equilibrium

ð5:7Þ

g I ¼ qI ð1 N I þ hÞ:

ð5:11Þ

g I ¼ 0:

2

However, if we replace density-dependent transmission with frequency-dependent transmission in the SI the system (5.8), (5.9)

<

ð1 þ hÞ þ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 þ hÞ2 3h 3

:

Table 5.1 Summary of diffusive instability for SI models and Prey–Predator models. Models SI-Model 2.7, 2.8 SI-Model (2.9)– (2.10) SI-Model 5.5, 5.6 SI-Model 5.8, 5.9 SI-Model 5.11, 5.12 PP-Model (5.14) PP-Model (5.14) with d¼0 PP-Model (5.16)

fI

fS h fS ¼ S 1 þ h 2N þ

bI 2 ðN Þ

i

>0

h g I ¼ I qð1 þ h 2N Þ

bS 2 ðN Þ

i

Diffusive Instability

Potential Mechanisms

Yes

Asymmetricity and nonlinearity arise from frequency-dependent horizontal transmission Symmetricity arises from certain forms of vertical transmission with Allee effects; Linearity arises from density-dependent vertical transmission Asymmetricity and nonlinearity of arise from certain forms of vertical transmission with Allee effects Linearity arises from density-dependent horizontal transmission

<0 fS ¼ S ð1 þ h 2N Þ < 0

g I ¼ qI ð1 þ h 2N Þ < 0

No

fS ¼ S ð1 N S þ hÞ < 0

g I ¼ qI ð1 N I þ hÞ > 0

Yes

gI ¼ 0

No

fS ¼ ½2S þ ða1 þ qÞI h ð1 S a2 I Þ ðS þ qI ÞðS þ a1 I hÞ bI < 0

fS ¼ S 1 þ h 2N þ bI 2 > 0

fS ¼ ax < 0

g I ¼ dy > 0

Yes

Asymmetricity and nonlinearity arise from frequency-dependent horizontal transmission Nonlinearity arises from Allee effects in predator

fS ¼ ax < 0

gI ¼ 0

No

Linearity arises from a Holling Type I functional response

Yes

Asymmetricity and nonlinearity arise from a Beddington-DeAngelis type functional response

ðN Þ

h fS ¼ x a þ

bh1 y 2 ð1þh1 x þh2 y Þ

i

>0

g I ¼ bS I 2 < 0

Yes

ðN Þ

y ð1þh1 x Þ ð1þh1 x þh2 y Þ2

g I ¼ ch2 x

<0

114

Y. Kang, C. Castillo-Chavez / Mathematical Biosciences 248 (2014) 97–116

Since SI-disease and prey–predator interaction models share structural similarities, we ﬁrst look at the following two patch prey–predator model (5.14) with differential migration coefﬁcients l; m introduced by Levin [41]:

dxi ¼ xi ðK axi byi Þ þ lðxj xi Þ ¼ f ðxi ; yi Þ þ lðxj xi Þ dt dyi ¼ yi ðL þ cxi þ dyi Þ þ mðyj yi Þ ¼ gðxi ; yi Þ þ mðyj yi Þ dt

6. Conclusion

ð5:14Þ

which supports the equilibrium:

x ¼ x1 ¼ x2 ¼

Lb Kd ; bc ad

1 ¼ y 2 ¼ y ¼ y

Kc La bc ad

and

@f @f ðx ; y Þ ¼ ax < 0; f I ¼ ðx ; y Þ ¼ bx < 0 @xi @yi @g @g gS ¼ ðx ; y Þ ¼ cy > 0; g I ¼ ðx ; y Þ ¼ dy > 0: @xi @yi

fS ¼

ð5:15Þ

According to Theorem 5.1, we conclude that diffusive instability arises if m < g2I ; lm is large enough and the following equalities hold

a K b aðb þ dÞ ; < < min ; c L d dða þ cÞ

bc > ad:

Notice that the positivity of g I comes from the assumption that yi is able to survive without xi , i.e., yi ! 1 if yi ð0Þ > L=d. If there is no Allee effects, i.e., d ¼ 0, then the prey–predator Model (5.14) does not have diffusive instability. However, if we replace a Holling-Type I functional response with a Beddington–DeAngelis type functional response in the prey–predator Model (5.14) with d ¼ 0, then we obtain the following two-patch prey predator model that can have diffusive instability:

dxi byi þ lðxj xi Þ ¼ f ðxi ; yi Þ þ lðxj xi Þ ¼ xi K axi dt 1 þ h1 xi þ h2 yi dyi cxi þ mðyj yi Þ ¼ gðxi ; yi Þ þ mðyj yi Þ ¼ yi L þ dt 1 þ h1 xi þ h2 yi ð5:16Þ which supports a unique locally asymptotically stable interior equilibrium ðx ; y Þ whenever

ch2 > bh1 ;

cK > L; a þ h1 K

l¼m¼0

and

fS ¼

! @f bh1 y ; ðx ; y Þ ¼ x a þ 2 @xi ð1 þ h1 x þ h2 y Þ

fI ¼

@f bx ð1 þ h1 x Þ ðx ; y Þ ¼ <0 2 @yi ð1 þ h1 x þ h2 y Þ

gS ¼

@g cy ð1 þ h2 y Þ ðx ; y Þ ¼ > 0; 2 @xi ð1 þ h1 x þ h2 y Þ

gI ¼

@g ch2 x y ð1 þ h1 x Þ ðx ; y Þ ¼ < 0: 2 @yi ð1 þ h1 x þ h2 y Þ

Prey–predator models (5.14), (5.16)] and models not supporting diffusive instability [SI-models 2.9, 2.10, 5.8, 5.9; the prey–predator model (5.14) with d ¼ 0] are summarized in Table 5.1.

ð5:17Þ

For example, if K ¼ :5; L ¼ 0:01; c ¼ 2:5; a ¼ :1; h1 ¼ 1:5; h2 ¼ 1:; b ¼ 1; l ¼ m ¼ 0, then prey–predator Model (5.14) has a unique locally asymptotically stable interior equilibrium ðx ; y ; x ; y Þ ¼ ð0:008084; 1:008; 0:008084; 1:008Þ with fS ¼ 0:0022 > 0; g I ¼ 0:005. In this section, we have discussed diffusive instability in the context of ﬁve different SI-models and three different Prey–predator models. So, what are the criterions and mechanisms leading to diffusive instabilities? Comparisons between models supporting diffusive instability [SI-Models 2.7, 2.8, 5.5, 5.6, 5.11, 5.12;

Parasites and pathogens are sometimes effective ‘‘regulators of natural populations’’ [3,19]. Hence, it is of theoretical and empirical interest to study when multiple transmission modes are preferred; or whether pathogen/disease transmission depends on either host population density or its frequency; or the role of small populations (Allee effects) on populations living under the selection pressures placed by pathogens or parasites. Answers to such questions are needed to assess the role and impact of selection on populations, communities and ecosystems. In this manuscript, we explore the contributions of some of these factors on the dynamics of host-parasite interactions within a controlled setting, namely a general SI model that includes: (a) Horizontally and vertically-transmitted disease modes, (b) Net reproduction terms that account for the limitations posed by Allee effects, (c) Disease induced death rates, and (d) Diseasedriven reductions in reproduction ability. The analyses carried out in the prior sections leads to the following conclusions and observations: Density- versus frequency-dependent horizontal transmission: From Theorem 3.3, we know that the system (2.7), (2.8) can have two interior equilibria, one a saddle and one a locally asymptotically stable equilibrium. the system (2.9), (2.10) can support stable limit cycles that emerge via Hopf-Bifurcation (see Fig. 4.4 and 4.10). In other words, the SI model with density-dependent horizontal transmission turns out to support more complicated outcomes than its frequency-dependent counterpart. Effects of q; b; d and h: Rh0 ¼ bd is identiﬁed as the horizontaltransmission reproduction number and Rv0 ¼ qd as the verticaltransmission reproduction number. 1. Theorem 2.1 and its Corollary 2.2 assert that for the SI systems (2.7), (2.8) and (2.9), (2.10), sufﬁciently large initial conditions (Nð0Þ ¼ Sð0Þ þ Ið0Þ) and Rv0 can prevent extinction. 2. Proposition 3.1, Theorem 3.2 and Theorem 3.3 imply that whenever Rh0 6 1 we can expect disease-free dynamics in the systems (2.7), (2.8) and (2.9), (2.10). 3. The SI-FD given by the system (2.7), (2.8) tends to support susceptible-free dynamics under large values of Rv0 ; coexistence of susceptibles and infectives under intermediate values of Rv0 ; and disease-driven extinction for small values of Rv0 . 4. The SI-DD system (2.9), (2.10) supports the following outcomes: (i) No boundary equilibrium on the I-axis and possibly disease-driven extinction for a range of parameter 4 values whenever 1 < Rh0 < 1b and Rv0 < ð1hÞ 2 . (ii) Suscepti-

ble-free dynamics for Rv0 -intermediate values; values that n o 4 < Rv0 < c2 . (iii) An satisfy the inequality max ð1hÞ 2 ; c1 unique interior equilibrium whenever Rh0 > 1b and possibly two interior equilibria if 1 < Rh0 < 1b. (iv) Large values of

q; h; b and small values of d can destabilize the system (bifurcation diagrams; Fig. 4.9). Horizontal versus vertical modes of transmission in the SI systems: Small values of the horizontal transmission rate can lead to the disease-free dynamics for both the systems (2.7), (2.8) and (2.9), (2.10) with low reproductive rates for infectives

Y. Kang, C. Castillo-Chavez / Mathematical Biosciences 248 (2014) 97–116

leading, under certain conditions, to disease-driven extinction. The systems (2.7), (2.8) and (2.9), (2.10) have similar dynamics to those of the SI-model

dS ¼ SðN hÞð1 NÞ dt dI ¼ qIðN hÞð1 NÞ dI dt

ð6:1Þ

whenever the horizontal-transmission reproductive number is not greater than 1 (Rh0 6 1) and the vertical-transmission reproduction number is dominant. In this case, the system (6.1) has E0;0 [ E1;0 as its global attractor (disease-free dynamics). If horizontal transmission is dominant, that is, the vertical transmission rate is small due to the low reproductive ability (q) of those in4 fected (Rv0 P ð1hÞ 2 ) then the systems (2.7), (2.8) and (2.9), (2.10) have similar dynamics to those supported by the SI-Models (6.2) and (6.3), respectively:

dS bSI ¼ SðN hÞð1 NÞ dt N dI bSI ¼ dI dt N

ð6:2Þ

dS ¼ SðN hÞð1 NÞ bSI dt dI ¼ bSI dI: dt

ð6:3Þ

and

Model (6.2) and (6.3) can in fact have the disease-driven extinction, under some conditions. Diffusive instability: Sufﬁcient conditions leading to diffusive instability (Theorem 5.1 in Section 5) require that SI/Prey–Predator models support a locally asymptotically stable interior equilibrium with the product of the diagonal entries of the Jacobian matrix (evaluated at this interior equilibrium) being negative. In this manuscript, we have investigated possible ways for diffusive instability to emerge in ﬁve different SI-models while contrasting their behavior with those of three different prey– predator models. The results of these comparisons have been summarized (Table 5.1). From Table 5.1 we conclude that: 1. In the context of our SI models, asymmetricity and nonlinearity that emerge as a result of frequency-dependent horizontal transmission or some forms of vertical transmission in populations under Allee effects, can generate diffusive instability. 2. In the context of prey-predator models, asymmetricity and nonlinearity arising from certain forms of functional responses such as Beddington-DeAngelis type functional response or Allee effects in the predator population, can generate diffusive instability. In conclusion, the presence of asymmetricity and nonlinearity such as nonlinear density dependent factors including Allee effects, could be critical for the generation of diffusive instabilities in both SI models and prey–predator models.

References [1] W.C. Allee, The Social Life of Animals, Norton, New York, 1938. [2] S. Altizer, R. Bartel, B.A. Han, Animal migration and infectious disease risk, Science 331 (2011) 296. [3] R.M. Anderson, R.M. May, Regulation and stability of host-parasite population interactions I: regulatory processes; II: destabilizing processes, J. Anita. Ecol. 47 (219–247) (1978) 249. [4] R.M. Anderson, R.M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, London, 1991.

115

[5] J. Antonovics, H.M. Alexander, Epidemiology of Anther-Smut infection of Silene-alba (= S-Latifolia) caused by Ustilago–Violacea-patterns of spore deposition in experimental populations, Proc. R. Soc. Lond. B 250 (1992) 157. [6] D.S. Boukal, L. Berec, Single-species models of the Allee effect: extinction boundaries, sex ratios and mate encounters, J. Theor. Biol. 218 (2002) 375. [7] F.S. Berezovskaya, G. Karev, B. Song, C. Castillo-Chavez, A simple epidemic model with surprising dynamics, Math. Biosci. Eng. 1 (2004) 1. [8] F.S. Berezovskaya, B. Song, C. Castillo-Chavez, Role of prey dispersal and refuges on predator–prey dynamics, SIAM J. Appl. Math. 70 (2010) 1821. [9] F. Brauer, C. Castillo-Chavez, Mathematical models in population biology and epidemiology, Texts in Applied Mathematics, second ed., Springer-Verlag, 2012. 530 p.. [10] F. de Castro, B. Bolker, Mechanisms of disease-induced extinction, Ecol. Lett. 8 (2005) 117. [11] A. Deredec, F. Courchamp, Combined impacts of Allee effects and parasitism, OIKOS 112 (2006) 667. [12] F. Courchamp, L. Berec, J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, 2009. [13] J.M. Cushing, Oscillations in age-structured population models with an Allee effect. Oscillations in nonlinear systems: applications and numerical aspects, J. Comput. Appl. Math. 52 (1994) 71. [14] P. Daszak, L. Berger, A.A. Cunningham, A.D. Hyatt, D.E. Green, R. Speare, Emerging infectious diseases and amphibian population declines, Emerging Infect. Dis. 5 (1999) 735. [15] E. Diaz, Dissertation: Diffusive Instability And Aggregation In Epidemiology, Arizona State University, 2010. [16] J.M. Drake, Allee effects and the risk of biological invasion, Risk Anal. 24 (2004) 795. [17] O. Diekmann, M. Kretzshmar, Patterns in the effects of infectious diseases on population growth, J. Math. Biol. 29 (1991) 539. [18] A. Drew, E.J. Allen, L.J.S. Allen, Analysis of climate and geographic factors affecting the presence of chytridiomycosis in Australia, Dis. Aquat. Org. 68 (2006) 245. [19] G. Dwyer, S.A. Levin, L. Buttel, A simulation model of the population dynamics and evolution of myxomatosis, Ecol. Monogr. 60 (1990) 423. [20] K.E. Emmert, L.J.S. Allen, Population persistence and extinction in a discretetime stage-structured epidemic model, J. Differ. Equ. Appl. 10 (2004) 1177. [21] M.J. Ferrari, S.E. Perkins, L.W. Pomeroy, O.N. Bjornstad, Pathogens, social networks, and the paradox of transmission scaling, Interdiscip. Perspect. Infect. Dis. 10 (2011), http://dx.doi.org/10.1155/2011/267049. 10 p.. [22] A. Friedman, A.-A. Yakubu, Fatal disease and demographic Allee effect: population persistence and extinction, J. Biol. Dyn. (2011), http://dx.doi.org/ 10.1080/17513758.2011.630489. [23] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, 1983. [24] F.M.D. Gulland, The role of nematode parasites in Soay sheep (Ovis-aries L) mortality during a population crash, Parasitology 105 (1992) 493. [25] F.M. Hilker, M.A. Lewis, H. Seno, M. Langlais, H. Malchow, Pathogens can slow down or reverse invasion fronts of their hosts, Biol. Invasions 7 (2005) 817. [26] F.M. Hilker, M. Langlais, S.V. Petrovskii, H. Malchow, A diffusive SI model with Allee effect and application to FIV, Math. Biosci. 206 (2007) 61. [27] F.M. Hilker, M.l. Langlais, H. Malchow, The Allee effect and infectious diseases: extinction, multistability, and the (dis-)appearance of oscillations, Am. Nat. 173 (2009) 72. [28] F.M. Hilker, Population collapse to extinction: the catastrophic combination of parasitism and Allee effect, J. Biol. Dyn. 4 (2010) 86. [29] W.M. Hochachka, A.A. Dhondt, Density-dependent decline of host abundance resulting from a new infectious disease, Proc. Natl. Acad. Sci., USA 97 (2000) 5303. [30] P.J. Hudson, A.P. Dobson, D. Newborn, Prevention of population cycles by parasite removal, Science 282 (1998) 2256. [31] P.J. Hudson, A. Rizzoli, B.T. Grenfell, H. Heesterbeek, A.P. Dobson, The Ecology of Wildlife Diseases, Oxford Press, Oxford, UK, 2002. [32] V. Hutson, A theorem on average Liapunov functions, Monatsh. Math. 98 (1984) 267. [33] S.R.-J. Jang, S.L. Diamond, A host-parasitoid interaction with Allee effects on the host, Comput. Math. Appl. 53 (2007) 89. [34] Y. Kang, D. Armbruster, Dispersal effects on a two-patch discrete model for plant-herbivore interactions, J. Theor. Biol. 268 (2011) 84. [35] Y. Kang, N. Lanchier, Expansion or extinction: deterministic and stochastic two-patch models with Allee effects, J. Math. Biol. 62 (2011) 925. [36] Y. Kang, C. Castillo-Chavez, Multiscale analysis of compartment models with dispersal, J. Biol. Dyn. 6 (2) (2012) 50. [37] Y. Kang, C. Castillo-Chavez, A simple epidemiological model for populations in the wild with Allee effects and disease-modiﬁed ﬁtness, J. Discrete Continuous Dyn.Syst. -B 19 (1) (2013) 89. [38] Y. Kang, C. Castillo-Chavez, A simple two-patch epidemiological model with Allee effects and disease-modiﬁed ﬁtness, The special AMS Contemporary Math Series in honor of Ronald Mickens’ 70th birthday, 2013 (in press). [39] A.M. Kilpatrick, S. Altizer, Disease ecology, Nat. Educ. Knowl. 3 (10) (2012) 55. [40] A.M. Kramer, B. Dennis, A.M. Liebhold, J.M. Drake, The evidence for Allee effects, Popul. Ecol. 51 (2009) 341. [41] S.A. Levin, Dispersion and population interactions, Am. Nat. 108 (1974) 207. [42] S.A. Levin, L.A. Segel, Pattern generation in space and aspect, SIAM Rev. 27 (1985) 45. [43] J.D. Murray, Mathematical Biology, I & II, Springer, New York, NY, 2003.

116

Y. Kang, C. Castillo-Chavez / Mathematical Biosciences 248 (2014) 97–116

[44] A. Potapov, E. Merrill, M. Lewis, Wildlife disease elimination and density dependence, Proc. R. Soc. B (2012), http://dx.doi.org/10.1098/rspb.2012.0520 (Published online May 16). [45] K.R. Rios-Soto, C. Castillo-Chavez, M. Neubert, E.S. Titi, A.-A. Yakubu, Epidemic spread in populations at demographic equilibrium, in: A. Gumel, C. CastilloChavez, D.P. Clemence, R.E. Mickens (Eds.), Mathematical Studies on Human Disease Dynamics: Emerging Paradigms and Challenges, American Mathematical Society, 2006, p. 297. [46] J. Santaella, C.B. Ocampo, N.G. Saravia, F. Mndez, R. Gngora, M.A. Gomez, L.E. Munstermann, R.J. Quinnell, Leishmania (Viannia) infection in the domestic dog in Chaparral, Colombia, Am. J. Trop. Med. Hyg. 84 (5) (2011) 674. [47] L.A. Segel, J.L. Jackson, Dissipative structure: an explanation and an ecological example, J. Theor. Biol. 37 (1972) 545. [48] L.A. Segel, S.A. Levin, Application of nonlinear stability theory to the study of the effects of diffusion on predator–prey interactions, in: R.A. Piccirelli (Ed.),

[49] [50] [51] [52]

[53] [54]

Topics in Statistical Mechanics in Biophysics: a memorial to Julius L. Jackson, AIP Conference Proceedings, vol. 27, 1976, pp. 123–152. K.F. Smith, D.F. Sax, K.D. Lafferty, Evidence for the role of infectious disease in species extinction and endangerment, Conserv. Biol. 20 (2006) 1349. P.A. Stephens, W.J. Sutherland, Consequences of the Allee effect for behaviour, ecology and conservation, Trends Ecol. Evol. 14 (1999) 401. P.A. Stephens, W.J. Sutherland, R.P. Freckleton, What is the Allee effect?, Oikos 87 (1999) 185. H.R. Thieme, T. Dhirasakdanon, Z. Han, R. Trevino, Species decline and extinction: synergy of infectious disease and Allee effect?, J Biol. Dyn. 3 (2009) 305. U. Timm, A. Okubo, Diffusion-driven instability in a predator–prey system with time-varying diffusivities, J. Math. Biol. 30 (1992) 307. A.-A. Yakubu, Allee effects in a discrete-time SIS epidemic model with infected newborns, J. Differ. Equ. Appl. 13 (2007) 341.

Dynamics of SI models with both horizontal and vertical transmissions as well as Allee effects

Dynamics of SI models with both horizontal and vertical transmissions as well as Allee effects

Recommend Documents