Abstract
A model for bacteriophage infections and bacteria defense is analyzed using the concepts of synergetics. The model order parameter is determined and the corresponding amplitude equations are derived. Within this framework it is shown how the order parameter defines a multi-species building block that captures the organization of infection outbreaks and the initial defense reaction and how the order parameter amplitude determines the corresponding temporal characteristics. Two approximative models with different domains of application are derived as well. In doing so, a supplementary perspective of bacteriophage infections that provides insights beyond the classical state space perspective is provided.
1 Introduction
In general, epidemiological systems are given by complex networks of interacting populations of different species (). While the isolated populations typically exhibit a relatively simple dynamics, a challenge in the field of network physiology is to understand how the interactions between the different types of species shape the overall network dynamics. In this context, an important first step is to consider mean field approximations in terms of ODE and coupled differential equation models (; ), which due to their relative simplicity frequently allow for analytical solution methods. For virus infections the ODE three-species TIV model captures the basic dynamics of interacting target cells, infected cells, and virus particles (). Likewise, in the context of bacteriophage infections, we are dealing with target bacteria, infected bacteria, and bacteriophages–the latter act as viruses. Studying bacterial infections and the role of bacteriophages is an important task (; ; ) and is an indispensable step when attempting to use phages in modern medicine to cure certain diseases in humans (; ). To this end, both simplified and generalized three-species models have been studied in the literature (; ; ). In particular, as part of their comprehensive study, considered bacteria that can exist as resistant and non-resistant phenotypes with respect to a given invading phage. Increasing the concentration of resistant phenotypes is a defense mechanism against phage attack because this mechanism decreases the effective contact rate between phage and non-resistant bacteria such that under appropriate conditions the infection dies out. While the dynamics systems perspective, in general, is an indispensable tool to analyze bacteriophage infection models, little attention has been paid to utilize the more specific dynamical systems concepts of synergetics (; ; ; ) in this regard. However, in the wake of the COVID-19 pandemic it has been shown that synergetics can supplement existing dynamic systems approaches to understand epidemiological and virus dynamic models (). In this Brief Report a parsimonious four-species model for bacteriophage infection and bacteria defense will be considered that involves resistant phenotypes as in . The relative simplicity of the model will allow for an analytical approach. The aim of the study is to derive explicitly the order parameter of the model and to show how it determines the initial organization of a phage attack and the corresponding defense reaction. Moreover, the aim is to derive the amplitude equations that determine the evolution of the system along the order parameter and the remaining directions. Two approximative models for the system dynamics in this context will be derived as well. An exemplary simulation will illustrate some of the analytical results.
2 Materials and methods
As reviewed above, our starting point is the three-species model that involves susceptible and infected bacteria and phage load . In line with the TIV model of virus dynamics () the model equations readwith , where denotes time, describes the -dependent transition rate of transitions, and denote the decay rates of infected cells and phages, respectively, and describes the production rate of phages per infected bacteria. The evolution equation for involves a logistic growth term with the growth rate and the capacity . Below we will consider both the case when the growth term can be neglected and the more general case . By comparing these two cases, we will see that the bacterial growth dynamics actually has no effect on the initial outbreaks dynamics captured by the order parameter. Therefore, the growth term may be neglected when (i) changes in are primarily due to transitions capture by the -term or (ii) the focus is on the initial phase of the outbreak dynamics. The transition rate in Equation 1 depends on the infecting species like , where denotes the effective contact rate (). In order to take the active defense mechanism mentioned in the introduction into account, the model (1) was modified in two ways. First, it was assumed that when infected bacteria emerge in the bacteria population then resistant bacteria mutations are grown likewhere and denote the growth rate and the maximal concentration of resistant bacteria, respectively. From a mechanistic point of view, Equation 2 captures the adaptive defense mechanism of bacteria via the so-called CRISPR system (; ). The CRISPR system allows bacteria to memorize attacking phages such that they become immune against future attacks. In doing so, in the presence of invading phages phage-resistant bacteria emerge. Second, in general, there are several mechanism with which resistant bacteria may slow down or stop a bacteriophage infection (). Again, for sake of brevity, only the effect of on the transition rate was considered. By doing so, the transition rate becomes a function of and and reads ()where measures the effectivity of the active defense mechanism. Basically, Equation 3 states that the presence of resistant cells lowers the chance of an effective contact between phages and susceptible bacteria, such that the -dependent effective contact rate reads .
Taking a synergetics perspective (; ), for the model (1–3) the order parameter and the amplitude equations were derived. To this end, using the state vector , bacteriophage infections were considered that start close to an initial fixed point (see Results and Discussions section) of the model. Subsequently, with the help of the eigenvectors obtained from a linear stability analysis the amplitudes were implicitly defined by
By constructing a bi-orthogonal basis (; ) spanned by the vectors with (Kronecker symbol), the amplitudes were explicitly expressed likewhere denotes the difference vector . From the model Equations 1–3 and Equation 5, eventually the model amplitude equations of the formwere derived for with constituting the amplitude vector . In Equation 6 denote the eigenvalues of the system and are nonlinear functions in the amplitudes. The order parameter and its order parameter amplitude were identified as the eigenvector and its amplitude corresponding to the potentially positive eigenvalue of the model (; ).
3 Results and discussions
3.1 Amplitude equation perspective
The fixed-point analysis showed that the model (1–3) exhibits the phage-free fixed points defined by
for . For Equation 7 holds with . As mentioned in the Methods section, it is assumed that at time , i.e., before the infection takes place, the system stays in a fixed point (7). The fixed point is referred to as initial fixed point and denoted by . At time the bacteria population is infected by phages of concentration such that the state is shifted out of its fixed point. Consequently, the model describes infection outbreaks that begin with an initial phage infection of at time and end in a phage-free state defined by Equation 7 or an endemic state if it exists (see below).
The linear stability analysis at showed that the model for exhibits the eigenvalues , , as well aswith , where the upper (lower) sign holds for . For in Equation 8 and in what follows we must substitute . It can be shown that for arbitrary model parameters holds. In contrast, can assume positive or negative values. In this context, note that using the next-generation method, the basic reproduction number of the model () can be obtained as . Case I is defined by , which is equivalent to , such that the fixed point is a neutrally stable/asymptotically stable fixed point for and , respectively. There is no infection outbreak. Tn contrast case II is characterized by , which is tantamount to say that holds. The fixed point is unstable. The infection dynamics describes an infection outbreak. The inequality means that the infection outbreak scenario, i.e., case II, occurs when the system parameters and are relatively large, the initial value is relatively large, while the initial concentration is relatively small. For the model exhibits only phage-free fixed points. A detailed calculation shows that for an endemic fixed point exists if the defense mechanism via the dynamics cannot stabilized the phage-free fixed point with . Mathematically speaking, if holds for , which is equivalent to say that holds (where ), then an endemic fixed point with and exists. Having said that since the objective of the study is examine initial infection outbreaks from the phage-free state, we will not dwell on the endemic state.
The linear stability analysis of the phage-free fixed point produced the eigenvectors and associated with the zero eigenvalues and . For the eigenvectors read as shown in Equation 9with , where is a normalization constant such that . It was found that the bi-orthogonal vectors of the model exhibit the well-known structure from other epidemiological models (): and , where and denote the and coordinates of the eigenvectors , respectively. Here . A detailed calculation showed that and associated with and , respectively, read as shown in Equation 10with . As in other virus dynamics models (), the nonlinear functions occurring in the amplitude Equation 6 can be expressed as projections of a nonlinear vector-valued function on the biorthogonal vectors likewith . That is, are the coordinates of the difference vector introduced in the Methods section. A detailed calculation showed the results shown in Equations 12, 13 thatand
As indicated in Equation 11, the variables are expressed in terms of . Explicitly, we have , see Equation 4. Consequently, the amplitude equations defined by Equation 6 and (11–13) form a closed set of coupled differential equations.
3.2 Implications
3.2.1 Order parameter: essential building-block and initial organization
The model exhibits maximally one positive eigenvalue. Consequently, under the case II scenario with the system exhibits an order parameter given by and the order parameter amplitude (; ). Let us split the state dynamics into two parts , where for and for describes the dynamics along the stable direction(s) and captures the remaining dynamics. Initially, i.e., for , we havewith constant and for , whereas for . Equation 14 describes the dynamics along the unstable direction away from the initial fixed point (i.e., the outwards dynamics). In contrast, describes the dynamics towards the unstable direction, i.e., towards the order parameter. Consequently, the order parameter describes the emergent organization of the multi-species physiological network and its amplitude describes how this organization evolves over time.
Since initially decays in magnitude over time, when considering the initial infection dynamics we may neglect its contribution to the state dynamics. If so, then any state change defined by approximately is given by
Equation 15 illustrates again that the order parameter describes the essential building-block that shapes an infection outbreak in the multi-species network under consideration including the defense reaction (see component ).
3.2.2 Stopping mechanisms
The exponential increase along as described by Equation 15 is slowed down and eventually stopped at some point in time. In line with the stability analysis let us assume that are small quantities of the order . Then when interpreting as a function of the difference variables can be expanded such that the amplitude equation for reads
Note that for any . Consequently, the network physiology produces two mechanisms that slow down the exponential increase of : the decay in susceptibles in the presence of phages as measured by the interaction term and the increase of the number of resistant bacteria again in the presence of phages as measured by the interaction term . The former mechanism is a passive mechanism that simply states that the exponential infection outbreak slows down due to a decay of the resources (i.e., susceptible bacteria). The latter mechanism is an active mechanism that states that the introduction of resistant bacteria mutations into the bacteria colony has the desired effect of slowing down the phages invasion.
3.2.3 Linear predictor equations
Equation 15 implies that all species initially satisfy linear regression equations of the form as shown in Equation 17
Accordingly, any species of the network can be used to predict any other network species (assuming for all ). The network components are coupled by linear order parameter links. For example, the bacteriophage population size may be used to construct regression models likewhere are intercept parameters depending on . As indicated in Equation 18, because of .
3.2.4 Limited impact of bacterial growth term
Clearly, the bacterial growth term may affect the dynamics. However, it does not affect the order parameter eigenvalue and it does not affect the orientation of the order parameter in the 3D space , which is of primary concern. Consequently, the initial outbreak dynamics in the space as determined by the order parameter dynamics (see Section 3.2.1) is completely unaffected by the bacterial growth term.
3.3 2D infected/infectious species dynamics and double exponential dynamics approximation
The dynamics of the infected and infectious species and is completely described by the amplitudes and . The reason for this is that the eigenvectors and do not exhibit components in the subspace. The mapping from and to and readswhere and denote the projections of and into the - subspace. From Equation 19 it follows that the initial evolution of and satisfies a double-exponential function as shown in Equation 20
3.4 Scaled model
Using the variable transformations , , , , the model (1–3) becomeswithand , , and . Among other things, the scaled model exhibits the following two properties. First, the variables are dimensionless. Second, the variable transformation turns the phage variable into a bacteria-like variable (). That is, just as the model describes that a susceptible bacteria turns into an infected bacteria, the scaled model describes that an infected bacteria turns into a phage unit in a 1:1 manner when expressing phages in the variable (rather than in ). Mathematically speaking, from Equation 21 it follows that increases due to the term at the same rate as the number of infected bacteria decays due to the term , which means that the model describes the aforementioned 1:1 transition.
3.5 Simulation
An Euler forward simulation scheme was used to solve Equations 21 and 22. In a first simulation, see Figure 1, only the passive defense mechanism was considered with and . In a second simulation, see Figure 2, the active defense mechanism was taken into account with and . The remaining parameters and initial conditions were /days (), /day, /day, , and . For sake of brevity, only the most relevant phase curves will be presented and only the first week of the initial outbreak stage will be considered. In this context note that in line with our discussion in Section 3.2.4. In both simulations was used.
FIGURE 1
FIGURE 2
As can be seen in panels (a) and (b) of Figures 1, 2, the phase curves quickly converge towards the order parameter and, subsequently, evolve along . By definition, the order parameter does not capture the dynamics along the stable direction . As can be seen in panels (c) and (d) of Figures 1, 2, the double exponential approximations can describe the transient initial dynamics towards the order parameter (i.e., the dynamics along ) as well as the subsequent dynamics along the order parameter .
Comparing Figures 1, 2, it can be seen that due to the impact of the phage resistant bacteria the actual dynamics differs from the order parameter dynamics to a greater extent. Likewise, the actual dynamics departs earlier from the double exponential approximations. These observations do not come as a surprise since the active defense mechanism slows down and eventually stops the infection outbreak, see Equation 16. Therefore, the actual dynamics will deviate earlier from the order parameter dynamics, on the one hand, and the double exponential dynamics, on the other hand. Roughly speaking, the active mechanisms weakens the linear order parameter link between the network components.
4 Conclusions and limitations
We conclude that under appropriate conditions as specified in the Methods and Simulation sections the order parameter and its amplitude characterize the (self-)organization of a bacteriophage infection and the corresponding bacterial defense. In physics various experiments have been conducted to test specifically predictions of the theory of self-organization (and synergetics) as presented above. Therefore, just as in physics, the results presented above may serve as a basis for conducting laboratory experiments on bacteriophage infections to test the order parameter hypothesis. Moreover, we conclude that linear regression models as derived above may be used to estimate species that are difficult to observe on the basis of species that can be measured more conveniently. For sake of brevity, in the current study, properties of the endemic fixed point as studied, e.g., by have not been examined in detail. Likewise, the current study was limited to consider one possible defense mechanism while alternative mechanisms (; ) were ignored. A more comprehensive study (which is beyond the scope of this Brief Report) may overcome such limitations by generalizing the results presented above.
Statements
Data availability statement
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.
Author contributions
TF: Writing – original draft, Writing – review and editing.
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References
1
AbedonS. T. (2012). Bacterial immunity against bacteriophages. Bacteriophage2, 50–54. 10.4161/bact.18609
2
BlochS.WegrzynA. (2024). Editorial: bacteriophage and host interactions. Front. Microbiol.15, 1422076. 10.3389/fmicb.2024.1422076
3
FrankT. D. (2022). COVID-19 epidemiology and virus dynamics: nonlinear physics and mathematical modeling. Berlin: Springer. 10.1007/978-3-030-97178-6
4
GengP.FlintE.BernsmeierC. (2022). Plasticity of monocytes and macrophages in cirrhosis of the liver. Front. Netw. Physiol.2, 937739. 10.3389/fnetp.2022.937739
5
GrangerT.MichelitschT. M.BestehornM.RiascosA. P.ColletB. A. (2024). Stochastic compartment model with mortality and its application to epidemic spreading in complex networks. Entropy26, 362. 10.3390/e26050362
6
HakenH. (2004). Synergetics: introduction and advanced topics. Berlin: Springer. 10.1007/978-3-662-10184-1
7
HuttA.HakenH. (2020). Synergetics. New York: Springer. 10.1007/978-1-0716-0421-2
8
LiX.HuangR.HeM. (2021). Dynamics model analysis of bacteriophage infection of bacteria. Adv. Differ. Equations2021, 488. 10.1186/s13662-021-03466-x
9
NowakM. A.MayR. M. (2000). Viral dynamics: mathematical principles of immunology and virology. New York: Oxford University Press.
10
Pastor-SatorrasR.CastellanoC.Van MieghemP.VespignaniA. (1998). Epidemic processes in complex networks. Rev. Mod. Phys.87, 925–979. 10.1103/RevModPhys.87.925
11
SkanataA.KussellE. (2021). Ecological memory preserves phage resistance mechanisms in bacteria. Nat. Commun.12, 6817. 10.1038/s41467-021-26609-w
12
UhlC. (1999). Analysis of neurophysiological brain functioning. Berlin: Springer. 10.1007/978-3-642-60007-4
13
WeitzJ. S.HartmanH.LevinS. A. (2005). Coevolutionary arms races between bacteria and bacteriophage. Proc. Natl. Acad. Sci. U. S. A.102, 9535–9540. 10.1073/pnas.0504062102
14
WunnerG.PelsterA. (2016). Self-organization in complex systems: the past, present, and future of synergetics. Berlin: Springer. 10.1007/978-3-319-27635-9
15
ZborowskyS.SeuratJ.BalacheffQ.EcomardS.MuletC.MinhC. N. N.et al (2025). Macrophage-induced reduction of bacteriophage density limits the efficacy of in vivo pulmonary phage therapy. Nat. Commun.16, 5725. 10.1038/s41467-025-61268-1
Summary
Keywords
network physiology, bacteriophages, infection dynamics, order parameters, synergetics
Citation
Frank TD (2025) Analysis of a model for bacteriophage infections and bacteria defense: a synergetics perspective. Front. Netw. Physiol. 5:1657313. doi: 10.3389/fnetp.2025.1657313
Received
01 July 2025
Accepted
08 September 2025
Published
19 September 2025
Volume
5 - 2025
Edited by
Eckehard Schöll, Technical University of Berlin, Germany
Reviewed by
Christian Uhl, Ansbach University of Applied Sciences, Germany
Marwa Ali, Purdue University, United States
Updates
Copyright
© 2025 Frank.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: T. D. Frank, till.frank@uconn.edu
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.