Models for an Arenavirus Infection in a Rodent Population: Consequences of Horizontal, Vertical and Sexual Transmission

Abstract

Most arenaviruses are associated with rodent-transmitted diseases in humans. Five arenaviruses are known to cause human illness. Lassa virus, Junin virus, Machupo virus, Guanarito virus, and Sabia virus. In this investigation, we model the spread of Machupo virus in its rodent host Calomys callosus. Machupo virus infection in humans is known as Bolivian hemorrhagic fever (BHF). The mortality rate associated with BHF is approximately 5-30% [30].

Machupo virus is transmitted among rodents through horizontal (direct contact), vertical (infected mother to offspring), and sexual transmission. The immune response differs among rodents infected with Machupo virus. Either rodents develop immunity and recover (immunocompetent) or they do not develop immunity and remain infected (immunotolerant). We formulate a general deterministic model for male and female rodents consisting of eight differential equations, four for females and four for males. The four states represent susceptible, immunocompetent, immunotolerant and recovered rodents, denoted as S, I^t, I^c and R, respectively. A unique disease-free equilibrium (DFE) is shown to exist and a basic reproduction number R₀ is computed using the next generation matrix approach. The DFE is shown to be locally asymptotically stable if R₀ < 1 and unstable if R₀ > 1. The dependence of R₀ on model parameters important in the control of Machupo virus is discussed.

Special cases of the general model are studied, where there is only one immune stage, either I^t or I^c. In the first model, SI^c R^c, it is assumed that all infected rodents are immunocompetent and recover. In the second model, SI^t, it is assumed that all infected rodents are immunotolerant. For each of these two models, the basic reproduction numbers R₀ are computed and their relationship to the basic reproduction number of the general model determined. For the SI^t model, it is shown that bistability may occur, where the DFE and an endemic equilibrium (EE), with all rodents infectious, are locally asymptotically stable for the same set of parameter values. A simplification of the SI^t model yields a third model, where the sexes are not differentiated, and therefore, there is no sexual transmission. For this third simplified model, the dynamics are completely analyzed. It is shown that there exists a DFE and possibly two endemic equilibria, one of which is globally asymptotically stable for any given set of parameter values, bistability does not occur. Numerical examples illustrate the dynamics of the models. The biological implications of the mathematical results and future research goals are discussed in the conclusion.

1. Introduction

Arenaviruses are a family of viruses, some of which are associated with rodent-transmitted diseases in humans. Each virus is associated with a particular rodent reservoir. The arenaviruses are divided into two groups: the New World and the Old World. Lassa virus is the only Old World virus known to cause human disease. New World viruses that cause human disease are Junin virus, Machupo virus, Guanarito virus, and Sabia virus [26]. Our goal is to model the spread of Machupo virus in its rodent reservoir, identified as the vesper mouse, Calomys callosus [13, 14]. Machupo virus infection in humans is known as Bolivian hemorrhagic fever (BHF), named because it was first isolated during an outbreak in San Joaquin, Bolivia near the Machupo River in 1963 [19, 24]. Cases of person to person spread are uncommon but spread among nursing and laboratory staff and family of infected individuals have been reported [17]. The mortality rate for BHF is approximately 5-30% [30].

Little is published about the ecology, immunology, or natural history of the reservoir species C. callosus. Kuns [18]. reported C. callosus as a pastoral species often located in grasslands and along forest edges around the town of San Joaquin. He also remarked on the commensalism of this species with humans, as many individuals were trapped in and around houses. Information about the natural history of C. callosus is based on several years of data from laboratory rearing of this species [16]. The report stated that breeding was continuous throughout the year. Sexual maturity was attained at about eight weeks, estrous cycle was six days, and the gestation period was estimated as 21 days [16]. Average litter size was approximately six, with high weaning ratios [16]. Laboratory studies where C. callosus individuals from San Joaquin (of known age) were challenged with known quantities of viral loads showed a split response into two groups of phenotypes [33]. Either they were characterized by persistent viremia, little or no antibody production and reduced fertility in females (i.e., immunotolerant) or completely (or nearly so) cleared viremia, presented antibodies, did not shed virus and showed no reduced fertility (i.e., immunocompetent) [33].

Arenaviruses have three modes of transmission in rodents: horizontal transmission (direct contact), vertical transmission (infected mother to offspring) and sexual transmission. These modes of transmission differ from those of other rodent-borne zoonoses such as hantaviruses. In hantaviruses, the primary mode of transmission is horizontal. Although differing in modes of transmission, hantaviruses are similar to arenaviruses in many respects. Hantaviruses, like arenaviruses, are generally associated with a single rodent reservoir population, where the disease is maintained [10]. Continuous contamination of the environment by infectious urine shed by infected peridomestic rodents is the primary mode of infection in humans for both diseases [10]. In addition, hantavirus or arenavirus infection in humans is associated with increased densities of the rodent reservoir [10]. Hantavirus infection in humans results in either hemorrhagic fever with renal syndrome, common in Europe and Asia, or hantavirus pulmonary syndrome (HPS), common in the Americas [22]. HPS and BHF are both recognized as emerging diseases and as agents with bioterrorism potential.

Mathematical models have been developed and analyzed for the spread of hantaviruses in rodent populations [1, 2, 4, 5, 6, 20, 21, 27, 28, 34], but few have been developed for arenaviruses [5, 7]. The models developed by Allen and colleagues [4, 6]. for hantavirus included the dynamics of susceptible, exposed, infective and recovered (SEIR) male and female rodents. Both deterministic (ordinary differential equations) and stochastic (stochastic differential equations) models were developed. Numerical examples and simulations were based on data for the hantavirus Bayou virus, present in the United States, whose reservoir host is the rice rat. Numerical simulations showed that environmental and seasonal variability in the carrying capacity may result in hantavirus outbreaks in rodent populations [4]. McCormack and Allen applied SI and SIR models to investigate the role of spillover species [20]. and the importance of multiple habitats or patches on hantavirus persistence [21]. The models of Abramson and colleagues [1, 2]. were applied to one of the most common hantaviruses in the United States, Sin Nombre virus, whose reservoir species is deer mice. Their models were reaction-diffusion systems of partial differential equations for susceptible and infected mice. Traveling wave solutions were studied as a function of a variable carrying capacity. The models of Sauvage and colleagues [27, 28]. were applied to a hantavirus common in Europe and Asia, Puumala virus, whose reservoir species is bank voles. Their model was a system of ordinary differential equations, where rodents were infected in two different habitats: optimal and suboptimal [27]. In addition, the indirect effect of viral contamination of the ground was considered in both rodents and humans. The effects of a periodic birth and carrying capacity and direct and indirect transmission in a rodent population were investigated [27, 28]. as well as the effect of indirect transmission on human infection [28]. The model of Allen et al. [5]. was applied to a hantavirus (Black Creek Canal virus) and to an arenavirus (Tamiami virus), present in the United States, both of whose reservoir species is cotton rats. Their model distinguished between the modes of transmission; the hantavirus was transmitted horizontally, whereas the arenavirus primary mode of transmission was vertical. Conditions were determined for the two strains to coexist in a cotton rat population. The hantavirus model developed by Wesley et al. [34]. was a system of difference equations, structured by the stages of the infection (susceptible and infected), the stages of development (juvenile, subadult, and adult) and the sex of the rodent (male and female). The basic reproduction number was calculated for this model and its sensitivity to model parameters was examined.

Our new model for Machupo virus includes some of the features of these hantavirus models. The rodent population is divided into the following disease stages: susceptible, infectious, and recovered. As was the case for most of the hantavirus models, we assume that horizontal transmission is density-dependent. Increases in rodent population density increase the probability of contact and consequently, increase horizontal transmission. Unlike the hantavirus models, we include vertical and sexual transmission and two immune states in the infectious stage, either immunocompetent or immunotolerant, as suggested by the data on Machupo virus [31, 32, 33]. Sexually transmitted diseases occur throughout the animal kingdom and they affect the host population dynamics very differently than other directly transmitted infectious diseases [25]. In sexual transmission, it is often assumed that the rate of spread depends on the proportion or frequency rather than the population density [23, 25]. But it has also been suggested that in sexually transmitted diseases, density-dependent transmission is more appropriate than frequency-dependent transmission [25]. In our model, we assume frequency-dependent sexual transmission.

In the next section, we formulate a general model for Machupo virus infection in a rodent population. The model is formulated in terms of a system of ordinary differential equations for the different stages of infection (susceptible, infectious and recovered), the sex of the rodent (male and female), and the immune response of infectious individuals (immunocompetent and immunotolerant). The general model consists of eight differential equations, four for males and four for females. The basic reproduction number is computed for this general model. Two special cases of the general model are studied in Sections 3 and 4, an SIR model, where all infected individuals are immunocompetent and eventually recover and an SI model where all infected individuals are immunotolerant and remain infected. For these two models, the basic reproduction numbers R₀ are computed and for the SI model, a second reproduction number R, associated with an endemic equilibrium, is computed. The SI model is simplified further so that males and females are not differentiated, only females are modeled so there is no sexual transmission (Section 5). In this simple model, the dynamics are completely analyzed. The dynamics of the models are illustrated in several numerical examples. The concluding section summarizes our findings, discusses control measures for reduction of BHF in humans, and proposes future research goals.

2. General Model

2.1. Model Formulation

The underlying assumptions in the mathematical models for Machupo virus infection in a population of Calomys callosus are based on published work [13, 14, 16, 18, 31, 32, 33]. Machupo virus is transmitted horizontally, sexually and vertically and rodents infected with Machupo virus have a split response. Infectious rodents either shed the virus continuously and are antibody negative I^t, or they shed virus for a short period of time, become antibody positive I^c and eventually recover.

The four disease stages of the general model are susceptible (uninfected) S, infectious and shedding virus (antibody negative) I^t, infectious and shedding virus (antibody positive) I^c, and recovered and not shedding virus (antibody positive) R^c. The susceptible rodents infected with Machupo virus follow one of two paths. S → I^t or S → I^c → R^c. The stages for males are denoted by a subscript m and females by a subscript f. For example, uninfected male rodents are denoted as S_m and recovered immunocompetent males are denoted as $I_m^c$.

To simplify the notation in the model, we let $N_m=S_m+R_m^c$ and $I_m=I_m^t+I_m^c$ denote the densities of noninfectious and infectious male rodents, respectively. Similar definitions apply to females (N_f and I_f). In addition, we let N = N_m + N_f and I = I_m + I_f denote the densities of noninfectious and infectious rodents, respectively. Finally, we let M = N_m + I_m, F = N_f + I_f, and T = M + F denote the total population densities of males, females, and both males and females, respectively.

Other model assumptions and model parameters are defined. The recovery rate for immunocompetent males is γ_m and for immunocompetent females it is γ_f. Disease-related death rates for male and female rodents in stages I^t are δ_m and δ_f, respectively. No disease-related deaths occur in stage I^c. Because of the split response, the transmission and birth rates of infectious rodents are split into two parts. The average number of rodents, born to infectious parents, that survive to reproductive age, is split between immunocompetent and immunotolerant newborns, b^t and b^c, where b^t + b^c ≤ b, 0 < b^t < b^c and b is the number of rodents, born to noninfectious parents, that survive to reproductive age. Studies show that contacts between two males (aggressive behavior to defend the territory) are greater than between two females. Therefore, we assume the horizontal transmission coefficient for males is β_m while all other horizontal transmission coefficients are β, and β_m ≫ β. Horizontal transmission is density-dependent with transmission split between immunocompetent and immunotolerant. The transmission coefficient is ${\beta }_m={\beta }_m^t+{\beta }_m^c$ (males) or β = β^t + β^c (females). For example, susceptible males can be infected by infectious males or infectious females:

$$S_m[({\beta }_m^t+{\beta }_m^c)I_m+({\beta }^t+{\beta }^c)I_f].$$

The immune response is not a genetic trait but an individual response, thus, contact with infectious rodents that are either immunocompetent or immunotolerant results in infectious rodents of one of the two types. The sexual transmission coefficient is λ = λ^t +λ^c (mating that results in a susceptible adult becoming infectious), where rate of sexual transmission is frequency-dependent:

$$({\lambda }_m^t+{\lambda }_m^c)\frac{S_m{I}_f}{T}\mathrm{and}({\lambda }_f^t+{\lambda }_f^c)\frac{S_f{I}_m}{T}.$$

Sexual transmission is included in one of the birth terms in that newborns from mating between an infectious male parent I_m and a noninfectious female parent N_f result in an infectious newborn, either immunocompetent or immunotolerant (equally divided among males and female newborns),

$$b^c\frac{I_m{N}_f}{T}\mathrm{or}{b}^t\frac{I_m{N}_f}{T}.$$

Vertical transmission from infectious mothers is also part of the birth terms,

$$b^c\frac{M{I}_f}{T}\mathrm{or}{b}^t\frac{M{I}_f}{T}$$

Therefore, the total birth rate for infectious rodents leads to

$$b^c\frac{N_m{I}_f+F{I}_m}{T}\mathrm{or}{b}^t\frac{N_m{I}_f+F{I}_m}{T}$$

and the birth rate for noninfectious rodents is

$$\frac{b{N}_m{N}_f}{T}.$$

The expression for the birth function is based on the harmonic mean (one of the most commonly used birth functions) in that the total number of births for a population that is not infected is 2bMF/T [9]. In this formulation, complete vertical transmission is assumed; all newborns from infectious parents are infectious. We assume that immunocompetent males and females may also develop persistent infections and become immunotolerant at a rate ε [15], a switch of immune response. Finally, in the absence of infection, the rodent population is assumed to follow logistic growth, dependent on the environmental carrying capacity K. The natural death rate d(T) is a linear increasing function of the total population size T, d(T) = a + cT, with the property that d(K) = b/2 or K = (b/2 − a)/c. This latter assumption can be used to show that in a male-female population with no infection, the population size will approach the carrying capacity K.

A compartmental diagram for one sex (either male or female) is graphed in Figure 1.

Figure 1.

Compartmental diagram of the SIR model for one sex (either males or females). The dashed arrows entering compartments S, I^t, and I^c are births, the dashed arrows leaving each compartment are natural deaths (plus disease-related deaths from compartment I^t), the dotted arrow is transition from immunocompetent to immunotolerant stage, and the solid arrows are transitions that occur between the various stages. Because only one sex is represented in this diagram, sexual transmission is not shown.

Based on the preceding assumptions, the following four differential equations model the dynamics of the male population:

$$\begin{array}{l}\frac{d{S}_m}{\mathit{dt}}=\frac{b{N}_m{N}_f}{T}-({\lambda }_m^t+{\lambda }_m^c)\frac{S_m{I}_f}{T}-S_m[({\beta }_m^t+{\beta }_m^c)I_m+({\beta }^t+{\beta }^c)I_f]-d(T)S_m\\ \frac{d{I}_m^t}{\mathit{dt}}=b^t\frac{N_m{I}_f+F{I}_m}{T}+{\lambda }_m^t\frac{S_m{I}_f}{T}+S_m({\beta }_m^t{I}_m+{\beta }^t{I}_f)-{\delta }_m{I}_m^t+\epsilon I_m^c-d(T)I_m^t\\ \frac{d{I}_m^c}{\mathit{dt}}=b^c\frac{N_m{I}_f+F{I}_m}{T}+{\lambda }_m^c\frac{S_m{I}_f}{T}+S_m({\beta }_m^c{I}_m+{\beta }^c{I}_f)-{\gamma }_m{I}_m^c-\epsilon I_m^c-d(T)I_m^c\\ \frac{d{R}_m^c}{\mathit{dt}}={\gamma }_m{I}_m^c-d(T)R_m^c.\end{array}$$ (1)

Similar equations apply to females, with the exception that horizontal transmission coefficients for females differ from males:

$$\begin{array}{l}\frac{d{S}_f}{\mathit{dt}}=\frac{b{N}_m{N}_f}{T}-({\lambda }_f^t+{\lambda }_f^c)\frac{S_f{I}_m}{T}-S_f({\beta }^t+{\beta }^c)I-d(T)S_f\\ \frac{d{I}_f^t}{\mathit{dt}}=b^t\frac{N_m{I}_f+F{I}_m}{T}+{\lambda }_f^t\frac{S_f{I}_m}{T}+{\beta }^t{S}_{f}I-{\delta }_f{I}_f^t+\epsilon I_f^c-d(T)I_f^t\\ \frac{d{I}_f^c}{\mathit{dt}}=b^c\frac{N_m{I}_f+F{I}_m}{T}+{\lambda }_f^c\frac{S_f{I}_m}{T}+{\beta }^c{S}_{f}I-{\gamma }_f{I}_f^c-\epsilon I_f^c-d(T)I_f^c\\ \frac{d{R}_f^c}{\mathit{dt}}={\gamma }_f{I}_f^c-d(T)R_f^c.\end{array}$$ (2)

Parameters are positive and initial conditions are nonnegative with M(0) > 0 and F(0) > 0. It is straightforward to show that solutions are nonnegative for t ≥ 0 and the total population size is positive for t ≥ 0.

In the absence of infection, $I_m^k\equiv 0\equiv I_f^k$, k = t, c, the total population size satisfies a logistic growth assumption for males and females:

$$\begin{array}{l}\frac{d{N}_m}{\mathit{dt}}=b\frac{N_m{N}_f}{T}-d(T)N_m\\ \frac{d{N}_f}{\mathit{dt}}=b\frac{N_m{N}_f}{T}-d(T)N_f,\end{array}$$ (3)

where T =N_m + N_f. The population model (3) has a unique, globally stable equilibrium at N̄_m = K/2 = N̄_f (see [6], pp. 522-523). Therefore, the total population size T(t) → K, the environmental carrying capacity, and $R_m^c(t),R_f^c(t)\to 0$ as t → ∞. Hence, in the absence of disease, the rodent population approaches a unique disease-free equilibrium (DFE), where

$${\overline{S}}_m=\frac{K}{2}={\overline{S}}_f$$ (4)

and the equilibrium value for all of the other disease stages is zero.

With infection, the total population size is reduced below the carrying capacity. This can be seen from the following inequality:

$$\frac{\mathit{dT}}{\mathit{dt}}\le 2b\frac{\mathit{MF}}{T}-d(T)T\le T[\frac{b}{2}-d(T)]=T[\frac{b}{2}-a-\mathit{cT}].$$

Comparison of this differential equation with the solution u(t) to the differential equation du/dt = u[b/2 − a − cu], where u(0) = T(0), it follows that T(t) ≤ u(t) for t > 0. But u(t) → K. Hence, when there is infection, the total population size T will be less than the carrying capacity K.

For the general male-female epidemic model (1)-(2) we will determine conditions for stability of the DFE by applying the next generation matrix approach [12, 29]. There exist other equilibria, the extinction equilibrium and endemic equilibria where some rodents are infectious, but explicit forms for the endemic equilibria are hard to compute for the general model (1)-(2). In the numerical examples (Section 6), we show existence and stability of an endemic equilibrium for model (1)-(2). The extinction or zero equilibrium exists for the general model as a limiting case. Letting all of the state variables approach zero (through positive values) forces the right side of the differential equations to approach zero. For example, note that

$$\underset{N_m,N_f\to 0^{+}}{\lim }\frac{b{N}_m{N}_f}{T}=0\underset{S_m,S_f\to 0^{+}}{\lim }\frac{S_m{I}_f}{T}.$$

We do not study the stability of the extinction equilibrium for any of the models except in a special case of the simple SI model in Section 5. Linear stability analyses cannot be applied to the extinction equilibrium because of the singularity at the origin. However, we found in all of the numerical examples that solutions did not approach the origin; the extinction equilibrium was not stable.

2.2 Analysis of the General Model

We calculate the basic reproduction R₀ for the general male-female model (1)-(2) based on the next generation matrix approach [12, 29]. The system is rearranged so that the first four equations represent the infectious states and the last four the noninfectious states:

$$\dot{x}=F(x)-V(x),$$

where $x={\left[I_m^t,I_m^c,I_f^t,I_f^c,S_m,S_f,R_m^c,R_f^c\right]}^T$,

$$F(x)=\left[\begin{array}{c}{b}^t\frac{N_m{I}_f+F{I}_m}{T}+{\lambda }_m^t\frac{S_m{I}_f}{T}+S_m({\beta }_m^t{I}_m+{\beta }^t{I}_f)\\ b^c\frac{N_m{I}_f+F{I}_m}{T}+{\lambda }_m^c\frac{S_m{I}_f}{T}+S_m({\beta }_m^c{I}_m+{\beta }^c{I}_f)\\ b^t\frac{N_m{I}_f+F{I}_m}{T}+{\lambda }_f^t\frac{S_f{I}_m}{T}+{\beta }^t{S}_{f}I\\ b^c\frac{N_m{I}_f+F{I}_m}{T}+{\lambda }_f^c\frac{S_f{I}_m}{T}+{\beta }^c{S}_{f}I\\ 0\\ 0\\ 0\\ 0\end{array}\right]$$

and

$$V(x)=\left[\begin{array}{c}{\delta }_m{I}_m^t+d(T)I_m^t-\epsilon I_m^c\\ {\gamma }_m{I}_m^c+d(T)I_m^c+\epsilon I_m^c\\ {\delta }_f{I}_f^t+d(T)I_f^t-\epsilon I_f^c\\ {\gamma }_f{I}_f^c+d(T)I_f^c+\epsilon I_f^c\\ -\frac{b{N}_m{N}_f}{T}+({\lambda }_m^t+{\lambda }_m^c)\frac{S_m{I}_f}{T}+S_m[({\beta }_m^t+{\beta }_m^c)I_m+({\beta }^t+{\beta }^c)I_f]+d(T)S_m\\ -\frac{b{N}_m{N}_f}{T}+({\lambda }_f^t+{\lambda }_f^c)\frac{S_f{I}_m}{T}+S_f({\beta }^t+{\beta }^c)I+d(T)S_f\\ -{\gamma }_m{I}_m^c+d(T)R_m^c\\ -{\gamma }_f{I}_f^c+d(T)R_f^c\end{array}\right].$$

Differentiation of the right side of ẋ and evaluation at the DFE (given by (4) but denoted x_o) leads to the following matrix:

$$D(F-V)(x_0)=\left[\begin{array}{cc}F-V& O\\ A& J\end{array}\right],$$ (5)

where O is the zero matrix and J equals

$$J=\left[\begin{array}{cccc}-\frac{b}{4}-c\frac{K}{2}& \frac{b}{4}& 0& 0\\ \frac{b}{4}& -\frac{b}{4}-c\frac{K}{2}& 0& 0\\ 0& 0& -\frac{b}{2}& 0\\ 0& 0& 0& -\frac{b}{2}\end{array}\right],$$ (6)

where c = d′(K). Matrix J is block diagonal with 2 × 2 block matrices along the diagonal. Each of the 2 × 2 matrices have traces that are negative and determinants that are positive. Hence by the Routh-Hurwitz criteria the eigenvalues of J have negative real parts [3]. Matrix A plays no role in stability of the DFE. Thus, the DFE is locally asymptotically if F − V has eigenvalues with negative real parts. van den Dreissche and Watmough [29]. (Theorem 2, p. 33) have shown that linear stability is equivalent to the spectral radius of the next generation matrix FV⁻¹ satisfying ρ(FV⁻¹) < 1. This latter expression is known as the basic reproduction number,

$$R_0=\rho (F{V}^{-1}).$$

Matrices F and V⁻¹ are as follows:

$$F=\left[\begin{array}{cccc}\frac{{\beta }_m^{t}K+b^t}{2}& \frac{{\beta }_m^{t}K+b^t}{2}& \frac{{\beta }^{t}K+b^t+{\lambda }_m^t}{2}& \frac{{\beta }^{t}K+b^t+{\lambda }_m^t}{2}\\ \frac{{\beta }_m^{c}K+b^c}{2}& \frac{{\beta }_m^{c}K+b^c}{2}& \frac{{\beta }^{c}K+b^c+{\lambda }_m^c}{2}& \frac{{\beta }^{c}K+b^c+{\lambda }_m^c}{2}\\ \frac{{\beta }^{t}K+b^t+{\lambda }_f^t}{2}& \frac{{\beta }^{t}K+b^t+{\lambda }_f^t}{2}& \frac{{\beta }^{t}K+b^t}{2}& \frac{{\beta }^{t}K+b^t}{2}\\ \frac{{\beta }^{c}K+b^t+{\lambda }_f^c}{2}& \frac{{\beta }^{c}K+b^t+{\lambda }_f^c}{2}& \frac{{\beta }^{c}K+b^c}{2}& \frac{{\beta }^{c}K+b^c}{2}\end{array}\right]$$

and

$$V^{-1}=\left[\begin{array}{cccc}\frac{1}{{\delta }_m+b/2}& \frac{\epsilon }{\left({\delta }_m+b/2\right)\left({\gamma }_m+b/2+\epsilon \right)}& 0& 0\\ 0& \frac{1}{{\gamma }_m+b/2+\epsilon }& 0& 0\\ 0& 0& \frac{1}{{\delta }_f+b/2}& \frac{\epsilon }{({\delta }_m+b/2)({\gamma }_f+b/2+\epsilon )}\\ 0& 0& 0& \frac{1}{{\gamma }_f+b/2+\epsilon }\end{array}\right].$$

Thus, for model (1)-(2), the next generation matrix FV⁻¹ is

$$\left[\begin{array}{cccc}\frac{({\beta }_m^{t}K+b^t)/2}{{\delta }_m+b/2}& \frac{{\mathrm{\Delta }}_m({\beta }_m^{t}K+b^t)/2}{{\gamma }_m+b/2+\epsilon }& \frac{({\beta }^{t}K+b^t+{\lambda }_m^t)/2}{{\delta }_f+b/2}& \frac{{\mathrm{\Delta }}_f({\beta }^{t}K+b^t+{\lambda }_m^t)/2}{{\gamma }_f+b/2+\epsilon }\\ \frac{({\beta }_m^{c}K+b^c)/2}{{\delta }_m+b/2}& \frac{{\mathrm{\Delta }}_m({\beta }_m^{c}K+b^c)/2}{{\gamma }_m+b/2+\epsilon }& \frac{({\beta }^{c}K+b^c+{\lambda }_m^c)/2}{{\delta }_f+b/2}& \frac{{\mathrm{\Delta }}_f({\beta }^{c}K+b^c+{\lambda }_m^c)/2}{{\gamma }_f+b/2+\epsilon }\\ \frac{({\beta }^{t}K+b^t+{\lambda }_f^t)/2}{{\delta }_m+b/2}& \frac{{\mathrm{\Delta }}_m({\beta }^{t}K+b^t+{\lambda }_f^t)}{{\gamma }_m+b/2+\epsilon }& \frac{({\beta }^{t}K+b^t)/2}{{\delta }_f+b/2}& \frac{{\mathrm{\Delta }}_f({\beta }^{t}K+b^t)/2}{{\gamma }_f+b/2+\epsilon }\\ \frac{({\beta }^{c}K+b^c+{\lambda }_f^c)/2}{{\delta }_m+b/2}& \frac{{\mathrm{\Delta }}_m({\beta }^{c}K+b^c+{\lambda }_f^c)/2}{{\gamma }_m+b/2+\epsilon }& \frac{({\beta }^{c}K+b^c)/2}{{\delta }_f+b/2}& \frac{{\mathrm{\Delta }}_f({\beta }^{c}K+b^c)/2}{{\gamma }_f+b/2+\epsilon }\end{array}\right],$$ (7)

where

$${\mathrm{\Delta }}_m=\frac{{\delta }_m+b/2+\epsilon }{{\delta }_m+b/2}\mathrm{and}{\mathrm{\Delta }}_f=\frac{{\delta }_f+b/2+\epsilon }{{\delta }_f+b/2}.$$

Matrix FV⁻¹ has two zero eigenvalues and the remaining eigenvalues are the solutions to a quadratic equation. Hence, the spectral radius of matrix FV⁻¹ or R₀ can be expressed as

$$R_0=\rho (F{V}^{-1})=B+\sqrt{B^2-C}.$$ (8)

For the case ε = 0, ∆_m = 1 = ∆_f, the coefficients B and C in (8) are

$$\begin{array}{lll}B& =& \frac{1}{4}\left[\frac{{\beta }_m^{t}K+b^t}{{\delta }_m+b/2}+\frac{{\beta }_m^{c}K+b^c}{{\gamma }_m+b/2}+\frac{{\beta }^{t}K+b^t}{{\delta }_f+b/2}+\frac{{\beta }^{c}K+b^c}{{\gamma }_f+b/2}\right]\\ C& =& \frac{1}{4}\frac{({\beta }^{t}K+b^t+{\lambda }_f^t)({\beta }^{t}K+b^t+{\lambda }_m^t)-({\beta }_m^{t}K+b^t)({\beta }^{t}K+b^t)}{({\delta }_f+b/2)({\delta }_m+b/2)}\\ & & +\frac{1}{4}\frac{({\beta }^{t}K+b^t+{\lambda }_m^t)({\beta }^{c}K+b^c+{\lambda }_f^c)-({\beta }_m^{t}K+b^t)({\beta }^{c}K+b^c)}{({\delta }_m+b/2)({\gamma }_f+b/2)}\\ & & +\frac{1}{4}\frac{({\beta }^{t}K+b^t+{\lambda }_f^t)({\beta }^{c}K+b^c+{\lambda }_m^c)-({\beta }_m^{c}K+b^c)({\beta }^{t}K+b^t)}{({\delta }_f+b/2)({\gamma }_m+b/2)}\\ & & +\frac{1}{4}\frac{({\beta }^{c}K+b^c+{\lambda }_f^c)({\beta }^{c}K+b^c+{\lambda }_m^c)-({\beta }_m^{c}K+b^c)({\beta }^{c}K+b^c)}{({\gamma }_f+b/2)({\gamma }_m+b/2)}\end{array}$$

In general, R₀ increases with ε (as more rodents become persistently infectious or immunotolerant). Applying Theorem 2 in [29]. (p. 33) to model (1)-(2) we have the following result concerning local stability of the DFE.

Theorem 1

The general male-female epidemic model (1)-(2) has a unique DFE given by (4) and a basic reproduction R₀ defined in (7) and (8). If R₀ < 1, then the DFE is locally asymptotically stable and if R₀ > 1, it is unstable.

The form of R₀ is quite complicated but specific terms can be interpreted. For example, note that each of the four terms in the expression for B(when ε = 0) is the sum of two terms, horizontal and vertical transmission. Hence, the four terms in the expression for B represent secondary transmission due to horizontal and vertical transmission from (1) the male immunotolerant class, (2) the male immunocompetent class, (3) the female immunotolerant class, and (4) the female immunocompetent class.

If R₀ > 1, there is a potential for disease outbreak and for the disease to persist in the population. If ε = 0, we consider several special cases. First, suppose there is no sexual transmission, ${\lambda }_j^k=0$, j = m, f; k = t, c, and equal male and female transmission, ${\beta }_m^t={\beta }^t$ and ${\beta }_m^c={\beta }^c$, then

$$R_0=\frac{1}{2}\left[\frac{{\beta }^{t}K+b^t}{{\delta }_m+b/2}+\frac{{\beta }^{t}K+b^t}{{\delta }_f+b/2}+\frac{{\beta }^{c}K+b^c}{{\gamma }_m+b/2}+\frac{{\beta }^{c}K+b^c}{{\gamma }_f+b/2}\right].$$ (9)

The basic reproduction number R₀ is just the sum of transmission from the four classes described previously. If, in addition, δ_m = δ_f = δ and γ_m = γ_f = γ so that the disease-related death rates and recovery rates are the same for both males and females, then the expression for R₀ simplifies even further,

$$R_0=\frac{{\beta }^{t}K+b^t}{\delta +b/2}+\frac{{\beta }^{c}K+b^c}{\gamma +b/2}.$$ (10)

It is clear from the previous expressions that R₀ can be greater than one if the horizontal and/or vertical transmission parameters are sufficiently large. But it is also possible for vertical or sexual transmission alone to result in disease outbreaks (with R₀ > 1). As a second case, suppose there is no horizontal nor sexual transmission, only vertical transmission, ${\beta }_j^k=0$ and ${\lambda }_j^k=0$, j = m, f; k = t, c. Then

$$R_0=\frac{b^t}{2}\left[\frac{1}{{\delta }_f+b/2}+\frac{1}{{\delta }_m+b/2}\right]+\frac{b^c}{2}\left[\frac{1}{{\gamma }_f+b/2}+\frac{1}{{\gamma }_m+b/2}\right].$$ (11)

If b^t and b^c are sufficiently large, then there is a possibility of disease outbreak due to vertical transmission. As a third case, suppose there is no horizontal nor vertical transmission, only sexual transmission, b^t = 0 = b^c and ${\beta }_j^k=0$, j = m, f; k = t, c (infectious individuals do not reproduce). Then R₀ simplifies to

$$\frac{1}{2}{\left[\frac{{\lambda }_m^t{\lambda }_f^t}{({\delta }_m+b/2)({\delta }_f+b/2)}+\frac{{\lambda }_m^t{\lambda }_f^c}{({\delta }_m+b/2)({\gamma }_f+b/2)}+\frac{{\lambda }_m^c{\lambda }_f^t}{({\gamma }_m+b/2)({\delta }_f+b/2)}+\frac{{\lambda }_m^c{\lambda }_f^c}{({\gamma }_m+b/2)({\gamma }_f+b/2)}\right]}^{1/2}.$$ (12)

If the parameters, ${\lambda }_j^k,j=m,f$; k = t, c, are sufficiently large, then there is a potential for disease outbreak due to sexual transmission.

To better understand the role played by the immunocompetent and immunotolerant disease stages, simplifications of the general male-female model (1)-(2) are studied in the next two sections. Two models are considered, where animals exhibit only one type of immune response, either immunocompetent or immunotolerant. For these two models, their basic reproduction numbers are calculated and their relationship to the basic reproduction number for the general male-female model is shown. In addition, it is shown that if there is no recovery, then vertical transmission in the male-female immunotolerant model (Section 4) may result in bistability.

3. SIR Model

3.1. Model Description

The general male-female model is simplified to one where there are only immunocompetent animals. In this case there are only three disease stages: susceptible (uninfected) S, infectious and shedding virus I^c, and recovered and not shedding virus R^c. The difference between this new SIR model and model (1)-(2) is that there is no I^t stage. All the infectious rodents ultimately recover. There are still three modes of transmission: horizontal, sexual, and vertical. We use the same notation as in the general male-female model (1)-(2). The horizontal transmission coefficients are ${\beta }_m^c$ (male-to-male transmission) and β^c (all other horizontal transmission). The sexual transmission coefficients are ${\lambda }_m^c$ for males and ${\lambda }_f^c$ for females. The recovery rates are γ_m for males and γ_f for females. In addition, let, $M=S_m+I_m^c+R_m^c$, $F=S_f+I_f^c+R_f^c$, $I=I_m^c+I_f^c$, $N_m=S_m+R_m^c$, $N_f=S_f+R_f^c$, and T = M + F.

The differential equations for the SI^cR^c model have the following form:

$$\begin{array}{l}\frac{d{S}_m}{\mathit{dt}}=\frac{b{N}_m{N}_f}{T}-{\lambda }_m^c\frac{S_m{I}_f^c}{T}-S_m[{\beta }_m^c{I}_m^c+{\beta }^c{I}_f^c]-d(T)S_m\\ \frac{d{I}_m^c}{\mathit{dt}}=b^c\frac{N_m{I}_f^c+F{I}_m^c}{T}+{\lambda }_m^c\frac{S_m{I}_f^c}{T}+S_m\left[{\beta }_m^c{I}_m^c+{\beta }^c{I}_f^c\right]-{\gamma }_m{I}_m^c-d(T)I_m^c\\ \frac{d{R}_m^c}{\mathit{dt}}={\gamma }_m{I}_m^c-d(T)R_m^c\end{array}$$ (13)

$$\begin{array}{l}\frac{d{S}_f}{\mathit{dt}}=\frac{b{N}_m{N}_f}{T}-{\lambda }_f^c\frac{S_f{I}_m^c}{T}-{\beta }^c{S}_{f}I-d(T)S_f\\ \frac{d{I}_f^c}{\mathit{dt}}=b^c\frac{N_m{I}_f^c+F{I}_m^c}{T}+{\lambda }_f^c\frac{S_f{I}_m^c}{T}+{\beta }^c{S}_{f}I-{\gamma }^f{I}_f^c-d(T)I_f^c\\ \frac{d{R}_f^c}{\mathit{dt}}={\gamma }_f{I}_f^c-d(T)R_f^c.\end{array}$$ (14)

Parameters are positive and initial conditions are nonnegative with M(0) > 0 and F(0) > 0. In addition, b ≥ b^c. Solutions to (13)-(14) are nonnegative for t ≥ 0 and T(t) > 0 for t ≥ 0. If there is no reduction in fertility, as noted by Webb et al. [33], then b^c = b^t and the male and female subpopulations, M and F, satisfy the differential equations given in (3). That is,

$$\begin{array}{l}\frac{dM}{\mathit{dt}}=b\frac{\mathit{MF}}{T}=d(T)M\\ \frac{\mathit{dF}}{\mathit{dt}}=b\frac{\mathit{MF}}{T}-d(T)F,\end{array}$$ (15)

where T = M + F. Hence, M(t), F(t) → K/2 and T(t) → K as t → ∞. There is no reduction in the population size.

3.2 Analysis of the SIR Model

For model (13)-(14) the unique DFE is given by (4), the same equilibrium as in the general male-female model. To calculate the basic reproduction number R₀ for the SIR model (13)-(14), the terms are rearranged so that

$$\dot{x}={\left[{\dot{I}}_m^c,{\dot{I}}_f^c,{\dot{S}}_m,{\dot{S}}_f,{\dot{R}}_m^c,{\dot{R}}_f^c\right]}^T=F(x)-V(x),$$

where

$$F(x)=\left[\begin{array}{c}{b}^c\frac{N_m{I}_f^c+F{I}_m^c}{T}+{\lambda }_m^c\frac{S_m{I}_f^c}{T}+S_m[{\beta }_m^c{I}_m^c+{\beta }^c{I}_f^c]\\ b^c\frac{N_m{I}_f^c+F{I}_m^c}{T}+{\lambda }_f^c\frac{S_f{I}_m^c}{T}+{\beta }^c{S}_{f}I\\ 0\\ 0\\ 0\\ 0\end{array}\right]$$

and

$$V(x)=\left[\begin{array}{c}{\gamma }_m{I}_m^c+d(T)I_m^c\\ {\gamma }_f{I}_f^c+d(T)I_f^c\\ -\frac{b{N}_m{N}_f}{T}+\frac{{\lambda }_m^c{S}_m{I}_f^c}{T}+S_m[{\beta }_m^c{I}_m^c+{\beta }^c{I}_f^c]+d(T)S_m\\ -\frac{b{N}_m{N}_f}{T}+\frac{{\lambda }_f^c{S}_f{I}_m^c}{T}+{\beta }_c{S}_{f}I+d(T)S_f\\ -{\gamma }_m{I}_m^c+d(T)R_m^c\\ -{\gamma }_f{I}_f^c+d(T)R_f^c\end{array}\right].$$

Differentiation and evaluation at the DFE x_o leads to a matrix of the form (5), where O is the zero matrix and matrix J is given by (6). Matrices F and V⁻¹ are

$$F=\left[\begin{array}{cc}\frac{{\beta }_m^{c}K+b^c}{2}& \frac{{\beta }^{c}K+b^c+{\lambda }_m^c}{2}\\ \frac{{\beta }^{c}K+b^c+{\lambda }_f^c}{2}& \frac{{\beta }^{c}K+b^c}{2}\end{array}\right]$$

and

$$V^{-1}=\left[\begin{array}{cc}\frac{1}{{\gamma }_m+b/2}& 0\\ 0& \frac{1}{{\gamma }_f+b/2}\end{array}\right]$$

so that

$$F{V}^{-1}=\left[\begin{array}{cc}\frac{({\beta }_m^{c}K+b^c)/2}{{\gamma }_m+b/2}& \frac{({\beta }^{c}K+b^c+{\lambda }_m^c)/2}{{\gamma }_f+b/2}\\ \frac{({\beta }^{c}K+b^c+{\lambda }_f^c)/2}{{\gamma }_m+b/2}& \frac{({\beta }^{c}K+b^c)/2}{{\gamma }_f+b/2}\end{array}\right].$$ (16)

The basic reproduction number is given by

$$R_0=\rho (F{V}^{-1})=\frac{1}{2}\left(B+\sqrt{B^2-4C}\right),$$ (17)

where

$$\begin{array}{c}B=\frac{({\beta }_m^{c}K+b^c)(2{\gamma }_f+b)+({\beta }^{c}K+b^c)(2{\gamma }_m+b)}{(2{\gamma }_m+b)(2{\gamma }_f+b)},\\ C=\frac{[({\beta }_m^{c}K+b^c)({\beta }^{c}K+b^c)-({\beta }^{c}K+b^c+{\lambda }_m^c)({\beta }^{c}K+b^t+{\lambda }_f^c)]}{{\left[(2{\gamma }_m+b)(2{\gamma }_f+b)\right]}^2}.\end{array}$$

Now we apply Theorem 2 in [29]. (p. 33) to obtain the stability result for model (13)-(14).

Theorem 2

The SIR male-female epidemic model (13)-(14) has a unique DFE given by (4) and a basic reproduction number R₀ defined in (16) and (17). If R₀ < 1, then the DFE is locally asymptotically stable and if R₀ > 1, it is unstable.

Suppose ${\beta }_m^c={\beta }^c$ and ${\lambda }_m^c=0={\lambda }_f^c$ (no distinction between male and female transmission and no sexual transmission). In this case,

$$R_0=\frac{({\beta }^{c}K+b^c)/2}{{\gamma }_m+b/2}+\frac{({\beta }^{c}K+b^c)/2}{{\gamma }_f+b/2}.$$ (18)

The two terms in formula (18) correspond to male and female reproduction numbers. They also correspond to the third and fourth terms in the expression for the basic reproduction number for the general male-female model, given in (9). If, in addition, γ_m = γ_f = γ in (18), then

$$\frac{R_0={\beta }^{c}K+b^c}{\gamma +b/2}.$$ (19)

Formula (19) corresponds to the second term in the expression for the basic reproduction number for the general male-female model, given in (10). If there is only vertical transmission or only sexual transmission, then the expressions for R₀ reduce to

$$R_0=\frac{b^c}{2}\left[\frac{1}{{\gamma }_f+b/2}+\frac{1}{{\gamma }_m+b/2}\right]$$

or

$$R_0=\frac{1}{2}{\left[\frac{{\lambda }_m^c{\lambda }_f^c}{({\gamma }_m+b/2)({\gamma }_f+b/2)}\right]}^{1/2},$$

respectively. These expressions agree with those given in (11) and in (12) when there is no immunotolerant stage.

4. SI Model

4.1. Model Description

Another simplification of the general male-female model (1)-(2) is considered, where only immunotolerant rodents are included. Two disease stages are included, susceptible S and infectious and shedding virus (antibody negative) I^t. The parameters have the same meaning as in the general male-female model. We assume b^t < b, infectious rodents have a lower birth rate than susceptible rodents. There is complete vertical transmission as in the other models; all rodents born to infectious mothers are infectious. Let $M=S_m+I_m^t$, $F=S_f+I_f^t$, $I=I_m^t+I_f^t$ and T = M + F. The SI^t model has the following form:

$$\begin{array}{l}\frac{d{S}_m}{\mathit{dt}}=\frac{b{S}_m{S}_f}{T}-{\lambda }_m^t\frac{S_m{I}_f^t}{T}-S_m[{\beta }_m^t{I}_m^t+{\beta }^t{I}_f^t]-d(T)S_m\\ \frac{d{I}_m^t}{\mathit{dt}}=b^t\frac{S_m{I}_f^t+F{I}_m^t}{T}+{\lambda }_m^t\frac{S_m{I}_f^t}{T}+S_m[{\beta }_m^t{I}_m^t+{\beta }^t{I}_f^t]-{\delta }_m{I}_m^t-d(T)I_m^t.\end{array}$$ (20)

$$\begin{array}{c}\frac{d{S}_f}{\mathit{dt}}=\frac{b{S}_m{S}_f}{T}-{\lambda }_f^t\frac{S_f{I}_m^t}{T}-{\beta }^t{S}_{f}I-d(T)S_f\\ \frac{d{I}_f^t}{\mathit{dt}}=b^t\frac{S_m{I}_f^t+F{I}_m^t}{T}+{\lambda }_f^t\frac{S_f{I}_m^t}{T}+{\beta }^t{S}_{f}I-{\delta }_f{I}_f^t-d(T)I_f^t.\end{array}$$ (21)

Parameters are positive and initial conditions are nonnegative with M(0) > 0 and F(0) > 0. Therefore, solutions are nonnegative for t ≥ 0 and T(t) > 0 for t ≥ 0.

4.2. Analysis of the SI Model

For model (20)-(21), there is a unique DFE given by (4) and a unique endemic equilibrium (EE), where all animals are infectious. In addition, there may exist one or more endemic equilibria, with both susceptible and infectious animals. These latter endemic equilibria are studied in numerical examples. Model (20)-(21) is analyzed near the DFE and near the EE, where all animals are infectious.

4.2.1. Disease-Free Equilibrium

The basic reproduction number R₀ for the male-female SI model (20)-(21) is calculated based on the next generation matrix approach [12, 29]. Let

$$\dot{x}={\left[{\dot{I}}_m^t,{\dot{I}}_f^t,{\dot{S}}_m,{\dot{S}}_f\right]}^T=F(x)-V(x),$$

where

$$F(x)=\left[\begin{array}{c}\frac{b^t{S}_m{I}_f^t+F{I}_m^t}{T}+{\lambda }_m^t\frac{S_m{I}_f^t}{T}+S_m[{\beta }_m^t{I}_m^t+{\beta }^t{I}_f^t]\\ b^t\frac{S_m{I}_f^t+F{I}_m^t}{T}+{\lambda }_f^t\frac{S_f{I}_m^t}{T}+{\beta }^t{S}_{f}I\\ 0\\ 0\end{array}\right]$$

and

$$V(x)=\left[\begin{array}{c}{\delta }_m{I}_m^t+d(T)I_m^t\\ {\delta }_f{I}_f^t+d(T)I_f^t\\ -\frac{b{S}_m{S}_f}{T}+\frac{{\lambda }_m^t{S}_m{I}_f^t}{T}+S_m\left[{\beta }_m^t{I}_m^t+{\beta }^t{I}_f^t\right]+d(T)S_m\\ -\frac{b{S}_m{S}_f}{T}+\frac{{\lambda }_f^t{S}_f{I}_m^t}{T}+{\beta }^t{S}_{f}I+d(T)S_f\end{array}\right].$$

Differentiation and evaluation at the DFE x_o leads to a matrix of the form (5), where O is the zero matrix and matrix J equals

$$J=\left[\begin{array}{cc}-\frac{b}{4}-c\frac{K}{2}& -\frac{b}{4}\\ -\frac{b}{4}& -\frac{b}{4}-c\frac{K}{2}\end{array}\right],$$

where c = d′(K) > 0. It is straightforward to see that the trace of J is negative and the determinant of J is positive. Hence, by the Routh-Hurwitz criteria, the eigenvalues of J have negative real parts. Matrices F and V⁻¹ are

$$F=\left[\begin{array}{cc}\frac{{\beta }_m^{t}K+b^t}{2}& \frac{{\beta }^{t}K+b^t+{\lambda }_m^t}{2}\\ \frac{{\beta }^{t}K+b^t+{\lambda }_f^t}{2}& \frac{{\beta }^{t}K+b^t}{2}\end{array}\right],$$

and

$$V^{-1}=\left[\begin{array}{cc}\frac{1}{{\delta }_m+b/2}& 0\\ 0& \frac{1}{{\delta }_f+b/2}\end{array}\right].$$

so that

$$F{V}^{-1}=\left[\begin{array}{cc}\frac{{\beta }_m^{t}K+b^t}{2{\delta }_m+b}& \frac{{\beta }^{t}K+b^t+{\lambda }_m^t}{2{\delta }_f+b}\\ \frac{{\beta }^{t}K+b^t+{\lambda }_f^t}{2{\delta }_m+b}& \frac{{\beta }^{t}K+b^t}{2{\delta }_f+b}\end{array}\right].$$ (22)

The basic reproduction number is given by

$$R_0=\rho (F{V}^{-1})=\frac{1}{2}(B+\sqrt{B^2-4C})$$ (23)

where

$$\begin{array}{c}B=\frac{({\beta }_m^{t}K+b^t)(2{\delta }_f+b)+({\beta }^{t}K+b^t)(2{\delta }_m+b)}{(2{\delta }_m+b)(2{\delta }_f+b)},\\ C=\frac{[({\beta }_m^{t}K+b^t)({\beta }^{t}K+b^t)-({\beta }^{t}K+b^t+{\lambda }_m^t)({\beta }^{t}K+b^t+{\lambda }_f^t)]}{{[(2{\delta }_m+b)(2{\delta }_f+b)]}^2}.\end{array}$$

Thus, a stability result for model (20)-(21) holds, a result similar to the previous models (Theorems 1 and 2).

Theorem 3

The SI male-female epidemic model (20)-(21) has a unique DFE given by (4) and a basic reproduction number R₀ defined in (22) and (23). If R₀ < 1, then the DFE is locally asymptotically stable and if R₀ > 1, it is unstable.

In the case that ${\beta }_m^t={\beta }^t$ and ${\lambda }_m^t=0={\lambda }_f^t$. That is, there is no distinction between male and female transmission and there is no sexual transmission, then

$$R_0=\frac{\left({\beta }^{t}K+b^t\right)/2}{{\delta }_m+b/2}+\frac{({\beta }^{t}K+b^t)/2}{{\delta }_f+b/2}.$$ (24)

Formula (24) corresponds to the first two terms in the expression for R₀ in the general male-female model, given in (9). In addition, if δ_m = δ_f = δ in (24), then

$$R_0=\frac{{\beta }^{t}K+b^t}{\delta +b/2}.$$ (25)

Formula (25) corresponds to the first term in the expression for the basic reproduction number for the general model, given in (10). If there is only vertical transmission or only sexual transmission, then the expressions for R₀ reduce to the ones computed for the general male-female model with no immunocompetent stage, i.e.,

$$R_0=\frac{b^t}{2}\left[\frac{1}{{\delta }_f+b/2}+\frac{1}{{\delta }_m+b/2}\right]$$

or

$$R_0=\frac{1}{2}{\left[\frac{{\lambda }_m^t{\lambda }_f^t}{({\delta }_m+b/2)({\delta }_f+b/2)}\right]}^{1/2},$$

respectively.

4.2.2. Endemic Equilibrium

To find an endemic equilibrium where the infectious states are positive, ${\overline{I}}_m^t>0$, ${\overline{I}}_f^t>0$, and S_m = 0 = S_f, the following equations must be satisfied:

$$\frac{b^t{\overline{I}}_f^t}{T}-(a+\mathit{cT})-{\delta }_m=0$$ (26)

and

$$\frac{b^t{\overline{I}}_m^t}{T}-(a+\mathit{cT})-{\delta }_f=0.$$ (27)

Adding equations (26) and (27) and substituting $T={\overline{I}}_m^t+{\overline{I}}_f^t$ yields

$$T=\frac{b^t-{\delta }_m-{\delta }_f-2a}{2c}.$$ (28)

Hence, the equilibrium values for an endemic equilibrium (EE) are

$${\overline{I}}_f^t=\frac{(a-cT+{\delta }_m)T}{b^t}\mathrm{and}{\overline{I}}_m^t=\frac{(a+cT+{\delta }_f)T}{b^t},$$ (29)

where T is defined in (28). For ${\overline{I}}_m^t>0$ and ${\overline{I}}_f^t>0$, then

$$b^t-{\delta }_m-{\delta }_f-2a>0.$$ (30)

To determine the stability of the equilibrium, defined in (29), for model (20)-(21), the Jacobian matrix J is evaluated at this equilibrium. Matrix J has the following form:

$$J=\left[\begin{array}{cc}{J}_1& 0\\ 0& J_2\end{array}\right].$$

After simplification, J₁ and J₂ satisfy

$$J_1=\left[\begin{array}{cc}-\frac{{\lambda }_m^{t}B}{A+B}-{\beta }_m^{t}A-{\beta }^{t}B-c(A+B)& 0\\ 0& -\frac{{\lambda }_f^{t}A}{A+B}-{\beta }^t(A+B)-a-c(A+B)\end{array}\right]$$

and

$$J_2=\left[\begin{array}{cc}-\frac{b^{t}AB}{{\left(A+B\right)}^2}-cA& \frac{b^t{A}^2}{{\left(A+B\right)}^2}-cA\\ \frac{b^t{B}^2}{{\left(A+B\right)}^2}-cB& \frac{b^{t}AB}{{\left(A+B\right)}^2}-cB\end{array}\right],$$

where $A={\overline{I}}_m^t$, $B={\overline{I}}_f^t$ and A + B = T. It is straightforward to see that the eigenvalues of J₁ are negative and the trace of J₂ is also negative if A and B are positive. Also the determinant of J₂ simplifies to

$$\det (J_2)=\frac{2c{b}^t\mathit{AB}}{A+B}.$$

Thus if A, B > 0, the trace of J₂ is negative and the determinant of J₂ is positive. Applying the Routh-Hurwitz criteria implies the eigenvalues of J₂ have negative real parts provided the equilibrium values are positive [3]. Whether the equilibrium values are positive and the equilibrium is locally asymptotically stable depends on inequality (30). Hence, the following ratio is defined as a reproduction number for the EE (29),

$$R=\frac{b^t}{{\delta }_m+{\delta }_f+2a}.$$ (31)

The expression for R is the ratio of the birth rate to death rates for the infectious stage. The results are summarized for the EE of the SI model in the next theorem.

Theorem 4

The SI male-female model (20)-(21) has a unique EE, where S̄_m = 0 = S̄_f and ${\overline{I}}_m^t$ and ${\overline{I}}_f^t$ are defined in (29). This EE is locally asymptotically stable if R > 1 and unstable if R < 1, where R is defined in (31).

It is interesting to note that there exist parameter values such that R₀ < 1 and R > 1 so that the DFE (4) and the EE (29) are both locally asymptotically stable, a case of bistability. A numerical example of bistability is given in Section 6.

5. Simple SI Model

5.1. Model Description

A simplification of model (20)-(21) is considered that does not differentiate between the sexes. Therefore, the differences in sexual behavior and sexual transmission are not included. In this model, there are approximately the same number of males and females so that only females are modeled (alternately, only males could be modeled). Let S = number of susceptible females, I = number of infectious, immunotolerant females, and T = S + I = total number of females (the superscript t on I is omitted for simplicity). The birth rates are simplified. The expression bS_mS_f /T is replaced by bS/2 and b^t(S_mI_f + FI_m)/T is replaced by b^tI/2. The fraction 1/2 is included because the litter size in the new model only includes females. The simplified SI model has the following form:

$$\begin{array}{l}\frac{\mathit{dS}}{\mathit{dt}}=S\left(\frac{b}{2}-d(T)-{\beta }^{t}I\right)\\ \frac{\mathit{dI}}{\mathit{dt}}=I\left(\frac{b^t}{2}-\delta -d(T)+{\beta }^{t}S\right),\end{array}$$ (32)

where d(T) = a + 2cT, 0 < a < b/2, c > 0, b > b^t and d (K/2) = b/2. Note that d(T) differs from previous models due to the fact that only half of the population is modeled, 2T=males + females. These are reasonable assumptions if males and females are approximately equal in number. Initial conditions and parameters are assumed to be positive. Therefore, solutions are positive for t ≥ 0.

Model (32) is similar to a model developed by Busenberg and Cooke [8]. But Busenberg and Cooke do not assume density-dependent deaths. They assume d(T) = r ≡ constant. In addition, Busenberg and Cooke assume incomplete vertical transmission. That is, b^tI/2 is split into two rates, pb^tI/2 is the rate at which new susceptibles are born (from infectious mothers) and qb^tI/2 is the rate at which new infectives are born, p + q = 1.

Although the simple model (32) does not distinguish between males and females and therefore, may be somewhat unrealistic for Machupo virus, the effect of vertical transmission can be more clearly seen in this simple model. In addition, model (32) can be completely analyzed and the results interpreted in terms of R₀.

5.2. Analysis of the Simple Model

Model (32) has four equilibria. These four equilibria are differentiated by subscripts on the variables, the zero equilibrium, (S̄₀, Ī₀) = (0, 0), the DFE,

$$({\overline{S}}_1,{\overline{I}}_1)=\left(\frac{b/2-a}{2c},0\right)=\left(\frac{K}{2},0\right),$$

and possibly two endemic equilibria,

$$({\overline{S}}_2,{\overline{I}}_2)=\left(0,\frac{b^t/2-(a+\delta )}{2c}\right),$$

and

$$({\overline{S}}_3,{\overline{I}}_3)=\left(\frac{2c({\overline{S}}_1-{\overline{I}}_2)-{\beta }^t{\overline{I}}_2}{{\left({\beta }^t\right)}^2/2c},\frac{{\beta }^t{\overline{S}}_1-2c({\overline{S}}_1-{\overline{I}}_2)}{{\left({\beta }^t\right)}^2/2c}\right).$$

The DFE is always positive since (b/2 − a)/(2c) = K/2 > 0.

At the endemic equilibrium (S̄₂, Ī₂), S̄₂ = 0 and Ī₂ > 0, if

$$\frac{b^t}{2}>a+\delta .$$ (33)

This inequality can be rewritten as b^t/(2a + 2δ) > 1 which is equivalent to the inequality R > 1 in (31) when δ_m = δ_f = δ.

At the endemic equilibrium (S̄₃, Ī₃), S̄₃ > 0 and Ī₃ > 0, if 2c(S̄₁ − Ī₂) > β^tĪ₂ and β^tS̄₁ > 2c(S̄₁ − Ī₂). Simplifying the latter two inequalities leads to

$${\overline{S}}_1>\left(\frac{{\beta }^t}{2c}+1\right){\overline{I}}_2>\left(1-{\left(\frac{{\beta }^t}{2c}\right)}^2\right){\overline{S}}_1.$$ (34)

Note that Ī₂ may not be feasible (Ī₂ ≤ 0) and the endemic equilibrium (S̄₃, Ī₃) could still be positive. In particular, Ī₂ ≤ 0, S̄₃ > 0, and Ī₃ > 0, provided β^t > 2c and −Ī₂ < (β^t/2c − 1) S̄₁. In addition, if Ī₂ ≤ 0 and if S̄₃ > 0, it follows from the definition of S̄₃ that S̄₃ < S̄₁. On the other hand, if Ī₃ > 0 its’ value may be less than or greater than Ī₂ depending on the magnitude of horizontal transmission. If Ī₃ > 0 and β^t > 2c, then the definition of Ī₃ implies Ī₃ > Ī₂. But if Ī₃ > 0 and β^t < 2c, then Ī₃ < Ī₂.

Next, the local stability of each equilibrium is studied. The Jacobian matrix of model (32) evaluated at the zero equilibrium is

$$J_0=\left[\begin{array}{cc}\frac{b}{2}-a& 0\\ 0& \frac{b^t}{2}-\delta -a\end{array}\right].$$

The eigenvalues of J₀ are b/2 − a > 0 and b^t/2 − δ − a. If b^t/2 > δ + a, then the zero equilibrium is an unstable node. If b^t/2 < δ + a, then the zero equilibrium is a saddle point.

The Jacobian matrix evaluated at the DFE (S̄₁, Ī₁) simplifies to

$$J_1=\left[\begin{array}{cc}-\frac{cK}{2}& -({\beta }^t+2c)\frac{K}{2}\\ 0& \frac{b^t-b+{\beta }^{t}K}{2}-\delta \end{array}\right].$$

The DFE is locally asymptotically stable if (b^t − b + β^tK)/2 < δ. This latter inequality is equivalent to

$$\frac{({\beta }^{t}K+b^t)/2}{\delta +b/2}<1.$$ (35)

The left side of (35) is the basic reproduction number for model (32)

$$R_0=\frac{({\beta }^{t}K+b^t)/2}{\delta +b/2}.$$ (36)

Note that the value of R₀ is equal to the female reproduction number for the male-female SI model, given in (24). Calculation of R₀ can also be found from the criteria of van den Driessche and Watmough [29].

At the endemic equilibrium (S̄₂, Ī₂), the Jacobian matrix has the form

$$J_2=\left[\begin{array}{cc}\frac{b}{2}-a-2cT-{\beta }^t{\overline{I}}_2& 0\\ {\overline{I}}_2{\beta }^t-2c{\overline{I}}_2& \frac{b^t}{2}-\delta -a-2cT-2c{\overline{I}}_2\end{array}\right].$$

Eigenvalues of J₂ are given by b/2 − a − (2c + β^t)Ī₂ and −2cĪ₂ where T = Ī₂. For the first eigenvalue to be negative requires that (2c + β^t)Ī₂ > 2cS̄₁ or equivalently

$${\overline{S}}_1<\left(\frac{{\beta }^t}{2c}+1\right){\overline{I}}_2,$$ (37)

where S̄₁ = K/2. Inequality (37) contradicts (34). Thus, if the endemic equilibrium (S̄₂, Ī₂) is stable, then the endemic equilibrium (S̄₃, Ī₃) is not feasible (not positive). Conversely, if the endemic equilibrium (S̄₃, Ī₃) is positive, the inequalities in (34) hold and equilibrium (S̄₂, Ī₂) is unstable.

Next, evaluating the Jacobian matrix at (S̄₃, Ī₃), leads to

$$J_3=\left[\begin{array}{cc}\frac{b}{2}-a-2\mathit{cT}-{\beta }^t{\overline{I}}_3-2c{\overline{S}}_3& -{\beta }^t{\overline{S}}_3-2c{\overline{S}}_3\\ {\beta }^t{\overline{I}}_3-2c{\overline{I}}_3& \frac{b^t}{2}-\delta -a-2\mathit{cT}+{\beta }^t{\overline{S}}_3-2c{\overline{I}}_3\end{array}\right],$$

where T = S̄₃ + Ī₃. The trace of J₃ simplifies to −(b − b^t + 2δ)c/β^t. It is negative since b + 2δ > b^t. The determinant of J₃ simplifies to (β^t)²S̄₃Ī₃. It is positive if Ī₃ > 0 and S̄₃ > 0. It follows from the Routh-Hurwitz criteria that the endemic equilibrium is locally asymptotically stable if Ī₃ > 0 and S̄₃ > 0.

The expression R₀ < 1, where R₀ is defined in (36), is equivalent to

$${\overline{S}}_1\left(1-\frac{{\beta }^t}{2c}\right)>{\overline{I}}_2.$$ (38)

This latter inequality is the reverse of the second inequality in (34). Therefore, for (S̄₃, Ī₃) to be positive, it must be the case that R₀ > 1. Also note that R₀ ≤ 1 and Ī₂ > 0 imply that Ī₂ cannot be locally asymptotically stable. This result comes from the following observations. If Ī₂ > 0 is locally asymptotically stable, then inequalities (37) and (38) imply

$${\overline{I}}_2<{\overline{S}}_1\left(1-\frac{{\beta }^t}{2c}\right)<\left(1-{\left(\frac{{\beta }^t}{2c}\right)}^2\right){\overline{I}}_2,$$

a contradiction. Consequently, this rules out bistability for the simple SI model when R₀ < 1. The preceding local asymptotic stability results together with Poincaré-Bendixson theory [11]. for autonomous two-dimensional systems will be used to verify that the local results are global results for model (32).

Theorem 5

Let R₀ be defined in (36).

1. If R₀ ≤ 1, then the DFE (S̄₁, Ī₁) of model (32) is globally asymptotically stable.

2. If R₀ > 1, then there are two mutually exclusive cases.

   1. If the inequalities (33) and (37) are satisfied, then the endemic equilibrium (S̄₂, Ī₂) of model (32) is globally asymptotically stable.

   2. If one of the inequalities in (33) or (37) is not satisfied, then inequality (34) is satisfied and the endemic equilibrium (S̄₃, Ī₃) of model (32) is globally asymptotically stable.

Proof

First, it is shown that solutions to model (32) are bounded. Solutions are bounded below by zero. To show solutions are bounded above, consider

$$\frac{\mathit{dS}}{\mathit{dt}}\le S\left(\frac{b}{2}-a-2\mathit{cS}\right).$$

The solution S can be compared to the solution u₁ that satisfies the following differential equation:

$$\frac{d{u}_1}{\mathit{dt}}=u_1\left(\frac{b}{2}-a-2c{u}_1\right).$$

For u₁(0) = S(0) > 0, the solution u₁(t) satisfies lim_t→∞ u₁(t) = K/2. In addition u₁(t) approaches K/2 monotonically. Thus, S(t) ≤ u₁(t) for t ≥ 0 which implies S(t) ≤ max_t≥0 {u₁(t)} ≤ max {S(0), K/2} = Ŝ. Applying this inequality to I(t) in model (32) yields

$$\frac{\mathit{dI}}{\mathit{dt}}\le I\left(\frac{b^t}{2}-\delta -a-2\mathit{cI}+{\beta }^t\widehat{S}\right).$$

Again the solution I can be compared to the solution u₂, where

$$\frac{d{u}_2}{\mathit{dt}}=u_2\left(\frac{b^t}{2}-\delta -a-2c{u}_2+{\beta }^t\widehat{S}\right)$$

with u₂(0) = I(0) > 0. Since the solution of u₂(t) is bounded, I(t) is bounded, I(t) ≤ u₂(t), t ≥ 0.

Note that three of the equilibria lie on the boundary of the S-I phase plane and (S₃, I₃) is the only one that may lie in the interior. The origin is always unstable and no solutions beginning in the interior of the S-I phase plane can approach (S₀, I₀).

1. The assumptions in part (a) imply there is no interior equilibrium (S̄₃, Ī₃). There are only three boundary equilibria. The origin (S̄₀, Ī₀) is an unstable node (if Ī₂ ≤ 0) or a saddle point (if Ī₂ > 0). Also, as noted previously, if R₀ < 1, (S̄₂, Ī₂) is not locally asymptotically stable. Hence, applying Poincaré-Bendixson theory, solutions must approach (S̄₁, Ī₁).

2. Assume R₀ > 1.

   1. The assumptions in part (i) imply Ī₂ > 0, (S̄₂, Ī₂) is locally asymptotically stable and (S̄₃, Ī₃) is not in the interior of the S-I phase plane. In addition, the two boundary equilibria (S̄₀, Ī₀) and (S̄₁, Ī₁) are unstable nodes. Hence, applying Poincaré-Bendixson theory, solutions must approach the stable equilibrium (S̄₂, Ī₂).

   2. Suppose inequality (33) does not hold. That is b^t/2 ≤ a + δ. This latter inequality with the assumption R₀ > 1 imply β^t > 2c. But this means that the inequalities in (34) are satisfied, (S₃, I₃) lies in the interior of the S-I phase plane. Now suppose inequality (37) does not hold. Then S̄₁ ≥ (β²/(2c) + 1)Ī₂ and R₀ > 1 imply that the inequalities (34) are satisfied and again (S₃, I₃) lies in the interior of the S-I phase plane.

It has already been shown that if the inequalities in (34) hold (also R₀ > 1) and if Ī₂ > 0, then the boundary equilibrium (0, Ī₂) is unstable (with unstable trajectory pointing into the interior of the S-I phase plane). In addition, the other two boundary equilibria (S̄, Ī₀) and (S̄₁, Ī₁) are unstable with trajectories pointing into the interior of the S-I phase plane.

Periodic solutions need to be ruled out. We apply Dulac’s criteria. Let D be an open region in the positive quadrant of the S-I phase plane and let f(S, I) and g(S, I) be the right sides of the differential equations for S and I in model (32), respectively. In addition, for the Dulac function B(S, I) = 1/(SI),

$$\frac{\partial (\mathit{Bf})}{\partial S}+\frac{\partial (\mathit{Bg})}{\partial I}=-\left(\frac{2c}{I}+\frac{2c}{S}\right)<0.$$

Dulac’s criteria implies there are no periodic solutions in D [3]. Thus, Poincaré-Bendixson theory implies solutions must approach an equilibrium, a homoclinic trajectory or a heteroclinic trajectory. Since (S̄₃, Ī₃) is locally asymptotically stable and the boundary equilibria are all unstable, there can be no heteroclinic nor homoclinic trajectories. Solutions must approach (S̄₃, Ī₃).

In Figure 2 the stable equilibria are graphed as a function of R₀(β^t). All parameters are fixed except β^t (b = 3, b^t = b/3, K = 10, 000, a = 0.25, and δ = 0.5). The basic reproduction number R₀ is an increasing function of β^t. Two transcritical bifurcations occur, one at R₀ = 1, where I switches from 0 to I₃, and a second one at R₀ = 7.88, where I switches from I₃ to I₂ ≈ 455.

Figure 2.

Bifurcation diagram of the stable equilibria as a function of R₀ for the SI model (32), where R₀ is an increasing function of β^t, R₀ ≡ R₀(β^t). Other parameter values are b = 3, b^t = b/3, K = 10, 000, a = 0.25, and δ = 0.5. The dashed curve is the stable total population size S + I and the solid curve is the stable infectious population size I.

The SI model with vertical transmission can result in the entire population becoming infectious (at the equilibrium value Ī₂). This result also occurs in the SI male-female model (Theorem 4) and is a consequence of vertical transmission. The results of Theorem 5 show that bistability, observed in the SI male-female model, does not occur in the simple SI model.

6. Numerical Examples

6.1. Basic Parameter Values

Several numerical examples are presented and discussed that illustrate the dynamics of the models. Basic parameter values for the numerical examples are listed in Table 1. Time units are measured in increments of three months, since the gestation period until sexual maturity is approximately three months [16]. Hence, 10 time units equals 2.5 years and 20 time units equals 5 years.

Table 1.

Basic Parameter Values

Parameter	Value	Parameter	Value
b	6	${\lambda }_m^t$	0.01
bt	b/3	${\lambda }_m^c$, ${\lambda }_f^t$, ${\lambda }_f^c$	${\lambda }_m^t$
bc	b/2	γm	5
${\beta }_m^t$	0.0002 to 0.004	γf	2γm
${\beta }_m^c$	${\beta }_m^t$	δm	0.5 to 3
βt, βc	${\beta }_m^t/10$	δf	δm/2
a	0.25	K	10, 000
ε	0.1	c	(b/2 − a)/K

The per capita natural birth rate (and survival for three months, until reproductive) is assumed to be b = 6 [16]. For the general model, b^t + b^c ≤ b and b^t < b^c. Therefore, we assume b^t = b/3 and b^c = b/2. Horizontal transmission between males is greater than between males and females or between females. Therefore, ${\beta }^t={\beta }_m^t/10$ and ${\beta }^c={\beta }_m^c/10$. A range of values is selected for ${\beta }_m^t$, ${\beta }_m^t\in [0.0002,0.004]$. The carrying capacity K is equal to 10,000 rodents. The per capita density-independent natural death rate is a = 0.25 and the per capita density-dependent natural death rate c = (b/2 − a)/K. Thus, the average life-expectancy in the absence of density-dependent effects is 1/a ≈ 1 year. The sexual transmission coefficients are assumed to be the same for males and females and both infectious stages; although unknown, relatively small values are assumed ${\lambda }_m^t={\lambda }_f^t={\lambda }_m^c={\lambda }_f^c=0.01$. The transition rate from the immunocompetent stage to the immunotolerant stage is assumed to be ε = 0.1. The per capita male recovery rate is γ_m = 5 with female recovery rate, γ_f = 2γ_m (male infectious period is greater than female, e.g., 1/γ_m = 1/5 = 2.4 weeks versus 1/γ_f = 1/10 = 1.2 weeks) and the per capita male disease-related death rate is in the range δ_m ∈ [0.5, 3]. with lower death rate for females δ_f = δ_m/2 (e.g., 1=δ_m = 1 = 3 months, death occurs 3 months after infection). The values for many of these parameters are unknown but the values chosen for the simulations are biologically reasonable. The dynamics of two of the models, general male-female model and male-female SI model, are illustrated using the baseline parameter values given in Table 1.

6.2 General Model

Two numerical examples illustrate the stability of the DFE (R₀ < 1) and of an endemic equilibrium (R₀ > 1) for the general male-female model (1)-(2).

Example 1

In the first numerical example, let ${\beta }_m^t=0.0002$ and δ_m = 3. The remaining parameters are given in Table 1. The basic reproduction number is R₀ = 0.911. According to Theorem 1, the DFE is locally asymptotically stable. Solutions approach the DFE very rapidly (see Figure 3), S̄_m = 5000 = S̄_f. As sexual or horizontal transmission rates increase or as the recovery rates or disease-related death rates decrease or as transition to the immunotolerant stage increases, then the basic reproduction number increases. For example, if ε is increased from 0.1 to 1 or to 10, then R₀ = 0.933 for ε = 1 and R₀ = 1.047 for ε = 10.

Figure 3.

Solutions corresponding to Example 1, the general male-female model (1)-(2), where R₀ =0.911 with the initial conditions S_m(0) = 3000 S_f (0) = 3500, $I_m^t(0)=150$, $I_f^t(0)=100$, $I_m^c(0)=50$, $I_f^c(0)=25$, $R_m^c(0)=1000$ and $R_f^c(0)=1500$. Solution approach the DFE, S̄_m = 5000 = S̄_f.

Example 2

In the second numerical example, the disease-related death rates are decreased to δ_m = 0.5 and δ_f = δ_m/2 = 0.25. The remaining parameter values are the same as in Example 1. The basic reproduction number is R₀ = 1.213. According to Theorem 1, the DFE is unstable. Anendemic equilibrium is reached,

$$\begin{array}{l}{\overline{S}}_m\approx 2303,{\overline{I}}_m^t\approx 826,{\overline{I}}_m^c\approx 460,{\overline{R}}_m^c\approx 847\\ {\overline{S}}_f\approx 2683,{\overline{I}}_f^t\approx 715,{\overline{I}}_f^c\approx 240,{\overline{R}}_f^c\approx 885.\end{array}$$ (39)

The total population size is less than the carrying capacity due to reduced fertility and disease-related deaths, T̄ = 8, 959 < 10, 000 = K. A linear stability analysis shows that this endemic equilibrium is locally asymptotically stable. Figure 4 shows rapid convergence to this endemic equilibrium.

Figure 4.

Solutions corresponding to Example 2, the general male-female model (1)-(2), where R₀ = 1.213 with the same initial conditions as in Example 1. Solutions approach the endemic equilibrium (39).

6.3 The SI Model

Three numerical examples illustrate some of the dynamics for the SI^t male-female model (20)-(21).

Example 3

Let ${\beta }_m^t=0.0002$ and δ_m = 0.5. The remaining parameter values are given in Table 1. The basic reproduction number for the DFE is R₀ = 0.803 < 1 and the reproduction number for the EE is R = 1.6 > 1. According to Theorems 3 and 4, the DFE, S̄_m = 5000 = S̄_f, and the EE, Ī_m ≈ 597, Ī_f ≈ 767, are locally asymptotically stable. Depending on the initial conditions solutions may tend to one of these equilibria. For initial conditions close to the DFE, solutions tend to DFE but for initial conditions close to the EE, solutions tend to the EE. This is a case of bistability.

Example 4

In this numerical example, we use the same parameter values as in Example 3 with the exception that ${\beta }_m^t$ is increased to ${\beta }_m^t=0.0006$. Then both reproduction numbers are greater than one, R₀ = 1.308 and R = 1.6. According to Theorems 3 and 4, the DFE is unstable and the EE, Ī_m ≈ 597, Ī_f ≈ 767, is locally asymptotically stable. For all of the numerical simulations, with I_m(0) > 0 and I_f (0) > 0, we found that solutions approached the EE (see Figure 5).

Figure 5.

Solution to the SI^t male-female model (20)-(21), corresponding to the parameter values given in Example 4. A stable endemic equilibrium is reached, where Ī_m ≈ 597, Ī_f ≈ 767, R₀ = 1.308 and R = 1.6.

Example 5

As a final example, we show the existence of an endemic equilibrium (EE) different from the one predicted in Theorem 4. This EE is stable for some parameter values if R₀ > 1 and R < 1 but becomes unstable as horizontal transmission ${\beta }_m^t$ increases. Let ${\beta }_m^t=0.0008$ and δ_m = 1.5 with the remaining parameter values as in Table 1. In this case, R₀ = 1.245 and R = 0.727. There exists a stable EE,

$${\overline{S}}_m\approx 2304,{\overline{I}}_m^t\approx 753,{\overline{S}}_f\approx 2889,{\overline{I}}_f^t\approx 529.$$

The total population size T̄ ≈ 6475 is much reduced from a carrying capacity of K = 10, 000. As ${\beta }_m^t$ increases, the endemic equilibrium loses its stability and solutions exhibit cyclic behavior. For example, if ${\beta }_m^t$ is increased to ${\beta }_m^t=0.004$ but δ_m = 1.5 remains the same with all other parameters as in Table 1, then R₀ = 4.8 and R = 0.727; solutions appear to converge to a periodic solution. Solutions are graphed over time and in the S_m-I_m phase plane in Figure 6. Because the infectious cycles are very close to zero for part of the cycle, the disease may not persist in the long run and cyclic behavior may not be observed, only an outbreak. For example, stochastic simulations show that the disease does not persist [7].

Figure 6.

Cyclic behavior for the SI^t male-female model (20)-(21) when R₀ = 4.8 and R = 0.727 ${\beta }_m^t=0.004$, δ_m = 1.5 and other parameter values are given in Table 1). The figure on the left is a graph of the solutions over time and the figure on the right is the graph of the solution in the S_m-I_m phase plane. The + in the figure on the right identifies the unstable endemic equilibrium S̄_m ≈ 81.3, Ī_m ≈ 208.5.

7 Conclusion

Deterministic mathematical models were formulated that describe the epizoology of Machupo virus in its reservoir host, Calomys callosus. The models follow the changes in the disease stages of the rodent population over time. Rodents are classified as susceptible, infectious or recovered. The models incorporate the differences in male and female behavior, vertical, horizontal and sexual modes of transmission, and differences in immune reponse, immunocompetent and immunotolerant. A general male-female model was formulated and analyzed, model (1)-(2). The possible roles played by the immunocompetent and immunotolerant infectious stages were investigated in two special cases of this general model, where only one type of immune response was considered, SI^cR^c model and SI^t model, respectively. For each of these models, formulas were derived for the basic reproduction number R₀, one of the most important parameters in mathematical epidemiology. When R₀ < 1, if a small number of infectious rodents are introduced into the population, the disease does not persist, but if R₀ > 1, then an outbreak may occur and the disease may approach an endemic level. It was shown that the basic reproduction number depends on a combination of parameters, including parameters for horizontal, vertical and sexual transmission, birth rates for susceptible and infected rodents, disease-related death rates, natural death rates, and the carrying capacity of the rodent population.

The dynamics of the models were studied analytically and numerically. In the general male-female model (1)-(2) and the SI^cR^c male-female model (13)-(14), for the chosen parameter values, the numerical results indicated global stability of the DFE when R₀ < 1. However, in the SI^t model, more complex behavior was observed–periodicity and bistability. In Example 5, the SI^t model exhibited cyclic behavior for large values of R₀ > 1 and R < 1 (Figure 6). In the cyclic behavior of Example 5, an outbreak occurs then disappears rapidly but then occurs again. But when R₀ < 1 and R > 1 in the SI^t model, bistability occurs. either the disease dies out or the disease persists at an endemic level (Example 3). If the disease persists, the entire population becomes infected. This type of behavior makes it very difficult to control a disease, it is not sufficient to reduce R₀ below unity, R must also be reduced below unity. In the simple SI model, where only females are modeled, bistability does not occur. Therefore, the male-female dynamics are important for bistability. Although these simpler models provide insight into the roles played by the immunocompetent and immunotolerant stages of infection, their individual dynamics may not be observed in the more general male-female model.

The general male-female model (1)-(2) is the one that best describes the epizoology and immunology of Calomys callosus infected with Machupo virus. For the range of parameter values used in the numerical experiments, the general model did not exhibit the behavior shown in the SI^t model - bistability or periodicity. The SI^cR^c model with only the immunocompetent stage did not exhibit these properties either. The immunocompetent stage may provide greater population stability to the general male-female model. The basic reproduction number for the general model male-female (1)-(2), given in (9), shows that disease outbreaks or disease persistence (R₀ greater than unity) are a result of both of these infectious stages. In addition, the population size in the general model will be reduced below carrying capacity when there is infection due to reduced fertility and disease-related deaths caused by the immunotolerant stage. Horizontal, vertical and sexual transmission all contribute to the persistence of the disease in the general male-female model. Individually or collectively, these three modes of transmission can lead to disease outbreaks or disease persistence in a rodent population (equations (9), (11), and (12)).

To control the disease in a rodent population and prevent human infection, rodent control is the obvious method and the one applied in Bolivia during the 1960s [17]. Rodent control programs have the effect of reducing the value of K, thereby reducing R₀. No cases of BHF were reported from 1973 to 1992 [17]. More recently, cases of BHF have been reported in Bolivia but may be due to lack of continued aggressive rodent control measures [17].

Our future goals are to obtain better estimates for the model parameters and a better understanding of the functional forms for transmission (frequency versus density-dependent transmission). Estimates of the model parameters are needed to determine the threshold parameter R₀ and to provide realistic model simulations. Analysis of these models has given us a greater understanding of the dynamics that are possible in the Machupo-rodent model. Whether all of these dynamics occur in nature require further study. The effects of spatial and environmental variations on the population and the disease dynamics need to be considered in a more complex model.