A bacteriophage (in short, phage) is a virus that can infect and replicate within bacteria. Bacteriophages were discovered by Frederick Twort in 1915 [1] and Felix d’Herelle in 1917 [2]. Bacteriophages consist of a core of genetic material (nucleic acid) surrounded by a protein capsid. Usually a phage follows one of two life cycles: lytic (virulent) or lysogenic (temperate). When lytic phages infect bacteria, they attach their tails to the bacterial surface and then inject DNA from their heads into the bacteria. The DNA that enters the bacteria, being biosynthesised of the raw material of the host bacteria, causes proliferation of new daughter phages. Reaching a certain amount, these phages give rise to bacterial cells bursting, after which they can multiply rapidly and form hundreds of daughter phage particles. Lysogenic phages incorporate their nucleic acid into the chromosome of the host cell and replicate with it as a unit without destroying the cell. Under certain conditions, lysogenic phages can be induced for which a lytic cycle takes place. Each daughter phage can infect another bacterial cell, and the process is repeated over and over again, so that the phages can kill many cells. It has long been noted that phage therapy can be used to treat pathogenic bacterial infections. In particular, phage therapy has been widely used in livestock and poultry breeding (see [3, 4]), aquaculture (see [5]), food production (see [6]), and other fields (see [7, 8]). Also, phage therapy proved itself as an important tool in the treatment of human diseases such as diarrheal diseases caused by e. coli, shigella or vibrio, and wound infections caused by skin facultative pathogens (for example, staphylococcus and streptococcus). In recent years, phage therapy has also been used for systemic and even intracellular infections. It has been 100 years since the discovery of phage, and the research on phage has never stopped. With the emergence of antibiotics, phage therapy was gradually ignored. However, the widespread antibiotic resistance of bacteria in recent years has made the research on bacteriophages a hot spot. To which extent phages can replace antibiotics as a new treatment for bacterial infections is still under debate, and probably it will take some time for them to appear in clinical trials. However, finding a theoretical basis for studying the relationship between phage and bacteria is a problem of great importance and formidable complexity. This paper attempts to model the dynamic stability of phage infected bacteria, to discuss the relationship between phage and bacteria, and to provide some theoretical basis for whether phages can effectively prevent, control, and treat infectious diseases.

Phages can be thought of as organisms that prey on bacteria, or parasites that host bacteria. To the best of our knowledge, there are many dynamics models of viral infection in host cells (see [9–20] and the references therein). For example, in [11], Ebert et al. considered a model of microparasite transmission for a horizontally transmitted parasite and established the host-born density-dependent cabin model by ignoring the possibility of host recovery. The model equations are

$$ \textstyle\begin{cases} \dot{x}=r(x+fy)[1-c(x+y)]-\mu x-\beta xy, \\ \dot{y}=\beta xy-(\mu +\nu )y, \end{cases} $$

(1)

where \({x,y}\) are the densities of uninfected (susceptible) and infected (infective) hosts at time *t*, respectively; *r* is the maximum per capita birth rate of uninfected hosts; *f* is the relative fecundity of infected hosts; *c* measures the per capita density-dependent reduction in birth rate; *μ* is the parasite-independent host background mortality; *β* is the infection rate constant; and *ν* is the parasite-induced excess death rate.

This model predicts the existence of a stable equilibrium of infected and uninfected hosts, and the population is predicted to approach this equilibrium either monotonically or by damped oscillations.

In this paper, we modify Ebert’s model by assuming that infected hosts (bacteria) are capable of reproducing with logistic law, and investigate a class of parasites (phages) infection models. The model is given by the following system of differential equations:

$$ \textstyle\begin{cases} \dot{S}=r_{1}S (1-\frac{S+I}{M} )-d_{1}S-\beta SI, \\ \dot{I}= \beta SI+r_{2}I (1-\frac{S+I}{M} )-(d_{1}+ \varepsilon )I. \end{cases} $$

(2)

Here, \(r_{1}\), \(r_{2}\) are the proliferation constants of uninfected and infected hosts, *M* is the environmental tolerance of a host population, \(d_{1}\) stands for the phage-independent bacteria background mortality, *β* is the proportionality coefficient of parasite infection, *ε* denotes the phage-induced excess death rate. In this model, we assume that the total hosts population *N* is composed of two population classes: the first one consists of uninfected hosts (denoted *S*), while the second one consists of the virus infected hosts (denoted *I*). Thus, \(N(t)=S(t)+I(t)\). We make the following assumption: both uninfected and infected hosts are capable of reproducing with logistic law, and the logistic growth of the uninfected bacteria and infected bacteria is given by \(r_{1}S(t)[1-(S(t) + I(t))/M]\) and \(r_{2 }I(t)[1- (S(t) + I(t))/M]\), respectively.

To establish our results, the existence and number of steady states, as well as local stability and global stability of uninfected and infected steady states, are analyzed by the Jacobian matrix and Bendixson–Dulac theory. Under certain assumptions (reasonable from the biological viewpoint), we derive the basic reproduction number \(R_{0}\): \(E_{1}\) is locally asymptotically stable if \(R_{0}<1\), and \(E_{1}\) is unstable if \(R_{0}>1\). Our theoretical discoveries are supported by numerical simulations.

At present, there is not much work on mathematical modeling of phage infection bacteria, so our study has certain significance. In particular, the parasite (phage) population is not explicitly modeled in this model, which is a striking feature of this model.

The organization of this paper is as follows. In Sect. 2, we discuss the positively invariant set and equilibria. In Sect. 3, we give local and global stability analysis. In Sect. 4, we derive the basic reproduction number. Then, in Sect. 5, we give numerical simulations to support our main result. Section 6 ends the paper with a discussion about phage therapy that could be a new savior for patients infected with superbugs.