Analysis of Risk-Structured Vaccination Model for the Dynamics of Oncogenic and Warts-Causing HPV Types

Abstract

A new deterministic model is designed and used to assess the community-wide impact of mass vaccination of new sexually active individuals on the dynamics of the oncogenic and warts-causing HPV types. Rigorous qualitative analyses of the model, which incorporates the two currently available anti-HPV vaccines, reveal that it undergoes competitive exclusion when the reproduction of one HPV risk type (low/high) exceeds unity, while that of the other HPV risk type is less than unity. For the case when the reproduction numbers of the two HPV risk types (low/high) exceed unity, the two risk types co-exist. It is shown that the sub-model with the low-risk HPV types only has at least one endemic equilibrium whenever the associated reproduction threshold exceeds unity. Furthermore, this sub-model undergoes a re-infection-induced backward bifurcation under certain conditions. In the absence of the re-infection of recovered individuals and cancer-induced mortality in males, the associated disease-free equilibrium of the full (risk-structured) model is shown to be globally asymptotically stable whenever the reproduction number of the model is less than unity (that is, the full model does not undergo backward bifurcation under this setting). It is shown, via numerical simulations, that the use of the Gardasil vaccine could lead to the effective control of HPV in the community if the coverage rate is in the range of 73–95 % (84 %). If 70 % of the new sexually active susceptible females are vaccinated with the Gardasil vaccine, additionally vaccinating 34–56 % (45 %) of the new sexually active susceptible males can lead to the effective community-wide control (or elimination) of the HPV types.

Appendices

Appendices

1 Description of the Model (3)

The population of unvaccinated susceptible females ($S_\mathrm{{f}}$) is increased by the recruitment of new sexually active females at a rate $\pi _\mathrm{{f}}$ (a fraction, $1-\varphi _\mathrm{{f}}^\mathrm{{b}}-\varphi _\mathrm{{f}}^\mathrm{{q}}$, with $0< \varphi _\mathrm{{f}}^\mathrm{{b}}+\varphi _\mathrm{{f}}^\mathrm{{q}}\le 1$, of which, is vaccinated; where $\varphi _\mathrm{{f}}^\mathrm{{b}}$ is the fraction of unvaccinated susceptible females vaccinated with the Cervarix vaccine, and $\varphi _\mathrm{{f}}^\mathrm{{q}}$ is the fraction vaccinated with the Gardasil vaccine). This population is further increased by the loss of infection-acquired immunity by infected females who recovered without developing cervical cancer (at a rate $\xi _\mathrm{{f}}$). The population is decreased by infection, following effective contacts with males infected with the high-risk and the low-risk HPV types (i.e. those in the $E_\mathrm{{m}}^{\ell }, E_\mathrm{{m}}^\mathrm{{h}}, I_\mathrm{{m}}^{\ell }, I_\mathrm{{m}}^\mathrm{{h}}, P_\mathrm{{m}}^{\ell }$ and $P_\mathrm{{m}}^\mathrm{{h}}$ classes), at the rates $\lambda _\mathrm{{m}}^{\ell }$ and $\lambda _\mathrm{{m}}^\mathrm{{h}}$ given, respectively, by

$$\begin{aligned} \lambda _\mathrm{{m}}^{\ell }&= \frac{\beta _\mathrm{{m}}^{\ell }c_\mathrm{{f}}\left( N_\mathrm{{m}},N_\mathrm{{f}}\right) \left( \eta _\mathrm{{m}}^{\ell }E_\mathrm{{m}}^{\ell }+I_\mathrm{{m}}^{\ell }+\theta _\mathrm{{m}}^{\ell }P_\mathrm{{m}}^{\ell }\right) }{N_\mathrm{{m}}},\end{aligned}$$

(14)

$$\begin{aligned} \lambda _\mathrm{{m}}^\mathrm{{h}}&= \frac{\beta _\mathrm{{m}}^\mathrm{{h}}c_\mathrm{{f}}\left( N_\mathrm{{m}},N_\mathrm{{f}}\right) \left( \eta _\mathrm{{m}}^\mathrm{{h}}E_\mathrm{{m}}^\mathrm{{h}}+I_\mathrm{{m}}^\mathrm{{h}}+\theta _\mathrm{{m}}^\mathrm{{h}}P_\mathrm{{m}}^\mathrm{{h}}\right) }{N_\mathrm{{m}}}. \end{aligned}$$

(15)

In (14), $\beta _\mathrm{{m}}^{\ell }$ is the probability of transmission of HPV infection from infected males (with the low-risk HPV types) to susceptible females per contact, and $c_\mathrm{{f}}\left( N_\mathrm{{m}},N_\mathrm{{f}}\right)$ is the average number of female partners per male per unit time. Thus, $\beta _\mathrm{{m}}^{\ell }c_\mathrm{{f}}\left( N_\mathrm{{m}},N_\mathrm{{f}}\right)$ is the effective contact rate for male-to-female transmission of the low-risk HPV types. Furthermore, $\eta _\mathrm{{m}}^{\ell }$ (with $0\le \eta _\mathrm{{m}}^{\ell } <1$) is the modification parameter accounting for the assumption that exposed males with the low-risk HPV types are less infectious than symptomatically infected males with the low-risk HPV types. Similarly, in (15), $\beta _\mathrm{{m}}^\mathrm{{h}}$ is the probability of transmission of HPV infection from infected males (with the high-risk HPV types) to susceptible females per contact, and $\beta _\mathrm{{m}}^\mathrm{{h}}c_\mathrm{{f}}\left( N_\mathrm{{m}},N_\mathrm{{f}}\right)$ is the effective contact rate for male-to-female transmission of the high-risk HPV types. The parameter $\eta _\mathrm{{m}}^\mathrm{{h}}$ (with $0\le \eta _\mathrm{{m}}^\mathrm{{h}} <1$) accounts for the assumption that exposed males with the high-risk HPV types are less infectious than symptomatically infected males with the high-risk HPV types, and $\theta _\mathrm{{m}}^{\ell } (\theta _\mathrm{{m}}^\mathrm{{h}})>0$ is the modification parameter accounting for the assumption that infected males with persistent infection with the low-risk (high-risk) HPV types transmit HPV at a different rate compared to infected males in the other infected classes $(E_\mathrm{{m}}^{\ell }, I_\mathrm{{m}}^{\ell } (E_\mathrm{{m}}^\mathrm{{h}}, I_\mathrm{{m}}^\mathrm{{h}}))$. The population of unvaccinated susceptible females is further decreased by natural death (at a rate $\mu _\mathrm{{f}}$; it is assumed that females in all epidemiological compartments suffer natural death at the rate $\mu _\mathrm{{f}}$). Standard incidence formulation is used in (14) and (15), where the contact rate is assumed to be constant, unlike in the case of the mass action incidence (where the contact rate increases linearly with the total size of the population Hethcote 2000; Lakshmikantham et al. 1989). Thus,

$$\begin{aligned} \frac{\mathrm{{d}}S_\mathrm{{f}}}{\mathrm{{d}}t} = (1-\varphi _\mathrm{{f}}^\mathrm{{b}}-\varphi _\mathrm{{f}}^\mathrm{{q}})\pi _\mathrm{{f}}+\xi _\mathrm{{f}}R_\mathrm{{f}}- \left( \lambda _\mathrm{{m}}^{\ell }+\lambda _\mathrm{{m}}^\mathrm{{h}}\right) S_\mathrm{{f}}-\mu _\mathrm{{f}}S_\mathrm{{f}}. \end{aligned}$$

(16)

The population of new sexually active susceptible females vaccinated with the bivalent Cervarix vaccine ($V_\mathrm{{f}}^\mathrm{{b}}$) is generated by the vaccination of a fraction, $\varphi _\mathrm{{f}}^\mathrm{{b}}$, of unvaccinated susceptible females with the Cervarix vaccine (at the rate $\pi _\mathrm{{f}}\varphi _\mathrm{{f}}^\mathrm{{b}}$). It is decreased by HPV infection, following effective contacts with males infected with high-risk HPV types (at the reduced rate $(1-\varepsilon _{b})\lambda _\mathrm{{m}}^\mathrm{{h}}$, where $0<\varepsilon _{b}\le 1$ represents the efficacy of the Cervarix vaccine against infection with the high-risk HPV types) and males infected with the low-risk HPV types (at the rate $\lambda _\mathrm{{m}}^{\ell }$; it should be emphasized that the Cervarix vaccine has no efficacy against the low-risk HPV types, HPV-6 and -11 Health Canada 2010; Public Health Agency of Canada 2010, 2007). This population is decreased by natural death. Since there is currently no evidence to the contrary, it is assumed that this vaccine (as well as Gardasil) does not wane Canadian Cancer Society 2010; Centres for Disease Control and Prevention 2012; Health Canada 2010; Public Health Agency of Canada 2010, 2007. Hence,

$$\begin{aligned} \frac{\mathrm{{d}}V_\mathrm{{f}}^\mathrm{{b}}}{\mathrm{{d}}t} = \varphi _\mathrm{{f}}^\mathrm{{b}} \pi _\mathrm{{f}}-(1-\varepsilon _{b})\lambda _\mathrm{{m}}^\mathrm{{h}}V_\mathrm{{f}}^\mathrm{{b}}- \lambda _\mathrm{{m}}^{\ell }V_\mathrm{{f}}^\mathrm{{b}}-\mu _\mathrm{{f}}V_\mathrm{{f}}^\mathrm{{b}}. \end{aligned}$$

(17)

The population of new sexually active susceptible females vaccinated with the quadrivalent Gardasil vaccine ($V_\mathrm{{f}}^\mathrm{{q}}$) is generated by the vaccination of a fraction, $\varphi _\mathrm{{f}}^\mathrm{{q}}$, of unvaccinated susceptible females with the Gardasil vaccine (at the rate $\pi _\mathrm{{f}}\varphi _\mathrm{{f}}^\mathrm{{q}}$). It is decreased by HPV infection, following effective contacts with males infected with the low- and high-risk HPV types (at a reduced rate $(1-\varepsilon _{q})\left( \lambda _\mathrm{{m}}^{\ell }+\lambda _\mathrm{{m}}^\mathrm{{h}}\right)$, where $0<\varepsilon _{q}\le 1$ represents the efficacy of Gardasil vaccine against infection with HPV-6, -11, -16 and -18). This population is decreased by natural death. Thus,

$$\begin{aligned} \frac{\mathrm{{d}}V_\mathrm{{f}}^\mathrm{{q}}}{\mathrm{{d}}t} = \varphi _\mathrm{{f}}^\mathrm{{q}}\pi _\mathrm{{f}}-(1-\varepsilon _{q})\left( \lambda _\mathrm{{m}}^\mathrm{{h}}+ \lambda _\mathrm{{m}}^{\ell }\right) V_\mathrm{{f}}^\mathrm{{q}}-\mu _\mathrm{{f}}V_\mathrm{{f}}^\mathrm{{q}}. \end{aligned}$$

(18)

The population of exposed females with the low-risk (high-risk) HPV types ($E_\mathrm{{f}}^{\ell } (E_\mathrm{{f}}^\mathrm{{h}})$) is generated by the infection of unvaccinated and vaccinated susceptible females with the low-risk HPV types (at the rate $\lambda _\mathrm{{m}}^{\ell } (\lambda _\mathrm{{m}}^\mathrm{{h}})$). This population is further increased by the re-infection of recovered females with the low-risk (high-risk) HPV types (at a rate $\rho _\mathrm{{f}}^{\ell }\lambda _\mathrm{{m}}^{\ell } (\rho _\mathrm{{f}}^\mathrm{{h}}\lambda _\mathrm{{m}}^\mathrm{{h}})$, where $0\le \rho _\mathrm{{f}}^{\ell } ( \rho _\mathrm{{f}}^\mathrm{{h}})<1$ accounts for the assumption that the re-infection of recovered females with low-risk (high-risk) HPV types occurs at a rate lower than the rate for primary infection of susceptible females). Exposed females develop clinical symptoms of the low-risk (high-risk) HPV types (at a rate $\sigma _\mathrm{{f}}^{\ell } (\sigma _\mathrm{{f}}^\mathrm{{h}})$) and suffer natural death. Thus,

$$\begin{aligned} \frac{\mathrm{{d}}E_\mathrm{{f}}^{\ell }}{\mathrm{{d}}t}&= \left[ S_\mathrm{{f}}+V_{b}+(1-\varepsilon _{q})V_{q}\right] \lambda _\mathrm{{m}}^{\ell }+\rho _\mathrm{{f}}^{\ell }\lambda _\mathrm{{m}}^{\ell }R_\mathrm{{f}}-(\sigma _\mathrm{{f}}^{\ell }+\mu _\mathrm{{f}})E_\mathrm{{f}}^{\ell },\nonumber \\ \frac{\mathrm{{d}}E_\mathrm{{f}}^\mathrm{{h}}}{\mathrm{{d}}t}&= \left[ S_\mathrm{{f}}+(1-\varepsilon _{b})V_{b}+(1-\varepsilon _{q})V_{q}\right] \lambda _\mathrm{{m}}^\mathrm{{h}}+\rho _\mathrm{{f}}^\mathrm{{h}} \lambda _\mathrm{{m}}^\mathrm{{h}}R_\mathrm{{f}}-(\sigma _\mathrm{{f}}^\mathrm{{h}}+\mu _\mathrm{{f}})E_\mathrm{{f}}^\mathrm{{h}}. \end{aligned}$$

(19)

The population of infected females with clinical symptoms of the low-risk (high-risk) HPV types ($I_\mathrm{{f}}^{\ell } (I_\mathrm{{f}}^\mathrm{{h}})$) is generated at the rate $\sigma _\mathrm{{f}}^{\ell } (\sigma _\mathrm{{f}}^\mathrm{{h}})$. This population is decreased by recovery (at a rate $\psi _\mathrm{{f}}^{\ell } (\psi _\mathrm{{f}}^\mathrm{{h}})$) and natural death. Hence,

$$\begin{aligned} \frac{\mathrm{{d}}I_\mathrm{{f}}^{\ell }}{\mathrm{{d}}t}&= \sigma _\mathrm{{f}}^{\ell }E_\mathrm{{f}}^{\ell }-(\psi _\mathrm{{f}}^{\ell }+\mu _\mathrm{{f}})I_\mathrm{{f}}^{\ell },\nonumber \\ \frac{\mathrm{{d}}I_\mathrm{{f}}^\mathrm{{h}}}{\mathrm{{d}}t}&= \sigma _\mathrm{{f}}^\mathrm{{h}}E_\mathrm{{f}}^\mathrm{{h}}-(\psi _\mathrm{{f}}^\mathrm{{h}}+\mu _\mathrm{{f}})I_\mathrm{{f}}^\mathrm{{h}}. \end{aligned}$$

(20)

The population of females with persistent infection with the low-risk HPV types ($P_\mathrm{{f}}^{\ell }$) is generated by the development of persistent infection, with the low-risk HPV types, by symptomatic females with the low-risk HPV types (at a rate $(1-r_\mathrm{{f}}^{\ell })\psi _\mathrm{{f}}^{\ell }$, where $0<r_\mathrm{{f}}^{\ell }\le 1$ is the fraction of symptomatic females with the low-risk HPV types, who recovered from HPV infection without developing genital warts; it is assumed that individuals infected with the low-risk HPV types do not progress to the CIN stages and/or develop cancer Canadian Cancer Society 2010; Centres for Disease Control and Prevention 2012; Public Health Agency of Canada 2010, 2007). Females with persistent infection with the low-risk HPV types move out of this epidemiological class (either through recovery or development of genital warts) at a rate $\alpha _\mathrm{{f}}^{\ell }$, and suffer natural death. Thus,

$$\begin{aligned} \frac{\mathrm{{d}}P_\mathrm{{f}}^{\ell }}{\mathrm{{d}}t} = (1-r_\mathrm{{f}}^{\ell })\psi _\mathrm{{f}}^{\ell }I_\mathrm{{f}}^{\ell }-(\alpha _\mathrm{{f}}^{\ell }+\mu _\mathrm{{f}})P_\mathrm{{f}}^{\ell }. \end{aligned}$$

(21)

The population of females with persistent infection with the high-risk HPV types ($P_\mathrm{{f}}^\mathrm{{h}}$) is generated at a rate $(1-r_\mathrm{{f}}^\mathrm{{h}})\psi _\mathrm{{f}}^\mathrm{{h}}$, where $0<r_\mathrm{{f}}^\mathrm{{h}}\le 1$ is the fraction of symptomatic females with the high-risk HPV types, who recovered from HPV without progressing to the low-grade CIN1 stage, and by a fraction, $1-\left( s_{1\mathrm{{m}}}+s_{2\mathrm{{f}}}\right)$, of infected females in the high-grade CIN2/3 stage, who develop persistent infection (at a rate $\left[ 1-\left( s_{1\mathrm{{m}}}+s_{2\mathrm{{f}}}\right) \right] z_\mathrm{{f}}$, where $s_{1\mathrm{{m}}}$ and $s_{2\mathrm{{f}}}$, with $0\le s_{1\mathrm{{m}}}+s_{2\mathrm{{f}}} \le 1$, are the fractions of infected females in the high-grade CIN2/3 stage, who naturally recovered from HPV infection, and of infected females in the high-grade CIN2/3 stage, who revert to the low-grade CIN1 stage, respectively). Females with persistent infection with the high-risk HPV types move out of this epidemiological class (either through recovery or development of pre-cancerous CIN lesions) at a rate $\alpha _\mathrm{{f}}^\mathrm{{h}}$, and suffer natural death. Hence,

$$\begin{aligned} \frac{\mathrm{{d}}P_\mathrm{{f}}^\mathrm{{h}}}{\mathrm{{d}}t} = (1-r_\mathrm{{f}}^\mathrm{{h}}) \psi _\mathrm{{f}}^\mathrm{{h}}I_\mathrm{{f}}^\mathrm{{h}}+\left[ 1-\left( s_{1\mathrm{{m}}}+s_{2\mathrm{{f}}}\right) \right] z_\mathrm{{f}}G_\mathrm{{fh}}-(\alpha _\mathrm{{f}}^\mathrm{{h}}+\mu _\mathrm{{f}})P_\mathrm{{f}}^\mathrm{{h}}. \end{aligned}$$

The population of females with genital warts ($W_\mathrm{{f}}$) is generated when infected females with persistent infection with the low-risk HPV types develop genital warts (at a rate $(1-k_\mathrm{{f}}^{\ell })\alpha _\mathrm{{f}}^{\ell }$, where $0<k_\mathrm{{f}}^{\ell }\le 1$ is the fraction of infected females with persistent low-risk HPV types, who recovered from HPV infection). Since genital warts do not cause cervical cancer (or any other type of cancer Public Health Agency of Canada 2010; World Health Organization 2009), it is assumed that genital warts do not cause death in females and males. This population decreases due to recovery (at a rate $n_\mathrm{{f}}$) and natural death, so that

$$\begin{aligned} \frac{\mathrm{{d}}W_\mathrm{{f}}}{\mathrm{{d}}t}=(1-k_\mathrm{{f}}^{\ell })\alpha _\mathrm{{f}}^{\ell }P_\mathrm{{f}}^{\ell }-\left( n_\mathrm{{f}}+\mu _\mathrm{{f}}\right) W_\mathrm{{f}}. \end{aligned}$$

(22)

The population of females with the low-grade CIN1 ($G_\mathrm{{f}\ell }$) is generated when infected females with persistent infection with the high-risk HPV types develop pre-cancerous CIN lesions (at a rate $(1-k_\mathrm{{f}}^\mathrm{{h}})\alpha _\mathrm{{f}}^\mathrm{{h}}$, where $0<k_\mathrm{{f}}^\mathrm{{h}}\le 1$ is the fraction of infected females with persistent infection with the high-risk HPV types, who recovered from HPV infection). This population is further increased by the reversion (or regression) of individuals in the high-grade CIN2/3 stage into the low-grade CIN1 stage (at a rate $s_{2\mathrm{{f}}}z_\mathrm{{f}}$). Individuals move out of this class at a rate $u_\mathrm{{f}}$ (due to progression to the high-grade CIN2/3 stage Elbasha et al. 2007; Elbasha and Dasbach 2010; Malik et al. 2013; World Health Organization 2009, at a rate $(1-d_\mathrm{{f}})u_\mathrm{{f}}$, or recovery, at a rate $d_\mathrm{{f}}u_\mathrm{{f}}$). It is assumed that individuals in the CIN stages do not suffer disease-induced death (until they develop cervical cancer). Thus,

$$\begin{aligned} \frac{\mathrm{{d}}G_\mathrm{{f}\ell }}{\mathrm{{d}}t}=(1-k_\mathrm{{f}}^\mathrm{{h}})\alpha _\mathrm{{f}}^\mathrm{{h}}P_\mathrm{{f}}^\mathrm{{h}}+s_{2\mathrm{{f}}}z_\mathrm{{f}}G_\mathrm{{fh}}-\left( u_\mathrm{{f}}+\mu _\mathrm{{f}}\right) G_\mathrm{{f}\ell }. \end{aligned}$$

(23)

The population of females in the high-grade CIN2/3 stage ($G_\mathrm{{fh}}$) is generated by the progression of infected females with low-grade CIN1 (at the rate $(1-d_\mathrm{{f}})u_\mathrm{{f}}$, where $0\le d_\mathrm{{f}} \le 1$ is the fraction of infected females in the low-grade CIN1 stage, who naturally recovered from HPV infection). Transition out of this class occurs at a rate $z_\mathrm{{f}}$ (where a fraction, $s_{1\mathrm{{m}}}z_\mathrm{{f}}$, recovers; another fraction, $s_{2\mathrm{{f}}}z_\mathrm{{f}}$, reverts to the low-grade CIN1 stage and the remaining fraction, $1-(s_{1\mathrm{{m}}}+s_{2\mathrm{{f}}})$, develops persistent infection). This population is decreased by the development of cervical cancer (at a rate $\omega _\mathrm{{f}}$) and natural death. Hence,

$$\begin{aligned} \frac{\mathrm{{d}}G_\mathrm{{fh}}}{\mathrm{{d}}t}=(1-d_\mathrm{{f}})u_\mathrm{{f}}G_\mathrm{{f}\ell }-\left( z_\mathrm{{f}}+\omega _\mathrm{{f}}+\mu _\mathrm{{f}}\right) G_\mathrm{{fh}}. \end{aligned}$$

(24)

The population of females with cervical cancer ($C_\mathrm{{f}}^\mathrm{{c}}$) is generated by the development of cervical cancer by infected females in the high-grade CIN2/3 stage (at the rate $\omega _\mathrm{{f}}$). This population decreases due to recovery (at a rate $\gamma _\mathrm{{f}}$), natural death and cancer-induced death (at a rate $\delta _\mathrm{{f}}$), so that

$$\begin{aligned} \frac{\mathrm{{d}}C_\mathrm{{f}}^\mathrm{{c}}}{\mathrm{{d}}t} = \omega _\mathrm{{f}}G_\mathrm{{fh}}-(\gamma _\mathrm{{f}}+\mu _\mathrm{{f}}+\delta _\mathrm{{f}})C_\mathrm{{f}}^\mathrm{{c}}. \end{aligned}$$

(25)

The population of infected females who recovered from cervical cancer ($R_\mathrm{{f}}^\mathrm{{c}}$) is generated at the rate $\gamma _\mathrm{{f}}$, and decreases by natural death. As in Malik et al. (2013), it is assumed that individuals in this class do not acquire HPV infection again (since these individuals require treatment/surgery, which, typically, result in the removal or damage to the cervix and some other normal tissues around it National Cancer Institute 2011). Thus,

$$\begin{aligned} \frac{\mathrm{{d}}R_\mathrm{{f}}^\mathrm{{c}}}{\mathrm{{d}}t} = \gamma _\mathrm{{f}}C_\mathrm{{f}}^\mathrm{{c}}-\mu _\mathrm{{f}}R_\mathrm{{f}}^\mathrm{{c}}. \end{aligned}$$

(26)

The population of infected females who recovered from HPV infection (and genital warts) without developing cervical cancer ($R_\mathrm{{f}}$) is generated at the rates $r_\mathrm{{f}}^{\ell }\psi _\mathrm{{f}}^{\ell }, r_\mathrm{{f}}^{\ell }{h}\psi _\mathrm{{f}}^\mathrm{{h}}, k_\mathrm{{f}}^{\ell }\alpha _\mathrm{{f}}^{\ell }, k_\mathrm{{f}}^\mathrm{{h}}\alpha _\mathrm{{f}}^\mathrm{{h}}, n_\mathrm{{f}}, d_\mathrm{{f}}u_\mathrm{{f}}$ and $s_{1\mathrm{{m}}}z_\mathrm{{f}}$, respectively. Recovered females acquire re-infection at the rates $\rho _\mathrm{{f}}^{\ell }\lambda _\mathrm{{m}}^{\ell }$ and $\rho _\mathrm{{f}}^\mathrm{{h}}\lambda _\mathrm{{m}}^\mathrm{{h}}$. This population is further decreased by the loss of infection-acquired immunity (at the rate $\xi _\mathrm{{f}}$) and natural death. This gives

$$\begin{aligned} \frac{\mathrm{{d}}R_\mathrm{{f}}}{\mathrm{{d}}t}&= r_\mathrm{{f}}^{\ell }\psi _\mathrm{{f}}^{\ell }I_\mathrm{{f}}^{\ell }+r_\mathrm{{f}}^\mathrm{{h}}\psi _\mathrm{{f}}^\mathrm{{h}}I_\mathrm{{f}}^\mathrm{{h}}+k_\mathrm{{f}}^{\ell }\alpha _\mathrm{{f}}^{\ell }P_\mathrm{{f}}^{\ell }+k_\mathrm{{f}}^\mathrm{{h}}\alpha _\mathrm{{f}}^\mathrm{{h}}P_\mathrm{{f}}^\mathrm{{h}}+n_\mathrm{{f}}W_\mathrm{{f}}+d_\mathrm{{f}}u_\mathrm{{f}}G_\mathrm{{f}\ell }+s_{1\mathrm{{m}}}z_\mathrm{{f}}G_\mathrm{{fh}}\nonumber \\&-\left( \rho _\mathrm{{f}}^\mathrm{{h}}\lambda _\mathrm{{m}}^\mathrm{{h}}+\rho _\mathrm{{f}}^{\ell }\lambda _\mathrm{{m}}^{\ell }\right) R_\mathrm{{f}}-\left( \xi _\mathrm{{f}}+\mu _\mathrm{{f}}\right) R_\mathrm{{f}}. \end{aligned}$$

(27)

The population of unvaccinated susceptible males ($S_\mathrm{{m}}$) is generated by the recruitment of new sexually active males at a rate $\pi _\mathrm{{m}}$ (a fraction, $\varphi _\mathrm{{m}}^\mathrm{{q}}$, of which, is vaccinated with the Gardasil vaccine; it is assumed that males are not vaccinated with the Cervarix vaccine Public Health Agency of Canada 2010, 2007; World Health Organization 2009). It is further increased by the loss of infection-acquired immunity by recovered males (at a rate $\xi _\mathrm{{m}}$). This population is diminished by infection, following effective contacts with infected females (with both the low-risk and high-risk HPV types), at rates $\lambda _\mathrm{{f}}^{\ell }$ and $\lambda _\mathrm{{f}}^\mathrm{{h}}$, where

$$\begin{aligned} \lambda _\mathrm{{f}}^{\ell }&= \frac{\beta _\mathrm{{f}}^{\ell }c_\mathrm{{m}}\left( N_\mathrm{{m}},N_\mathrm{{f}}\right) \left( \eta _\mathrm{{f}}^{\ell }E_\mathrm{{f}}^{\ell }+I_\mathrm{{f}}^{\ell }+\theta _\mathrm{{f}}^{\ell }P_\mathrm{{f}}^{\ell }\right) }{N_\mathrm{{f}}},\end{aligned}$$

(28)

$$\begin{aligned} \lambda _\mathrm{{f}}^\mathrm{{h}}&= \frac{\beta _\mathrm{{f}}^\mathrm{{h}}c_\mathrm{{m}}\left( N_\mathrm{{m}},N_\mathrm{{f}}\right) \left( \eta _\mathrm{{f}}^\mathrm{{h}}E_\mathrm{{f}}^\mathrm{{h}}+I_\mathrm{{f}}^\mathrm{{h}}+\theta _\mathrm{{f}}^\mathrm{{h}}P_\mathrm{{f}}^\mathrm{{h}}\right) }{N_\mathrm{{f}}}. \end{aligned}$$

(29)

In (28) and (29), $\beta _\mathrm{{f}}^{\ell } (\beta _\mathrm{{f}}^\mathrm{{h}})$ is the probability of transmission of HPV infection from infected females with the low-risk (high-risk) HPV types to males per contact, and $c_\mathrm{{m}}\left( N_\mathrm{{m}},N_\mathrm{{f}}\right)$ is the average number of male partners per female per unit time. Furthermore, $\eta _\mathrm{{f}}^{\ell }$ ($\eta _\mathrm{{f}}^\mathrm{{h}}$) (with $0\le \eta _\mathrm{{f}}^{\ell } (\eta _\mathrm{{f}}^\mathrm{{h}})<1$) is the modification parameter accounting for the assumption that exposed females with the low-risk (high-risk) HPV types (i.e. those in the $E_\mathrm{{f}}^{\ell } (E_\mathrm{{f}}^\mathrm{{h}})$ class) are less infectious than symptomatically infected females (i.e. those in the $I_\mathrm{{f}}^{\ell } (I_\mathrm{{f}}^\mathrm{{h}})$ class), and $\theta _\mathrm{{f}}^{\ell } (\theta _\mathrm{{f}}^\mathrm{{h}})>0$ is the modification parameter accounting for the assumption that infected females with persistent infection with the low-risk (high-risk) HPV types transmit HPV at a different rate compared to infected females in the $E_\mathrm{{f}}^{\ell }, I_\mathrm{{f}}^{\ell } (E_\mathrm{{f}}^\mathrm{{h}}, I_\mathrm{{f}}^\mathrm{{h}})$ classes. This population is further decreased by natural death (at a rate $\mu _\mathrm{{m}}$, it is assumed that males in all epidemiological compartments suffer natural death at this rate, $\mu _\mathrm{{m}}$). Thus,

$$\begin{aligned} \frac{\mathrm{{d}}S_\mathrm{{m}}}{\mathrm{{d}}t} = \left( 1-\varphi _\mathrm{{m}}^\mathrm{{q}}\right) \pi _\mathrm{{m}}+\xi _\mathrm{{m}}R_\mathrm{{m}}- \left( \lambda _\mathrm{{f}}^\mathrm{{h}}+\lambda _\mathrm{{f}}^{\ell }\right) S_\mathrm{{m}}-\mu _\mathrm{{m}}S_\mathrm{{m}}. \end{aligned}$$

(30)

The population of new sexually active susceptible males vaccinated with the Gardasil vaccine ($V_\mathrm{{m}}^\mathrm{{q}}$) is generated by the vaccination of the fraction, $\varphi _\mathrm{{m}}^\mathrm{{q}}$, of unvaccinated susceptible males (at the rate $\pi _\mathrm{{m}}\varphi _\mathrm{{m}}^\mathrm{{q}}$). It is decreased by HPV infection, following effective contacts with females infected with the high-risk HPV types (at a reduced rate $(1-\varepsilon _{q})\lambda _\mathrm{{f}}^\mathrm{{h}}$, where $0<\varepsilon _{q} \le 1$ is the efficacy of the Gardasil vaccine) and females infected with the low-risk HPV types (at the rate $(1-\varepsilon _{q})\lambda _\mathrm{{f}}^{\ell })$. This population is reduced by natural death. Hence,

$$\begin{aligned} \frac{\mathrm{{d}}V_\mathrm{{m}}^\mathrm{{q}}}{\mathrm{{d}}t}= \varphi _\mathrm{{m}}^\mathrm{{q}} \pi _\mathrm{{m}}-(1-\varepsilon _{q})\left( \lambda _\mathrm{{f}}^\mathrm{{h}}+ \lambda _\mathrm{{f}}^{\ell }\right) V_\mathrm{{m}}^\mathrm{{q}}-\mu _\mathrm{{m}}V_\mathrm{{m}}^\mathrm{{q}}. \end{aligned}$$

(31)

The population of exposed males with the low-risk (high-risk) HPV types ($E_\mathrm{{m}}^{\ell } (E_\mathrm{{m}}^\mathrm{{h}})$) is generated by the infection of unvaccinated and vaccinated susceptible males with the low-risk (high-risk) HPV types (at the rate $\lambda _\mathrm{{f}}^{\ell } (\lambda _\mathrm{{f}}^\mathrm{{h}})$). This population is further increased by the re-infection of recovered males (at a rate $\rho _\mathrm{{m}}^{\ell }\lambda _\mathrm{{f}}^{\ell } (\rho _\mathrm{{m}}^\mathrm{{h}}\lambda _\mathrm{{f}}^\mathrm{{h}})$, where $0\le \rho _\mathrm{{m}}^{\ell } (\rho _\mathrm{{m}}^\mathrm{{h}})<1$ also accounts for the assumption that re-infection of recovered females occurs at a rate lower than the primary infection). Exposed males develop clinical symptoms of the low-risk (high-risk) HPV types (at a rate $\sigma _\mathrm{{m}}^{\ell } (\sigma _\mathrm{{m}}^\mathrm{{h}})$) and suffer natural death. Hence,

$$\begin{aligned} \frac{\mathrm{{d}}E_\mathrm{{m}}^{\ell }}{\mathrm{{d}}t}&= \left[ S_\mathrm{{m}}+(1-\varepsilon _{q})V_\mathrm{{m}}^\mathrm{{q}}\right] \lambda _\mathrm{{f}}^{\ell }+\rho _\mathrm{{m}}^{\ell }\lambda _\mathrm{{f}}^{\ell }R_\mathrm{{m}}-(\sigma _\mathrm{{m}}^{\ell }+\mu _\mathrm{{m}})E_\mathrm{{m}}^{\ell },\end{aligned}$$

(32)

$$\begin{aligned} \frac{\mathrm{{d}}E_\mathrm{{m}}^\mathrm{{h}}}{\mathrm{{d}}t}&= \left[ S_\mathrm{{m}}+(1-\varepsilon _{q})V_\mathrm{{m}}^\mathrm{{q}}\right] \lambda _\mathrm{{f}}^\mathrm{{h}}+\rho _\mathrm{{m}}^\mathrm{{h}}\lambda _\mathrm{{f}}^\mathrm{{h}}R_\mathrm{{m}}-(\sigma _\mathrm{{m}}^\mathrm{{h}}+\mu _\mathrm{{m}})E_\mathrm{{m}}^\mathrm{{h}}. \end{aligned}$$

(33)

The population of infected males with clinical symptoms of the low-risk (high-risk) HPV types ($I_\mathrm{{m}}^{\ell } (I_\mathrm{{m}}^\mathrm{{h}})$) is generated at the rate $\sigma _\mathrm{{m}}^{\ell } (\sigma _\mathrm{{m}}^\mathrm{{h}})$. It is reduced by recovery (at a rate $\psi _\mathrm{{m}}^{\ell } (\psi _\mathrm{{m}}^\mathrm{{h}})$) and natural death. Thus,

$$\begin{aligned} \frac{\mathrm{{d}}I_\mathrm{{m}}^{\ell }}{\mathrm{{d}}t}&= \sigma _\mathrm{{m}}^{\ell }E_\mathrm{{m}}^{\ell }-(\psi _\mathrm{{m}}^{\ell }+\mu _\mathrm{{m}})I_\mathrm{{m}}^{\ell },\end{aligned}$$

(34)

$$\begin{aligned} \frac{\mathrm{{d}}I_\mathrm{{m}}^\mathrm{{h}}}{\mathrm{{d}}t}&= \sigma _\mathrm{{m}}^\mathrm{{h}}E_\mathrm{{m}}^\mathrm{{h}}-(\psi _\mathrm{{m}}^\mathrm{{h}}+\mu _\mathrm{{m}})I_\mathrm{{m}}^\mathrm{{h}}. \end{aligned}$$

(35)

The population of males with persistent infection with the low-risk HPV types ($P_\mathrm{{m}}^{\ell }$) is generated by the development of persistent infection, with the low-risk HPV types, by symptomatic males with the low-risk HPV types (at a rate $(1-r_\mathrm{{m}}^{\ell })\psi _\mathrm{{m}}^{\ell }$, where $0<r_\mathrm{{m}}^{\ell }\le 1$ is the fraction of symptomatic males with the low-risk HPV types, who recovered from HPV infection without developing genital warts). Males with persistent infection with the low-risk HPV types move out of this epidemiological class (either through recovery or development of genital warts) at a rate $\alpha _\mathrm{{m}}^{\ell }$, and suffer natural death. Thus,

$$\begin{aligned} \frac{\mathrm{{d}}P_\mathrm{{m}}^{\ell }}{\mathrm{{d}}t} = (1-r_\mathrm{{m}}^{\ell })\psi _\mathrm{{m}}^{\ell }I_\mathrm{{m}}^{\ell }-(\alpha _\mathrm{{m}}^{\ell }+\mu _\mathrm{{m}})P_\mathrm{{m}}^{\ell }. \end{aligned}$$

(36)

The population of males with persistent infection with the high-risk HPV types ($P_\mathrm{{m}}^\mathrm{{h}}$) is generated at a rate $(1-r_\mathrm{{m}}^\mathrm{{h}})\psi _\mathrm{{m}}^\mathrm{{h}}$, where $0<r_\mathrm{{m}}^\mathrm{{h}}\le 1$ is the fraction of symptomatic males with the high-risk HPV types, who recovered from HPV without progressing to the low-grade INM1 stage, and by a fraction, $1-\left( s_{1\mathrm{{m}}}+s_{2\mathrm{{m}}}\right)$, of infected males in the high-grade INM2/3 stage, who develop persistent infection (at a rate $\left[ 1-\left( s_{1\mathrm{{m}}}+s_{2\mathrm{{m}}}\right) \right] z_\mathrm{{m}}$, where $s_{1\mathrm{{m}}}$ and $s_{2\mathrm{{m}}}$, with $0\le s_{1\mathrm{{m}}}+s_{2\mathrm{{m}}}\le 1$, are the fractions of infected males in the high-grade INM2/3 stage, who naturally recovered from HPV infection, and of infected males in the high-grade INM2/3 stage that reverts to the low-grade INM1 stage, respectively). Males with persistent infection with the high-risk HPV types move out of this epidemiological class (either through recovery or development of pre-cancerous lesions) at a rate $\alpha _\mathrm{{m}}^\mathrm{{h}}$, and suffer natural death. Hence,

$$\begin{aligned} \frac{\mathrm{{d}}P_\mathrm{{m}}^\mathrm{{h}}}{\mathrm{{d}}t} = (1-r_\mathrm{{m}}^\mathrm{{h}}) \psi _\mathrm{{m}}^\mathrm{{h}}I_\mathrm{{m}}^\mathrm{{h}}+\left[ 1-\left( s_{1\mathrm{{m}}}+s_{2\mathrm{{m}}}\right) \right] z_\mathrm{{m}}G_\mathrm{{mh}} -(\alpha _\mathrm{{m}}^\mathrm{{h}}+\mu _\mathrm{{m}})P_\mathrm{{m}}^\mathrm{{h}}. \end{aligned}$$

The population of males with genital warts ($W_\mathrm{{m}}$) is generated when infected males with persistent infection with the low-risk HPV types develop genital warts (at a rate $(1-k_\mathrm{{m}}^{\ell })\alpha _\mathrm{{m}}^{\ell }$, where $0<k_\mathrm{{m}}^{\ell }\le 1$ is the fraction of infected males with low-risk persistent HPV types, who recovered from HPV infection). This population decreases due to recovery (at a rate $n_\mathrm{{m}}$) and natural death, so that

$$\begin{aligned} \frac{\mathrm{{d}}W_\mathrm{{m}}}{\mathrm{{d}}t}=(1-k_\mathrm{{m}}^{\ell })\alpha _\mathrm{{m}}^{\ell }P_\mathrm{{m}}^{\ell }-\left( n_\mathrm{{m}}+\mu _\mathrm{{m}}\right) W_\mathrm{{m}}. \end{aligned}$$

(37)

The population of males in the low-grade HPV-related INM1 stage ($G_\mathrm{{m}\ell }$) is generated when infected males with persistent infection with the high-risk HPV types develop pre-cancerous lesions (at a rate $(1-k_\mathrm{{m}}^\mathrm{{h}})\alpha _\mathrm{{m}}^\mathrm{{h}}$, where $0<k_\mathrm{{m}}^\mathrm{{h}}\le 1$ is the fraction of infected males with persistent infection with the high-risk HPV types, who recovered from HPV infection). This population is further increased by the reversion of individuals in the high-grade HPV-related INM2/3 stage (at a rate $s_{2\mathrm{{m}}}z_\mathrm{{m}}$). Individuals move out of this class at a rate $u_\mathrm{{m}}$ (due to progression to the high-grade INM2/3 stage, at a rate $(1-d_\mathrm{{m}})u_\mathrm{{m}}$, or recovery, at a rate $d_\mathrm{{m}}u_\mathrm{{m}}$). It is assumed that individuals in INM stages do not suffer disease-induced death (until they develop HPV-related cancer). Thus,

$$\begin{aligned} \frac{\mathrm{{d}}G_\mathrm{{m}\ell }}{\mathrm{{d}}t}=(1-k_\mathrm{{m}}^\mathrm{{h}})\alpha _\mathrm{{m}}^\mathrm{{h}}P_\mathrm{{m}}^\mathrm{{h}}+s_{2\mathrm{{m}}}z_\mathrm{{m}}G_\mathrm{{mh}}-\left( u_\mathrm{{m}}+\mu _\mathrm{{m}}\right) G_\mathrm{{ml}}. \end{aligned}$$

(38)

The population of males in the high-grade HPV-related INM2/3 stage ($G_\mathrm{{mh}}$) is generated by the progression of infected males in the low-grade HPV-related INM1 stage (at the rate $(1-d_\mathrm{{m}})u_\mathrm{{m}}$, where $0\le d_\mathrm{{m}} \le 1$ is the fraction of infected males in the low-grade INM1 stage, who naturally recovered from HPV infection). Transition out of this class occurs at a rate $z_\mathrm{{m}}$ (where a fraction, $s_{1\mathrm{{m}}}z_\mathrm{{m}}$, recovers; another fraction, $s_{2\mathrm{{m}}}z_\mathrm{{m}}$, reverts to the low-grade INM1 stage, and the remaining fraction, $1-(s_{1\mathrm{{m}}}+s_{2\mathrm{{m}}})$, develops persistent infection). This population is decreased by the development of HPV-related cancer (at a rate $\omega _\mathrm{{m}}$) and natural death. Hence,

$$\begin{aligned} \frac{\mathrm{{d}}G_\mathrm{{mh}}}{\mathrm{{d}}t}=(1-d_\mathrm{{m}})u_\mathrm{{m}}G_\mathrm{{m}\ell }-\left( z_\mathrm{{m}}+\omega _\mathrm{{m}}+\mu _\mathrm{{m}}\right) G_\mathrm{{mh}}. \end{aligned}$$

(39)

The population of males with HPV-related cancers ($C_\mathrm{{m}}^\mathrm{{c}}$) is generated by the development of HPV-related cancers by infected males in the high-grade IN2/3 stage (at the rate $\omega _\mathrm{{m}}$). This population decreases due to recovery (at a rate $\gamma _\mathrm{{m}}$), natural death and cancer-induced death (at a rate $\delta _\mathrm{{m}}$), so that

$$\begin{aligned} \frac{\mathrm{{d}}C_\mathrm{{m}}^\mathrm{{r}}}{\mathrm{{d}}t} = \omega _\mathrm{{m}}G_\mathrm{{mh}}-(\gamma _\mathrm{{m}}+\mu _\mathrm{{m}}+\delta _\mathrm{{m}})C_\mathrm{{m}}^\mathrm{{r}}. \end{aligned}$$

(40)

The population of males who recovered from HPV-related cancers ($R_\mathrm{{m}}^\mathrm{{c}}$) is generated at the rate $\gamma _\mathrm{{m}}$, and decreases by natural death, so that

$$\begin{aligned} \frac{\mathrm{{d}}R_\mathrm{{m}}^\mathrm{{c}}}{\mathrm{{d}}t} = \gamma _\mathrm{{m}}C_\mathrm{{m}}^\mathrm{{c}}-\mu _\mathrm{{m}}R_\mathrm{{m}}^\mathrm{{c}}. \end{aligned}$$

(41)

The population of males who recovered from HPV infection (and genital warts) without developing cancer ($R_\mathrm{{m}}$) is generated at the rates $r_\mathrm{{m}}^{\ell }\psi _\mathrm{{m}}^{\ell }, r_\mathrm{{m}}\psi _\mathrm{{m}}^\mathrm{{h}}, k_\mathrm{{m}}^{\ell }\alpha _\mathrm{{m}}^{\ell }, k_\mathrm{{m}}^\mathrm{{h}}\alpha _\mathrm{{m}}^\mathrm{{h}}, n_\mathrm{{m}},$ $d_\mathrm{{m}}u_\mathrm{{m}}$ and $s_{1\mathrm{{m}}}z_\mathrm{{m}}$. It is decreased by re-infection (at the rates $\rho _\mathrm{{m}}^{\ell }\lambda _\mathrm{{f}}^{\ell }$ and $\rho _\mathrm{{m}}^\mathrm{{h}}\lambda _\mathrm{{f}}^\mathrm{{h}}$), loss of infection-acquired immunity (at the rate $\xi _\mathrm{{m}}$) and natural death, so that

$$\begin{aligned} \frac{\mathrm{{d}}R_\mathrm{{m}}}{\mathrm{{d}}t}&= r_\mathrm{{m}}^{\ell }\psi _\mathrm{{m}}^{\ell }I_\mathrm{{m}}^{\ell }+r_\mathrm{{f}}\psi _\mathrm{{m}}^\mathrm{{h}}I_\mathrm{{m}}^\mathrm{{h}}+k_\mathrm{{m}}^{\ell } \alpha _\mathrm{{m}}^{\ell }P_\mathrm{{m}}^{\ell }+k_\mathrm{{m}}^\mathrm{{h}}\alpha _\mathrm{{m}}^\mathrm{{h}}P_\mathrm{{m}}^\mathrm{{h}}+n_\mathrm{{m}}W_\mathrm{{m}}+d_mu_mG_\mathrm{{m}\ell }\nonumber \\&\quad +\, s_{1\mathrm{{m}}}z_\mathrm{{m}}G_\mathrm{{mh}}-\left( \rho _\mathrm{{m}}^\mathrm{{h}}\lambda _\mathrm{{f}}^\mathrm{{h}}+\rho _\mathrm{{m}}^{\ell } \lambda _\mathrm{{f}}^{\ell }\right) R_\mathrm{{m}}-\left( \xi _\mathrm{{m}}+\mu _\mathrm{{m}}\right) R_\mathrm{{m}}. \end{aligned}$$

(42)

For the sex-structured model, such as {(14)–(42)}, to be mechanistically and epidemiologically consistent, it is crucial that the associated conservation law of sexual contacts is applied. In other words, the total number of sexual contacts females make with males must equal the total number of sexual contacts males make with females (see also, for instant, Castillo-Chavez et al. 1997; Elbasha 2008; Mukandavire and Garira 2007; Sharomi and Gumel 2009; Xiaodong et al. 1993 and some of the references therein). Thus, the following conservation law must hold:

$$\begin{aligned} c_\mathrm{{m}}\left( N_\mathrm{{m}},N_\mathrm{{f}}\right) N_\mathrm{{m}}=c_\mathrm{{f}}\left( N_\mathrm{{m}},N_\mathrm{{f}}\right) N_\mathrm{{f}}. \end{aligned}$$

(43)

It is assumed that male sexual partners are abundant, so that females can always have enough number of male sexual contacts per unit time. Hence, it is assumed that $c_\mathrm{{f}}\left( N_\mathrm{{m}},N_\mathrm{{f}}\right) =c_\mathrm{{f}}$, a constant, and $c_\mathrm{{m}}\left( N_\mathrm{{m}},N_\mathrm{{f}}\right)$ is calculated from the relation (obtained from (43))

$$\begin{aligned} c_\mathrm{{m}}\left( N_\mathrm{{m}},N_\mathrm{{f}}\right) =\frac{c_\mathrm{{f}}N_\mathrm{{f}}}{N_\mathrm{{m}}}. \end{aligned}$$

(44)

For mathematical convenience, it is assumed, from now on, that the efficacies of the two anti-HPV vaccines (Cervarix and Gardasil) are the same (so that $\varepsilon _{b}=\varepsilon _{q}=\varepsilon _{v}$) (Canadian Cancer Society 2010; Food and Drug Administration 2010; Public Health Agency of Canada 2007; World Health Organization 2009). The model is obtained by (44) in {(14), (15), (28) and (29)}.

2 Proof of Theorem 2.1

Proof

Let $t_1$ = sup$\{t>0: S_\mathrm{{f}}(t)>0,V_\mathrm{{f}}^\mathrm{{b}}(t)>0, V_\mathrm{{f}}^\mathrm{{q}}(t)>0, E_\mathrm{{f}}^{\ell }(t)>0, E_\mathrm{{f}}^\mathrm{{h}}(t)>0, I_\mathrm{{f}}^{\ell }(t)>0, I_\mathrm{{f}}^\mathrm{{h}}(t)>0, P_\mathrm{{f}}^{\ell }(t)>0, P_\mathrm{{f}}^\mathrm{{h}}(t)>0, W_\mathrm{{f}}(t)>0, G_\mathrm{{f}\ell }(t)>0, G_\mathrm{{fh}}(t)>0, C_\mathrm{{f}}^\mathrm{{c}}(t)>0, R_\mathrm{{f}}^\mathrm{{c}}(t)>0, R_\mathrm{{f}}(t)>0, S_\mathrm{{m}}(t)>0, V_\mathrm{{m}}^\mathrm{{q}}(t)>0, E_\mathrm{{m}}^{\ell }(t)>0, E_\mathrm{{m}}^\mathrm{{h}}(t)>0, I_\mathrm{{m}}^{\ell }(t)>0, I_\mathrm{{m}}^\mathrm{{h}}(t)>0, P_\mathrm{{m}}^{\ell }(t)>0, P_\mathrm{{m}}^\mathrm{{h}}(t)>0, W_\mathrm{{m}}(t)>0, G_\mathrm{{m}\ell }(t)>0, G_\mathrm{{mh}}(t)>0, C_\mathrm{{m}}^\mathrm{{r}}(t)>0, R_\mathrm{{m}}^\mathrm{{c}}(t)>0, R_\mathrm{{m}}(t)>0\}$. Thus, $t_1>0$. It follows from the first equation of the model (3) that

$$\begin{aligned} \frac{\mathrm{{d}}S_\mathrm{{f}}}{\mathrm{{d}}t}&= \pi _\mathrm{{f}}(1-\varphi _\mathrm{{f}}^\mathrm{{b}}-\varphi _\mathrm{{f}}^\mathrm{{q}})+\xi _\mathrm{{f}}R_\mathrm{{f}}(t)- \left[ \lambda _\mathrm{{m}}^{\ell }(t)+\lambda _\mathrm{{m}}^\mathrm{{h}}(t)\right] S_\mathrm{{f}}(t)-\mu _\mathrm{{f}}S_\mathrm{{f}}(t)\nonumber \\&\ge \pi _\mathrm{{f}}(1-\varphi _\mathrm{{f}}^\mathrm{{b}}-\varphi _\mathrm{{f}}^\mathrm{{q}})-\left[ \lambda _\mathrm{{m}}^{\ell }(t) +\lambda _\mathrm{{m}}^\mathrm{{h}}(t)+\mu _\mathrm{{f}}\right] S_\mathrm{{f}}(t). \end{aligned}$$

(45)

For simplicity, let (note that $0<\varphi _\mathrm{{f}}^\mathrm{{b}}+\varphi _\mathrm{{f}}^\mathrm{{q}}\le 1$)

$$\begin{aligned} \varphi _\mathrm{{f}}=\varphi _\mathrm{{f}}^\mathrm{{b}}+\varphi _\mathrm{{f}}^\mathrm{{q}},\quad \mathrm{and} \quad \lambda _\mathrm{{m}}(t)=\lambda _\mathrm{{m}}^{\ell }(t)+\lambda _\mathrm{{m}}^\mathrm{{h}}(t), \end{aligned}$$

it follows from

(45), which can be re-written as

$$\begin{aligned} \frac{\mathrm{{d}}}{\mathrm{{d}}t}\left\{ S_\mathrm{{f}}(t)\; \mathrm{exp}\left[ \mu _\mathrm{{f}}t+\int \limits _0^t \! \lambda _\mathrm{{m}}(\tau ) \, \mathrm {\mathrm{{d}}}\tau \right] \right\} \ge \pi _\mathrm{{f}}(1-\varphi _\mathrm{{f}})\;\mathrm{exp}\left[ \mu _\mathrm{{f}}t+\int \limits _0^t \! \lambda _\mathrm{{m}}(\tau ) \, \mathrm {\mathrm{{d}}}\tau \right] . \end{aligned}$$

Hence,

$$\begin{aligned} S_\mathrm{{f}}(t_1)\; \mathrm{exp}\left[ \mu _\mathrm{{f}}t_{1}+\int \limits _0^{t_1}\! \lambda _\mathrm{{m}}(\tau ) \, \mathrm {\mathrm{{d}}}\tau \right] -S_\mathrm{{f}}(0) \ge \int \limits _0^{t_1}\! \pi _\mathrm{{f}}(1-\varphi _\mathrm{{f}})\;\mathrm{exp}\left[ \mu _\mathrm{{f}}y+\int \limits _0^y \! \lambda _\mathrm{{m}}(\tau ) \, \mathrm {\mathrm{{d}}}\tau \right] \mathrm {\mathrm{{d}}}y, \end{aligned}$$

so that

$$\begin{aligned}&S_\mathrm{{f}}(t_1) \ge S_\mathrm{{f}}(0)\;\mathrm{exp}\left[ -\mu _\mathrm{{f}}t_{1}-\int \limits _0^{t_1}\! \lambda _\mathrm{{m}}(\tau ) \, \mathrm {\mathrm{{d}}}\tau \right] \\&\quad + \left\{ \mathrm{exp}\left[ -\mu _\mathrm{{f}}t_{1}-\int \limits _0^{t_1} \! \lambda _\mathrm{{m}}(\tau ) \, \mathrm {d}\tau \right] \right\} \int \limits _0^{t_1}\! \pi _\mathrm{{f}}(1-\varphi _\mathrm{{f}})\;\mathrm{exp}\left[ \mu _\mathrm{{f}}y+\int \limits _0^y \! \lambda _\mathrm{{m}}(\tau ) \, \mathrm {d}\tau \right] \mathrm {d}y>0. \end{aligned}$$

Similarly, it can be shown that $V_\mathrm{{f}}^\mathrm{{b}}(t)>0, V_\mathrm{{f}}^\mathrm{{q}}(t)>0, E_\mathrm{{f}}^{\ell }(t)\ge 0, E_\mathrm{{f}}^\mathrm{{h}}(t)\ge 0, I_\mathrm{{f}}^{\ell }(t)\ge 0, I_\mathrm{{f}}^\mathrm{{h}}(t)\ge 0, P_\mathrm{{f}}^{\ell }(t)\ge 0, P_\mathrm{{f}}^\mathrm{{h}}(t)\ge 0, W_\mathrm{{f}}(t)\ge 0, G_\mathrm{{f}\ell }(t)\ge 0, G_\mathrm{{fh}}(t)\ge 0, C_\mathrm{{f}}^\mathrm{{c}}(t)\ge 0, R_\mathrm{{f}}^\mathrm{{c}}(t)\ge 0, R_\mathrm{{f}}(t)\ge 0, S_\mathrm{{m}}(t)>0, V_\mathrm{{m}}^\mathrm{{q}}(t)>0, E_\mathrm{{m}}^{\ell }(t)\ge 0, E_\mathrm{{m}}^\mathrm{{h}}(t)\ge 0, I_\mathrm{{m}}^{\ell }(t)\ge 0, I_\mathrm{{m}}^\mathrm{{h}}(t)\ge 0, P_\mathrm{{m}}^{\ell }(t)\ge 0, P_\mathrm{{m}}^\mathrm{{h}}(t)\ge 0, W_\mathrm{{m}}(t)\ge 0, G_\mathrm{{m}\ell }(t)\ge 0, G_\mathrm{{mh}}(t)\ge 0, C_\mathrm{{m}}^\mathrm{{r}}(t)\ge 0, R_\mathrm{{m}}^\mathrm{{c}}(t)\ge 0$ and $R_\mathrm{{m}}(t)\ge 0$ for all time $t>0$. Hence, all solutions remain positive for all non-negative initial conditions.$\square$

Theorem 2.1 can also be proved using the approach given in Appendix 1 of Thieme (2003).

3 Positivity of $\mathcal {R}_\mathrm{{f}\ell }, \mathcal {R}_\mathrm{{m}\ell }, \mathcal {R}_\mathrm{{fh}}$ and $\mathcal {R}_\mathrm{{mh}}$

Recall from Sect. 3.1 (with all the associated variables as defined in Sect. 3.1) that

$$\begin{aligned} \mathcal {R}_\mathrm{{f}\ell }&= \frac{\beta _\mathrm{{m}}^{\ell }c_\mathrm{{f}}\pi _\mathrm{{f}}\mu _\mathrm{{m}} \left( 1-\varepsilon _{v} \varphi _\mathrm{{f}}^\mathrm{{q}}\right) B_{1}}{\mu _\mathrm{{f}}\pi _\mathrm{{m}}D_{1}D_{2}D_{3}},\;\; \mathcal {R}_\mathrm{{fh}}=\frac{\beta _\mathrm{{m}}^\mathrm{{h}}c_\mathrm{{f}} \pi _\mathrm{{f}}\mu _\mathrm{{m}}\left[ 1-\varepsilon _{v} \left( \varphi _\mathrm{{f}}^\mathrm{{b}}+\varphi _\mathrm{{f}}^\mathrm{{q}}\right) \right] \left( Q_{1}+Q_{2}+Q_{3}\right) }{\mu _\mathrm{{f}}\pi _\mathrm{{m}}D_{5}D_{6}Q_{4}},\\ \mathcal {R}_\mathrm{{m}\ell }&= \frac{\beta _\mathrm{{f}}^{\ell }c_\mathrm{{f}}\left( 1-\varepsilon _{v}\varphi _\mathrm{{m}}^\mathrm{{q}}\right) B_{2}}{A_{1}A_{2}A_{3}},\;\; \mathcal {R}_\mathrm{{mh}}=\frac{\beta _\mathrm{{f}}^\mathrm{{h}}c_\mathrm{{f}}\left( 1-\varepsilon _{v}\varphi _\mathrm{{m}}^\mathrm{{b}}-\varepsilon _{q}\varphi _\mathrm{{m}}^\mathrm{{q}}\right) \left( Q_{5}+Q_{6}+Q_{7}\right) }{A_{5}A_{6}Q_{8}}. \end{aligned}$$

The following steps are taken to prove that the quantities above are positive:

$$\begin{aligned} B_{1}&= \eta _\mathrm{{m}}^{\ell }D_{3}D_{2}+k_{1}D_{3}+\theta _\mathrm{{m}}^{\ell }k_{2}k_{1}>0,\\ Q_{1}&= \eta _\mathrm{{m}}^\mathrm{{h}}\left( D_{6}D_{7}D_{8}D_{9}-k_{8}j_{2}D_{6}D_{7}-k_{8}k_{7}j_{1}D_{6}\right) \\&= \eta _\mathrm{{m}}^\mathrm{{h}}D_{6}\left( \omega _\mathrm{{m}}+\mu _\mathrm{{m}}\right) \left( u_\mathrm{{m}}+\mu _\mathrm{{m}}\right) \left( \alpha _\mathrm{{m}}^\mathrm{{h}}+\mu _\mathrm{{m}}\right) \\&\quad +\, \eta _\mathrm{{m}}^\mathrm{{h}}D_{6}z_\mathrm{{m}}\mu _\mathrm{{m}}\left( \alpha _\mathrm{{m}}^\mathrm{{h}}+\mu _\mathrm{{m}}\right) +\eta _\mathrm{{m}}^\mathrm{{h}}D_{6}d_\mathrm{{m}}u_\mathrm{{m}}z_\mathrm{{m}}\left( \alpha _\mathrm{{m}}^\mathrm{{h}}+s_{2\mathrm{{m}}}\mu _\mathrm{{m}}\right) \\&\quad +\,\eta _\mathrm{{m}}^\mathrm{{h}}D_{6}u_\mathrm{{m}}z_\mathrm{{m}}\mu _\mathrm{{m}}\left( 1-s_{2\mathrm{{m}}}\right) \\&\quad +\,\eta _\mathrm{{m}}^\mathrm{{h}}D_{6}u_\mathrm{{m}}z_\mathrm{{m}}\left( 1-d_\mathrm{{m}}\right) \alpha _\mathrm{{m}}^\mathrm{{h}}\left[ s_{1\mathrm{{m}}}\left( 1-k_\mathrm{{m}}^\mathrm{{h}}\right) +k_\mathrm{{m}}^\mathrm{{h}}\left( 1-s_{2\mathrm{{m}}}\right) \right] >0,\\ Q_{2}&= k_{5}D_{7}D_{8}D_{9}-k_{5}k_{8}j_{2}D_{7}-k_{5}k_{7}k_{8}j_{1}\\&= k_{5}\left( \omega _\mathrm{{m}}+\mu _\mathrm{{m}}\right) \left( u_\mathrm{{m}}+\mu _\mathrm{{m}}\right) \left( \alpha _\mathrm{{m}}^\mathrm{{h}}+\mu _\mathrm{{m}}\right) \\&\quad +\, k_{5}z_\mathrm{{m}}\mu _\mathrm{{m}}\left( \alpha _\mathrm{{m}}^\mathrm{{h}}+\mu _\mathrm{{m}}\right) +k_{5}d_\mathrm{{m}}u_\mathrm{{m}}z_\mathrm{{m}}\left( \alpha _\mathrm{{m}}^\mathrm{{h}}+s_{2\mathrm{{m}}}\mu _\mathrm{{m}}\right) +k_{5}u_\mathrm{{m}}z_\mathrm{{m}}\mu _\mathrm{{m}}\left( 1-s_{2\mathrm{{m}}}\right) \\&\quad +\, k_{5}u_\mathrm{{m}}z_\mathrm{{m}}\left( 1-d_\mathrm{{m}}\right) \alpha _\mathrm{{m}}^\mathrm{{h}}\left[ s_{1\mathrm{{m}}}\left( 1-k_\mathrm{{m}}^\mathrm{{h}}\right) +k_\mathrm{{m}}^\mathrm{{h}}\left( 1-s_{2\mathrm{{m}}}\right) \right] >0, \end{aligned}$$

$$\begin{aligned} Q_{3}&= \theta _\mathrm{{m}}^\mathrm{{h}}\left( k_{5}k_{6}D_{8}D_{9}-k_{5}k_{6}k_{8}j_{2}\right) \\&= \theta _\mathrm{{m}}^\mathrm{{h}}k_{5}k_{6}\left( \omega _\mathrm{{m}}+\mu _\mathrm{{m}}\right) \left( u_\mathrm{{m}}+\mu _\mathrm{{m}}\right) +\theta _\mathrm{{m}}^\mathrm{{h}}k_{5}k_{6}z_\mathrm{{m}}u_\mathrm{{m}}\left( 1-s_{2\mathrm{{m}}}\right) \\&\quad +\,\theta _\mathrm{{m}}^\mathrm{{h}}k_{5}k_{6}z_\mathrm{{m}}\left( d_\mathrm{{m}}u_\mathrm{{m}}s_{2\mathrm{{m}}}+\mu _\mathrm{{m}}\right) >0,\\ Q_{4}&= D_{7}D_{8}D_{9}-k_{8}j_{2}D_{7}-k_{7}k_{8}j_{1}\\&= \left( \omega _\mathrm{{m}}+\mu _\mathrm{{m}}\right) \left( u_\mathrm{{m}}+\mu _\mathrm{{m}}\right) \left( \alpha _\mathrm{{m}}^\mathrm{{h}}+\mu _\mathrm{{m}}\right) +z_\mathrm{{m}}\mu _\mathrm{{m}}\left( \alpha _\mathrm{{m}}^\mathrm{{h}}+\mu _\mathrm{{m}}\right) \\&\quad +\, d_\mathrm{{m}}u_\mathrm{{m}}z_\mathrm{{m}}\left( \alpha _\mathrm{{m}}^\mathrm{{h}}+s_{2\mathrm{{m}}}\mu _\mathrm{{m}}\right) +u_\mathrm{{m}}z_\mathrm{{m}}\mu _\mathrm{{m}}\left( 1-s_{2\mathrm{{m}}}\right) \\&\quad +\, u_\mathrm{{m}}z_\mathrm{{m}}\left( 1-d_\mathrm{{m}}\right) \alpha _\mathrm{{m}}^\mathrm{{h}}\left[ s_{1\mathrm{{m}}}\left( 1-k_\mathrm{{m}}^\mathrm{{h}}\right) +k_\mathrm{{m}}^\mathrm{{h}}\left( 1-s_{2\mathrm{{m}}}\right) \right] >0,\\ B_{2}&= \eta _\mathrm{{f}}^{\ell }A_{3}A_{2}+b_{1}A_{3}+\theta _\mathrm{{f}}^{\ell }b_{2}b_{1}>0,\\ Q_{5}&= \eta _\mathrm{{f}}^\mathrm{{h}}\left( A_{6}A_{7}A_{8}A_{9}-b_{8}g_{2}A_{6}A_{7}-b_{8}b_{7}g_{1}A_{6}\right) \\&= \eta _\mathrm{{f}}^\mathrm{{h}}A_{6}\left( \omega _\mathrm{{f}}+\mu _\mathrm{{f}}\right) \left( u_\mathrm{{f}}+\mu _\mathrm{{f}}\right) \left( \alpha _\mathrm{{f}}^\mathrm{{h}}+\mu _\mathrm{{f}}\right) \\&\quad +\, \eta _\mathrm{{f}}^\mathrm{{h}}A_{6}z_\mathrm{{f}}\mu _\mathrm{{f}}\left( \alpha _\mathrm{{f}}^\mathrm{{h}}+\mu _\mathrm{{f}}\right) +\eta _\mathrm{{f}}^\mathrm{{h}}A_{6}d_\mathrm{{f}}u_\mathrm{{f}}z_\mathrm{{f}}\left( \alpha _\mathrm{{f}}^\mathrm{{h}}+s_{2\mathrm{{f}}}\mu _\mathrm{{f}}\right) +\eta _\mathrm{{f}}^\mathrm{{h}}A_{6}u_\mathrm{{f}}z_\mathrm{{f}}\mu _\mathrm{{f}}\left( 1-s_{2\mathrm{{f}}}\right) \\&\quad +\, \eta _\mathrm{{f}}^\mathrm{{h}}A_{6}u_\mathrm{{f}}z_\mathrm{{f}}\left( 1-d_\mathrm{{f}}\right) \alpha _\mathrm{{f}}^\mathrm{{h}}\left[ s_{1\mathrm{{m}}}\left( 1-k_\mathrm{{f}}^\mathrm{{h}}\right) +k_\mathrm{{f}}^\mathrm{{h}}\left( 1-s_{2\mathrm{{f}}}\right) \right] >0, \end{aligned}$$

$$\begin{aligned} Q_{6}&= b_{5}A_{7}A_{8}A_{9}-b_{5}b_{8}g_{2}A_{7}-b_{5}b_{7}b_{8}g_{1}\\&= b_{5}\left( \omega _\mathrm{{f}}+\mu _\mathrm{{f}}\right) \left( u_\mathrm{{f}}+\mu _\mathrm{{f}}\right) \left( \alpha _\mathrm{{f}}^\mathrm{{h}}+\mu _\mathrm{{f}}\right) \\&+ b_{5}z_\mathrm{{f}}\mu _\mathrm{{f}}\left( \alpha _\mathrm{{f}}^\mathrm{{h}}+\mu _\mathrm{{f}}\right) +b_{5}d_\mathrm{{f}}u_\mathrm{{f}}z_\mathrm{{f}}\left( \alpha _\mathrm{{f}}^\mathrm{{h}}+s_{2\mathrm{{f}}}\mu _\mathrm{{f}}\right) +b_{5}u_\mathrm{{f}}z_\mathrm{{f}}\mu _\mathrm{{f}}\left( 1-s_{2\mathrm{{f}}}\right) \\&+ b_{5}u_\mathrm{{f}}z_\mathrm{{f}}\left( 1-d_\mathrm{{f}}\right) \alpha _\mathrm{{f}}^\mathrm{{h}}\left[ s_{1\mathrm{{m}}}\left( 1-k_\mathrm{{f}}^\mathrm{{h}}\right) +k_\mathrm{{f}}^\mathrm{{h}}\left( 1-s_{2\mathrm{{f}}}\right) \right] >0,\\ Q_{7}&= \theta _\mathrm{{f}}^\mathrm{{h}}\left( b_{5}b_{6}A_{8}A_{9}-b_{5}b_{6}b_{8}g_{2}\right) \\&= \theta _\mathrm{{f}}^\mathrm{{h}}b_{5}b_{6}\left( \omega _\mathrm{{f}}+\mu _\mathrm{{f}}\right) \left( u_\mathrm{{f}}+\mu _\mathrm{{f}}\right) +\theta _\mathrm{{f}}^\mathrm{{h}}b_{5}b_{6}z_\mathrm{{f}}u_\mathrm{{f}}\left( 1-s_{2\mathrm{{f}}}\right) \\&\quad +\,\theta _\mathrm{{f}}^\mathrm{{h}}b_{5}b_{6}z_\mathrm{{f}}\left( d_\mathrm{{f}}u_\mathrm{{f}}s_{2\mathrm{{f}}}+\mu _\mathrm{{f}}\right) >0,\\ Q_{8}&= A_{7}A_{8}A_{9}-b_{8}g_{2}A_{7}-b_{7}b_{8}g_{1}\\&= \left( \omega _\mathrm{{f}}+\mu _\mathrm{{f}}\right) \left( u_\mathrm{{f}}+\mu _\mathrm{{f}}\right) \left( \alpha _\mathrm{{f}}^\mathrm{{h}}+\mu _\mathrm{{f}}\right) +z_\mathrm{{f}}\mu _\mathrm{{f}}\left( \alpha _\mathrm{{f}}^\mathrm{{h}}+\mu _\mathrm{{f}}\right) \\&\quad +\,d_\mathrm{{f}}u_\mathrm{{f}}z_\mathrm{{f}}\left( \alpha _\mathrm{{f}}^\mathrm{{h}}+s_{2\mathrm{{f}}}\mu _\mathrm{{f}}\right) +u_\mathrm{{f}}z_\mathrm{{f}}\mu _\mathrm{{f}}\left( 1-s_{2\mathrm{{f}}}\right) \\&\quad +\,u_\mathrm{{f}}z_\mathrm{{f}}\left( 1-d_\mathrm{{f}}\right) \alpha _\mathrm{{f}}^\mathrm{{h}}\left[ s_{1\mathrm{{m}}}\left( 1-k_\mathrm{{f}}^\mathrm{{h}}\right) +k_\mathrm{{f}}^\mathrm{{h}}\left( 1-s_{2\mathrm{{f}}}\right) \right] >0. \end{aligned}$$

Thus,

$$\begin{aligned} \mathcal {R}_\mathrm{{f}\ell }>0,\;\; \mathcal {R}_\mathrm{{m}\ell }>0, \;\; \mathcal {R}_\mathrm{{fh}}>0 \;\; \mathrm{and} \;\; \mathcal {R}_\mathrm{{mh}}>0. \end{aligned}$$

4 Coefficients of the Polynomial (3.8)

$$\begin{aligned} Y_{0}&= b_{02}^{3}a_{33}+a_{02}a_{22}b_{02}b_{33}+b_{33}^{3}a_{00}+b_{33}^{2}b_{02}a_{11}>0,\\ Y_{1}&= 2b_{22}b_{33}\left( a_{00}b_{33}+a_{11}b_{02}\right) +b_{33}^{2}\left( b_{22}a_{00}-b_{02}a_{0}+b_{01}a_{11}\right) \\&\quad +\, a_{02}a_{22}\left( b_{02}b_{22}+b_{01}b_{33}\right) +2b_{02}^{2}b_{01}a_{33}\\&\quad +\,b_{02}b_{33}\left( b_{01}a_{22}-b_{02}a_{01}\right) +b_{02}^{2}\left( b_{01}a_{33}-b_{02}a_{02}\right) ,\\ Y_{2}&= b_{02}^{2}\left( b_{0}a_{33}-b_{01}a_{02}\right) +2b_{02}b_{01}\left( b_{01}b_{33}-b_{02}a_{02}\right) +b_{02}b_{33}\left( 2b_{0}b_{02}+b_{01}^{2}\right) \\&\quad +\, b_{02}b_{33}\left( b_{0}a_{22}-b_{01}a_{01}\right) +a_{02}a_{22}\left( b_{02}b_{11}+b_{01}b_{22}+b_{0}b_{33}\right) \\&\quad +\,\left( b_{02}b_{22}+b_{01}b_{33}\right) \left( b_{01}a_{22}-b_{02}a_{01}\right) +\left( 2b_{11}b_{33}+b_{22}^2\right) \left( a_{00}b_{33}+a_{11}b_{02}\right) \\&\quad +\, 2b_{33}b_{22}\left( b_{22}a_{00}-b_{02}a_{0}+b_{01}a_{11}\right) +b_{33}^{2}\left( b_{11}a_{00}+b_{0}a_{11}-b_{01}a_{0}\right) ,\\ Y_{3}&= -b_{02}^{2}b_{0}a_{02}+2b_{02}b_{01}\left( b_{0}a_{33}-b_{01}a_{02}\right) +\left( b_{02}b_{0}+b_{01}^{2}+b_{0}b_{02}\right) \\&\quad \times \left( b_{01}a_{33}-b_{02}a_{02}\right) -a_{01}b_{0}b_{02}b_{33}+2b_{0}b_{01}b_{02}a_{33}\\&\quad +\,a_{02}a_{22}\left( b_{02}b_{00}+b_{01}b_{11}+b_{0}b_{22}\right) + 2b_{22}b_{33}\left( a_{11}b_{0}+a_{00}b_{11}-b_{01}a_{0}\right) \\&\quad +\,\left( b_{01}a_{22}-b_{02}a_{01}\right) \left( b_{02}b_{11}+b_{01}b_{22}+b_{0}b_{33}\right) \\&\quad +\,\left( a_{00}b_{33}+a_{11}b_{02}\right) \left( 2b_{33}b_{00}+2b_{22}b_{11}\right) +\left( b_{0}a_{22}-b_{01}a_{01}\right) \\&\quad \times \left( b_{02}b_{22}+b_{01}b_{33}\right) +\left( b_{22}a_{00}-b_{02}a_{0}+b_{01}a_{11}\right) \left( 2b_{33}b_{11}+b_{22}^{2}\right) \\&\quad +\,b_{33}^{2}\left( a_{00}b_{00}-a_{0}b_{0}\right) , \end{aligned}$$

$$\begin{aligned} Y_{4}&= -2b_{0}a_{02}b_{02}b_{01}+\left( 2b_{0}b_{02}+b_{01}^{2}\right) \left( b_{0}a_{33}-b_{01}a_{02}\right) \\&\quad +\,2b_{0}b_{01}\left( b_{01}a_{33}-a_{02}b_{02}\right) +a_{02}a_{22}b_{01}b_{00} +a_{02}a_{22}b_{0}b_{11}\\&\quad +\,\left( b_{01}a_{22}-b_{02}a_{01}\right) \left( b_{02}b_{00}+b_{01}b_{11}+b_{0}b_{22}\right) -a_{01}b_{0}\left( b_{02}b_{22}+b_{01}b_{33}\right) \\&\quad +\,\left( b_{0}a_{22}-b_{01}a_{01}\right) \left( b_{02}b_{11}+b_{01}b_{22}+b_{0}b_{33}\right) +\left( a_{00}b_{33}+a_{11}b_{02}\right) \\&\quad \times \left( 2b_{22}b_{00}+b_{11}^{2}\right) +\left( b_{22}a_{00}-a_{0}b_{02}\right) \left( 2b_{22}b_{11}+2b_{33}b_{00}\right) \\&\quad +\,\left( a_{11}b_{0}+a_{00}b_{11}-b_{01}a_{0}\right) \left( 2b_{11}b_{33}+b_{22}^{2}\right) +2b_{22}b_{33}\left( a_{00}b_{00}-a_{0}b_{0}\right) ,\\ Y_{5}&= -2b_{0}a_{02}\left( 2b_{0}b_{02}+b_{01}^{2}\right) +b_{01}^{2}\left( b_{01}a_{33}-a_{02}b_{02}\right) +2b_{0}b_{01}\left( b_{0}a_{33}-b_{01}a_{02}\right) \\&\quad +\,2b_{00}b_{11}a_{00}b_{33}+ 2b_{00}b_{11}a_{11}b_{02}+\left( b_{01}a_{22}-b_{02}a_{01}\right) \left( b_{01}b_{00}+b_{0}b_{11}\right) \\&\quad -\,a_{01}b_{0}\left( b_{02}b_{11}+b_{01}b_{22}+b_{0}b_{33}\right) +\left( b_{22}a_{00}-a_{0}b_{02}+b_{01}a_{11}\right) \\&\quad \times \left( 2b_{00}b_{22}+b_{11}^{2}\right) +\left( b_{0}a_{22}-b_{01}a_{01}\right) \left( b_{02}b_{00}+b_{01}b_{11}+b_{0}b_{22}\right) \\&\quad +\,\left( a_{11}b_{0}+a_{00}b_{11}-b_{01}a_{0}\right) \left( 2b_{00}b_{33}+2b_{22}b_{11}\right) \\&\quad +\,\left( a_{00}b_{00}-a_{0}b_{0}\right) \left( 2b_{11}b_{33}+b_{22}^{2}\right) , \end{aligned}$$

$$\begin{aligned} Y_{6}&= -2b_{0}^{2}b_{01}a_{02}+b_{01}^{2}\left( b_{0}a_{33}-b_{01}a_{02}\right) +b_{0}b_{00}\left( b_{01}a_{22}-b_{02}a_{01}\right) \\&\quad +\,\left( b_{0}a_{22}-b_{01}a_{01}\right) \left( b_{01}b_{00}+b_{0}b_{11}\right) +2b_{00}b_{11}\left( b_{22}a_{00}-a_{0}b_{02}+b_{01}a_{11}\right) \\&\quad +\,b_{00}^{2}\left( a_{00}b_{33}+a_{11}b_{02}\right) -b_{0}^{2}a_{01}b_{00}-b_{01}a_{0}\left( 2b_{22}b_{00}+b_{11}^{2}\right) \\&\quad +\,\left( a_{11}b_{0}+b_{11}a_{00}\right) \left( 2b_{22}b_{00}+b_{11}^{2}\right) +\left( a_{00}b_{00}-a_{0}b_{0}\right) \left( 2b_{33}b_{00}+2b_{22}b_{11}\right) ,\\ Y_{7}&= -b_{0}a_{02}b_{01}^{2}+b_{0}b_{00}\left( b_{0}a_{22}-b_{01}a_{01}\right) -a_{01}b_{0}\left( b_{01}b_{00}+b_{0}b_{11}\right) \\&\quad +\,2b_{00}b_{11}\left( a_{00}b_{11}-b_{01}a_{0}+a_{11}b_{0}\right) +b_{00}^{2}\left( b_{22}a_{00}-a_{0}b_{02}+b_{01}a_{11}\right) \\&\quad +\left( a_{00}b_{00}-a_{0}b_{0}\right) \left( 2b_{22}b_{00}+b_{11}^{2}\right) ,\\ Y_{8}&= -a_{01}b_{0}^{2}b_{00}+b_{00}^{2}\left( a_{11}b_{0}+a_{00}b_{11}-a_{0}b_{01}\right) +2b_{00}b_{11}\left( a_{00}b_{00}-a_{0}b_{0}\right) ,\\ Y_{9}&= b_{00}^{3}a_{00}\left[ 1-\left( \mathcal {R}_{0}^{\ell }\right) ^{2}\right] >0\;\;\left( \mathrm{if}\;\; \mathcal {R}_{0}^{\ell }<1\right) . \end{aligned}$$

5 Proof of Theorem 3.3

Proof

The proof is based on using Centre Manifold theory (Carr 1981; Castillo-Chavez and Song 2004). It is convenient to use the change of variables:

$$\begin{aligned} S_\mathrm{{f}}&= x_{1},\;\; V_\mathrm{{f}}^\mathrm{{q}}=x_{2},\;\;E_\mathrm{{f}}^{\ell }=x_{3},\;\;I_\mathrm{{f}}^{\ell }=x_{4},\;\;P_\mathrm{{f}}^{\ell }=x_{5}, \;\;W_\mathrm{{f}}=x_{6},\;\;R_\mathrm{{f}}=x_{7},\;\;S_\mathrm{{m}}=x_{8},\nonumber \\ V_\mathrm{{m}}^\mathrm{{q}}&= x_{9},\;\;E_\mathrm{{m}}^{\ell }=x_{10},\;\;I_\mathrm{{m}}^{\ell }=x_{11}, \;\;P_\mathrm{{m}}^{\ell }=x_{12},\;\;W_\mathrm{{m}}=x_{13},\;\;R_\mathrm{{m}}=x_{14}. \end{aligned}$$

(46)

Let $\hat{f} = \left[ f_1,\ldots , f_{14}\right]$ denote the vector field of the low-risk-only model (8) in the notation (46), so that the low-risk-only model (8) is re-written in the following form:

$$\begin{aligned} \frac{\mathrm{{d}}x_{1}}{\mathrm{{d}}t}&= f_{1}=(1-\varphi _\mathrm{{f}}^\mathrm{{q}})\pi _\mathrm{{f}}-\frac{\beta _\mathrm{{m}}^{\ell }c_\mathrm{{f}}\mu _\mathrm{{m}}x_{11}x_{1}}{\pi _\mathrm{{m}}}-\mu _\mathrm{{f}}x_{1},\nonumber \\ \frac{\mathrm{{d}}x_{2}}{\mathrm{{d}}t}&= f_{2}=\varphi _\mathrm{{f}}^\mathrm{{q}}\pi _\mathrm{{f}}-(1-\varepsilon _{v})\frac{\beta _\mathrm{{m}}^{\ell }c_\mathrm{{f}}\mu _\mathrm{{m}}x_{11}x_{2}}{\pi _\mathrm{{m}}}-\mu _\mathrm{{f}}x_{2},\nonumber \\ \frac{\mathrm{{d}}x_{3}}{\mathrm{{d}}t}&= f_{3}=\left[ x_{1}+(1-\varepsilon _{v})x_{2}+\rho _\mathrm{{f}}^{\ell }x_{7}\right] \frac{\beta _\mathrm{{m}}^{\ell }c_\mathrm{{f}}\mu _\mathrm{{m}}x_{11}}{\pi _\mathrm{{m}}}-A_{1}x_{3},\nonumber \\ \frac{\mathrm{{d}}x_{4}}{\mathrm{{d}}t}&= f_{4}=\sigma _\mathrm{{f}}^{\ell }x_{3}-A_{2}x_{4}, \nonumber \\ \frac{\mathrm{{d}}x_{5}}{\mathrm{{d}}t}&= f_{5}=b_{2}x_{4}-A_{3}x_{5},\\ \frac{\mathrm{{d}}x_6}{\mathrm{{d}}t}&= f_{6}=b_{3}x_{5}-A_{4}x_6,\nonumber \\ \frac{\mathrm{{d}}x_{7}}{\mathrm{{d}}t}&= f_{7}=m_{1}x_{4}+m_{2}x_{5}+n_\mathrm{{f}}x_6-\rho _\mathrm{{f}}^{\ell }\frac{\beta _\mathrm{{m}}^{\ell }c_\mathrm{{f}}\mu _\mathrm{{m}}x_{11}x_{7}}{\pi _\mathrm{{m}}}-\mu _\mathrm{{f}}x_{7},\nonumber \\ \frac{\mathrm{{d}}x_{8}}{\mathrm{{d}}t}&= f_{8}=\left( 1-\varphi _\mathrm{{m}}^\mathrm{{q}}\right) \pi _\mathrm{{m}}-\frac{\beta _\mathrm{{f}}^{\ell }c_\mathrm{{f}}\mu _\mathrm{{m}}x_{3}x_{8}}{\pi _\mathrm{{m}}}-\mu _\mathrm{{m}}x_{8},\nonumber \\ \frac{\mathrm{{d}}x_{9}}{\mathrm{{d}}t}&= f_{9}=\varphi _\mathrm{{m}}^\mathrm{{q}}\pi _\mathrm{{m}}-(1-\varepsilon _{v})\frac{\beta _\mathrm{{f}}^{\ell }c_\mathrm{{f}}\mu _\mathrm{{m}}x_{3}x_{9}}{\pi _\mathrm{{m}}}-\mu _\mathrm{{m}}x_{9},\nonumber \\ \frac{\mathrm{{d}}x_{10}}{\mathrm{{d}}t}&= f_{10}=\left[ x_{8}+(1-\varepsilon _{v})x_{9}+\rho _\mathrm{{m}}^{\ell }x_{14}\right] \frac{\beta _\mathrm{{f}}^{\ell }c_\mathrm{{f}}\mu _\mathrm{{m}}x_{3}}{\pi _\mathrm{{m}}}-D_{1}x_{10},\nonumber \\ \frac{\mathrm{{d}}x_{11}}{\mathrm{{d}}t}&= f_{11}=\sigma _\mathrm{{m}}^{\ell }x_{10}-D_{2}x_{11},\nonumber \\ \frac{\mathrm{{d}}x_{12}}{\mathrm{{d}}t}&= f_{12}=k_{2}x_{11}-D_{3}x_{12},\nonumber \\ \frac{\mathrm{{d}}x_{13}}{\mathrm{{d}}t}&= f_{13}=k_{3}x_{13}-D_{4}x_{13},\nonumber \\ \frac{\mathrm{{d}}x_{14}}{\mathrm{{d}}t}&= f_{14}=m_{4}x_{11}+m_{5}x_{12}+n_\mathrm{{m}}x_{13}-\rho _\mathrm{{m}}^{\ell }\frac{\beta _\mathrm{{f}}^{\ell }c_\mathrm{{f}}\mu _\mathrm{{m}}x_{3}x_{14}}{\pi _\mathrm{{m}}}-\mu _\mathrm{{m}}x_{14},\nonumber \end{aligned}$$

(47)

where $A_{i},D_{i}$ ($i=1,\ldots ,4$), $b_{j},k_{j}$ ($j=1,\ldots ,3$) and $m_{1}, m_{2}, m_{4}, m_{5}$ are as defined in Sects. 3.1 and 3.2.

The Jacobian of the system (47) at the DFE ($\mathcal {E}_{0}^{\ell }$) is given by

$$\begin{aligned} J^{\ell }(\mathcal {E}_{0}^{\ell })=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} -\mu _\mathrm{{f}} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{}0&{}0&{} -U_{1}&{} 0 &{} 0&{}0\\ 0&{}-\mu _\mathrm{{f}} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{}0&{}0&{} -U_{2}&{} 0 &{} 0&{}0\\ 0 &{} 0 &{} U_{4}&{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} U_{3}&{} 0 &{} 0&{}0\\ 0 &{} 0 &{} \sigma _\mathrm{{f}}^{\ell } &{} U_{5} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0&{} 0 &{} 0&{}0\\ 0 &{} 0 &{} 0 &{} b_{2} &{} U_{6} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0&{}0&{}0&{}0\\ 0 &{} 0 &{} 0 &{}0&{} b_{3} &{}U_{7} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0&{}0&{}0\\ 0 &{} 0 &{} 0 &{} m_{1} &{} m_{2} &{} n_\mathrm{{f}} &{} -\mu _\mathrm{{f}} &{} 0 &{} 0 &{} 0 &{} 0&{}0&{}0&{}0\\ 0&{} 0 &{} 0 &{} -U_{8} &{} 0 &{} 0 &{} 0&{} -\mu _\mathrm{{m}} &{}0&{}0&{} 0&{} 0 &{} 0&{}0\\ 0&{} 0 &{} 0 &{} -U_{9} &{} 0 &{} 0 &{} 0&{} 0&{} -\mu _\mathrm{{m}} &{}0&{}0&{} 0&{} 0 &{} 0\\ 0&{} 0 &{} 0 &{} U_{10} &{} 0 &{} 0 &{} 0&{} 0&{}0&{} U_{11} &{}0&{}0&{} 0&{} 0 \\ 0&{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0&{} 0&{}0&{} \sigma _\mathrm{{m}}^{\ell }&{}U_{12}&{}0&{} 0&{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{}0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} k_{2}&{}U_{13}&{}0&{}0\\ 0 &{} 0 &{} 0 &{} 0 &{}0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{}0&{} k_{3}&{}U_{14}&{}0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0&{} m_{4}&{}m_{5}&{}n_\mathrm{{m}}&{}-\mu _\mathrm{{m}}\end{array}\right) , \end{aligned}$$

where

$$\begin{aligned} U_{1}&= \frac{\beta _\mathrm{{m}}^{\ell }c_\mathrm{{f}}\pi _\mathrm{{f}}\mu _\mathrm{{m}}\left( 1-\varphi _\mathrm{{f}}^\mathrm{{q}}\right) }{\mu _\mathrm{{f}}\pi _\mathrm{{m}}},\;\; U_{2}=\frac{\beta _\mathrm{{m}}^{\ell }c_\mathrm{{f}}\pi _\mathrm{{f}}\mu _\mathrm{{m}}\varphi _\mathrm{{f}}^\mathrm{{q}}\left( 1-\varepsilon _{v}\right) }{\mu _\mathrm{{f}}\pi _\mathrm{{m}}},\\ U_{3}&= \frac{\beta _\mathrm{{m}}^{\ell }c_\mathrm{{f}}\pi _\mathrm{{f}}\mu _\mathrm{{m}}\left( 1-\varepsilon _{q}\varphi _\mathrm{{f}}^\mathrm{{q}}\right) }{\pi _\mathrm{{m}}},\;\; U_{4}=-A_{1},\;\; U_{5}=-A_{2},\;\; U_{6}=-A_{3},\\ U_{7}&= -A_{4},\;\; U_{8}=\beta ^{*}c_\mathrm{{f}}\left( 1-\varphi _\mathrm{{m}}^\mathrm{{q}}\right) ,\;\; U_{9}=\beta ^{*}c_\mathrm{{f}}\varphi _\mathrm{{m}}^\mathrm{{q}}\left( 1-\varepsilon _{v}\right) ,\\ U_{10}&= \beta ^{*}c_\mathrm{{f}}\left( 1-\varepsilon _{v}\varphi _\mathrm{{m}}^\mathrm{{q}}\right) ,\;\; U_{11}=-D_{1},\;\; U_{12}=-D_{2},\\ U_{13}&= -D_{3},\;\; U_{14}=-D_{4}. \end{aligned}$$

Consider the case when $\mathcal {R}_{0}^{\ell }=1$, and choose $\beta _\mathrm{{f}}^{\ell }$ as a bifurcation parameter. Solving for $\beta _\mathrm{{f}}^{\ell }$ from $\mathcal {R}_{0}^{\ell }=1$ gives

$$\begin{aligned} \beta _\mathrm{{f}}^{\ell }=\beta ^{*}=\frac{A_{1}A_{2}D_{1}D_{2}\pi _\mathrm{{m}} \mu _\mathrm{{f}}}{\beta _\mathrm{{m}}^{\ell }c_\mathrm{{f}}^{2}\pi _\mathrm{{f}}\mu _\mathrm{{m}}k_{1}b_{1} \left( 1-\varepsilon _{v}\varphi _\mathrm{{f}}^\mathrm{{q}}\right) \left( 1-\varepsilon _{v}\varphi _\mathrm{{m}}^\mathrm{{q}}\right) }. \end{aligned}$$

(48)

The transformed system (47), with $\beta _\mathrm{{f}}^{\ell }=\beta ^{*}$, has a hyperbolic equilibrium point (i.e. the linearization has an eigenvalue with zero real part, while the other eigenvalues have negative real part; hence, the Centre Manifold theory can be used Carr 1981).$\square$

1.1 Eigenvectors of $J^{\ell }(\mathcal {E}_{0}^{\ell })\mid _{\beta _\mathrm{{f}}^{\ell }=\beta ^{*}}$:

It can be shown that the Jacobian of (47) at $\beta _\mathrm{{f}}^{\ell }=\beta ^{*}$ (denoted by $J_{\beta ^{*}}^{\ell }$) has a left eigenvector (associated with the zero eigenvalue) given by

$$\begin{aligned} \varvec{v}=\left[ v_{1},v_{2},v_{3},v_{4},v_{5},v_{6},v_{7},v_{8},v_{9},v_{10},v_{11},v_{12},v_{13},v_{14}\right] ^{T}, \end{aligned}$$

with

$$\begin{aligned} v_{1}&= 0,\;\; v_{2}=0,\;\; v_{3}=\frac{\sigma _\mathrm{{f}}^{\ell }v_{4}}{A_{1}},\;\; v_{4} = \frac{\beta ^{*}c_\mathrm{{f}}\left( 1-\varepsilon _{v}\varphi _\mathrm{{m}}^\mathrm{{q}}\right) v_{10}}{A_{2}}, \;\; v_{5}=0,\;\; v_{6}=0,\\ v_{7}&= 0,\;\; v_{8}=0,\;\; v_{9} = 0,\;\; v_{10} =v_{10}>0,\;\; v_{11}=\frac{\beta _\mathrm{{m}}^{\ell }c_\mathrm{{f}}\mu _\mathrm{{m}}\pi _\mathrm{{f}}\left( 1-\varepsilon _{v}\varphi _\mathrm{{f}}^\mathrm{{q}}\right) v_{3}}{D_{2}\pi _\mathrm{{m}}\mu _\mathrm{{f}}},\\ v_{12}&= 0,\;\; v_{13}=0,\;\; v_{14}=0. \end{aligned}$$

Furthermore, the matrix $J_{\beta ^{*}}^{\ell }$ has a right eigenvector (associated with the zero eigenvalue) given by

$$\begin{aligned} \varvec{w}=\left[ w_{1},w_{2},w_{3},w_{4},w_{5},w_{6},w_{7},w_{8},w_{9},w_{10},w_{11},w_{12},w_{13},w_{14}\right] ^{T}, \end{aligned}$$

where

$$\begin{aligned} w_{1}&= -\frac{\beta _\mathrm{{m}}^{\ell }c_\mathrm{{f}}\mu _\mathrm{{m}}\pi _\mathrm{{f}}\left( 1-\varphi _\mathrm{{f}}^\mathrm{{q}}\right) w_{11}}{\mu _\mathrm{{f}}^{2}\pi _\mathrm{{m}}},\;\;w_{2}=-\frac{\beta _\mathrm{{m}}^{\ell }c_\mathrm{{f}}\mu _\mathrm{{m}}\pi _\mathrm{{f}}\left( 1-\varepsilon _{v}\right) \varphi _\mathrm{{f}}^\mathrm{{q}}w_{11}}{\pi _\mathrm{{m}}\mu _\mathrm{{f}}^{2}},\nonumber \\ w_{3}&= \frac{\beta _\mathrm{{m}}^{\ell }c_\mathrm{{f}}\mu _\mathrm{{m}}\pi _\mathrm{{f}}\left( 1-\varepsilon _{v}\varphi _\mathrm{{f}}^\mathrm{{q}}\right) w_{11}}{A_{1}\pi _\mathrm{{m}}\mu _\mathrm{{f}}},\;\; w_{4}=\frac{\sigma _\mathrm{{f}}^{\ell }w_{3}}{A_{2}},\;\; w_{5}=\frac{b_{2}w_{4}}{A_{3}},\;\; w_{6}=\frac{b_{3}w_{5}}{A_{4}}, \nonumber \\ w_{7}&= \frac{m_{1}w_{4}+m_{2}w_{5}+n_\mathrm{{f}}w_{6}}{\mu _\mathrm{{f}}},\;\; w_{8}=-\frac{\beta ^{*}c_\mathrm{{f}}\left( 1-\varphi _\mathrm{{m}}^\mathrm{{q}}\right) w_{4}}{\mu _\mathrm{{m}}},\nonumber \\ w_{9}&= -\frac{\beta ^{*}c_\mathrm{{f}}\left( 1-\varepsilon _{v}\right) \varphi _\mathrm{{m}}^\mathrm{{q}}w_{4}}{\mu _\mathrm{{m}}},\;\; w_{10}=w_{10}>0,\;\; w_{11}=\frac{\sigma _\mathrm{{m}}^{\ell }w_{10}}{D_{2}},\;\; w_{12}=\frac{k_{2}w_{11}}{D_{3}},\nonumber \\ w_{13}&= \frac{k_{3}w_{12}}{D_{4}},\;\; w_{14}=\frac{m_{4}w_{11}+m_{5}w_{12}+n_\mathrm{{m}}w_{13}}{\mu _\mathrm{{m}}}.\nonumber \end{aligned}$$

Thus, using Theorem 4.1 of Castillo-Chavez and Song (2004), the associated bifurcation coefficients, $a$ and $b$, can be computed as below.

1.2 Computations of Bifurcation Coefficients, $a$ and $b$:

It can be shown, by computing the non-zero partial derivatives of the model (47) at the DFE ($\mathcal {E}_{0}^{\ell }$) and simplifying, that (Castillo-Chavez and Song 2004)

$$\begin{aligned} a = \sum _{k,i,j=1}^{14}v_{k}w_{i}w_{j}\frac{\partial ^{2}f_{k}}{\partial x_{i}\partial x_{j}}(0,0)=\frac{2c_\mathrm{{f}}}{\mu _\mathrm{{f}}^{2}\pi _\mathrm{{m}}^{2}}\left( M_{11}-M_{22}\right) , \end{aligned}$$

(49)

and

$$\begin{aligned} b = \sum _{k,i=1}^{14}v_{k}w_{i}\frac{\partial ^{2}f_{k}}{\partial x_{i}\partial \beta ^{*}}(0,0)= c_\mathrm{{f}}v_{10}w_{4}\left( 1-\varepsilon _{v}\varphi _\mathrm{{m}}^\mathrm{{q}}\right) >0, \end{aligned}$$

where

$$\begin{aligned} M_{11}&= v_{3}w_{7}w_{11}\rho _\mathrm{{f}}^{\ell }\beta _\mathrm{{m}}^{\ell }\mu _\mathrm{{m}}\mu _\mathrm{{f}}^{2}\pi _\mathrm{{m}}+v_{10}w_{4}w_{14}\rho _\mathrm{{m}}^{\ell }\beta ^{*}\mu _\mathrm{{m}}\mu _\mathrm{{f}}^{2}\pi _\mathrm{{m}}+v_{3}w_{11}^{2}\beta _\mathrm{{m}}^{\ell }\mu _\mathrm{{m}}^{2}\pi _\mathrm{{f}}\mu _\mathrm{{f}}\varepsilon _{v}\varphi _\mathrm{{f}}^\mathrm{{q}}\\&+ v_{3}w_{4}w_{11}\beta ^{*}c_\mathrm{{f}}\beta _\mathrm{{m}}^{\ell }\mu _\mathrm{{m}}\pi _\mathrm{{f}}\mu _\mathrm{{f}}\left( 1-\varepsilon _{v}\varphi _\mathrm{{f}}^\mathrm{{q}}\right) +v_{3}w_{11}^{2}\beta _\mathrm{{m}}^{\ell ^{2}}c_\mathrm{{f}}\mu _\mathrm{{m}}^{2}\pi _\mathrm{{f}}\varepsilon _{v}\varphi _\mathrm{{f}}^\mathrm{{q}}\left( 2-\varepsilon _{v}\right) ,\\ M_{22}&= v_{3}w_{11}^{2}\beta _\mathrm{{m}}^{\ell ^{2}}c_\mathrm{{f}}\mu _\mathrm{{m}}^{2}\pi _\mathrm{{f}}+v_{3}w_{11}^{2}\beta _\mathrm{{m}}^{\ell }\mu _\mathrm{{m}}^{2}\pi _\mathrm{{f}}\mu _\mathrm{{f}}+v_{3}w_{4}w_{11}\beta ^{*}c_\mathrm{{f}}\beta _\mathrm{{m}}^{\ell }\mu _\mathrm{{m}}\pi _\mathrm{{f}}\mu _\mathrm{{f}}\varphi _\mathrm{{m}}^\mathrm{{q}}\varepsilon _{v}\left( 1-\varepsilon _{v}\varphi _\mathrm{{f}}^\mathrm{{q}}\right) \\&\quad +\, v_{3}w_{10}w_{11}\beta _\mathrm{{m}}^{\ell }\mu _\mathrm{{m}}^{2}\pi _\mathrm{{f}}\mu _\mathrm{{f}}\left( 1-\varepsilon _{v}\varphi _\mathrm{{f}}^\mathrm{{q}}\right) +v_{3}w_{11}w_{12}\beta _\mathrm{{m}}^{\ell }\mu _\mathrm{{m}}^{2}\pi _\mathrm{{f}}\mu _\mathrm{{f}}\left( 1-\varepsilon _{v}\varphi _\mathrm{{f}}^\mathrm{{q}}\right) \\&\quad +\,v_{3}w_{11}w_{13}\beta _\mathrm{{m}}^{\ell }\mu _\mathrm{{m}}^{2}\pi _\mathrm{{f}}\mu _\mathrm{{f}}\left( 1-\varepsilon _{v}\varphi _\mathrm{{f}}^\mathrm{{q}}\right) +v_{3}w_{11}w_{14}\beta _\mathrm{{m}}^{\ell }\mu _\mathrm{{m}}^{2}\pi _\mathrm{{f}}\mu _\mathrm{{f}}\left( 1-\varepsilon _{v}\varphi _\mathrm{{f}}^\mathrm{{q}}\right) \\&\quad +\,v_{10}w_{4}^{2}\beta ^{*^{2}}c_\mathrm{{f}}\pi _\mathrm{{m}}\mu _\mathrm{{f}}^{2}\varphi _\mathrm{{m}}^\mathrm{{q}}\varepsilon _{v}^2\left( 1-\varphi _\mathrm{{m}}^\mathrm{{q}}\right) +v_{10}w_{4}w_{10}\beta ^{*}\pi _\mathrm{{m}}\mu _\mathrm{{f}}^{2}\mu _\mathrm{{m}}\left( 1-\varepsilon _{v}\varphi _\mathrm{{m}}^\mathrm{{q}}\right) \\&\quad +\,v_{10}w_{4}w_{11}\beta ^{*}\pi _\mathrm{{m}}\mu _\mathrm{{f}}^{2}\mu _\mathrm{{m}}\left( 1-\varepsilon _{v}\varphi _\mathrm{{m}}^\mathrm{{q}}\right) +v_{10}w_{4}w_{12}\beta ^{*}\pi _\mathrm{{m}}\mu _\mathrm{{f}}^{2}\mu _\mathrm{{m}}\left( 1-\varepsilon _{v}\varphi _\mathrm{{m}}^\mathrm{{q}}\right) \\&\quad +\, v_{10}w_{4}w_{13}\beta ^{*}\pi _\mathrm{{m}}\mu _\mathrm{{f}}^{2}\mu _\mathrm{{m}}\left( 1-\varepsilon _{v}\varphi _\mathrm{{m}}^\mathrm{{q}}\right) +v_{10}w_{4}w_{14}\beta ^{*}\pi _\mathrm{{m}}\mu _\mathrm{{f}}^{2}\mu _\mathrm{{m}}\left( 1-\varepsilon _{v}\varphi _\mathrm{{m}}^\mathrm{{q}}\right) \!. \end{aligned}$$

Thus, the result below follows from Theorem 4.1 of Castillo-Chavez and Song (2004).

Theorem 5.1

The transformed model (47) (or, equivalently, the low-risk-only model (8)) undergoes a backward bifurcation at $\mathcal {R}_{0}^{\ell }=1$ if the bifurcation coefficient $a$, given by (49), is positive.

It is clear from the expression for the bifurcation coefficient, $a$, given by (49), that $a>0$ whenever (since all the low-risk-only model parameters are positive)

$$\begin{aligned} M_{11}>M_{22}. \end{aligned}$$

(50)

Furthermore, consider the low-risk-only model (8) in the absence of re-infection ($\rho _\mathrm{{f}}^{\ell }=\rho _\mathrm{{m}}^{\ell }=0$). Setting $\rho _\mathrm{{f}}^{\ell }=\rho _\mathrm{{m}}^{\ell }=0$ into the expression for the bifurcation coefficient, $a$ in (49), and using $\beta ^*$ for $\beta _\mathrm{{f}}^{\ell }$ in (48), shows that (here, the eigenvectors $v_{10}$ and $w_{10}$ are given the value unity)

$$\begin{aligned} a&= - \frac{2\left( \sigma _\mathrm{{m}}^{\ell }+\mu _\mathrm{{m}}\right) }{\mu _\mathrm{{f}}\pi _\mathrm{{m}}\left( \psi _\mathrm{{m}}^{\ell }+\mu _\mathrm{{m}}\right) \left( 1-\varepsilon _{v}\varphi _\mathrm{{f}}^\mathrm{{q}}\right) ^{2}}\left[ \varepsilon _{v}\sigma _\mathrm{{m}}^{\ell }\varphi _\mathrm{{f}}^\mathrm{{q}}\psi _\mathrm{{m}}^{\ell }\mu _\mathrm{{m}} \left( 2-\varepsilon _{v}\varphi _\mathrm{{f}}^\mathrm{{q}}\right) +\varepsilon _{v}\sigma _\mathrm{{m}}^{\ell }\varphi _\mathrm{{f}}^\mathrm{{q}}\mu _\mathrm{{m}}^{2} \left( 2-\varepsilon _{v}\varphi _\mathrm{{f}}^\mathrm{{q}}\right) \right] \nonumber \\&\quad -\, \frac{2\left( \sigma _\mathrm{{m}}^{\ell }+\mu _\mathrm{{m}}\right) }{\mu _\mathrm{{f}}\pi _\mathrm{{m}}\left( \psi _\mathrm{{m}}^{\ell }+\mu _\mathrm{{m}}\right) \left( 1-\varepsilon _{v}\varphi _\mathrm{{f}}^\mathrm{{q}}\right) ^{2}} \left[ \varepsilon _{v}\sigma _\mathrm{{m}}^{\ell ^{2}}\varphi _\mathrm{{f}}^\mathrm{{q}}\left( \psi _\mathrm{{m}}^{\ell }+\mu _\mathrm{{m}}\right) \left( 2-\varepsilon _{v}\varphi _\mathrm{{f}}^\mathrm{{q}}\right) +\mu _\mathrm{{f}}\mu _\mathrm{{m}} \sigma _\mathrm{{m}}^{\ell ^{2}}\left( 1-\varepsilon _{v}\varphi _\mathrm{{f}}^\mathrm{{q}}\right) \right] \nonumber \\&\quad -\,\frac{2\mu _\mathrm{{m}}\sigma _\mathrm{{m}}^{\ell }\left( \sigma _\mathrm{{m}}^{\ell }+\mu _\mathrm{{m}}\right) \left( n_\mathrm{{m}}+\mu _\mathrm{{m}}\right) \psi _\mathrm{{m}}^{\ell }}{\pi _\mathrm{{m}}\mu _\mathrm{{f}}\left( \psi _\mathrm{{m}}^{\ell }+\mu _\mathrm{{m}}\right) \left( \alpha _\mathrm{{m}}^{\ell }+\mu _\mathrm{{m}}\right) \left( n_\mathrm{{m}}+\mu _\mathrm{{m}}\right) }\left[ r_\mathrm{{m}}^{\ell }\left( \alpha _\mathrm{{m}}^{\ell }+\mu _\mathrm{{m}}\right) +k_\mathrm{{m}}^{\ell }\alpha _\mathrm{{m}}^{\ell }\left( 1-r_\mathrm{{m}}^{\ell }\right) \right] \nonumber \\&\quad -\,\frac{2\mu _\mathrm{{m}}\sigma _\mathrm{{m}}^{\ell }\left( \sigma _\mathrm{{m}}^{\ell }+\mu _\mathrm{{m}}\right) }{\pi _\mathrm{{m}}\mu _\mathrm{{f}}\left( \psi _\mathrm{{m}}^{\ell }+\mu _\mathrm{{m}}\right) \left( \alpha _\mathrm{{m}}^{\ell }+\mu _\mathrm{{m}}\right) \left( n_\mathrm{{m}}+\mu _\mathrm{{m}}\right) }\left[ n_\mathrm{{m}}\left( 1-r_\mathrm{{m}}^{\ell }\right) \psi _\mathrm{{m}}^{\ell }\left( 1-k_\mathrm{{m}}^{\ell }\right) \alpha _\mathrm{{m}}^{\ell }\right] \nonumber \\&\quad -\, \frac{2\left( \sigma _\mathrm{{m}}^{\ell }+\mu _\mathrm{{m}}\right) }{\mu _\mathrm{{f}}\pi _\mathrm{{m}}\left( \psi _\mathrm{{m}}^{\ell }+\mu _\mathrm{{m}}\right) \left( 1-\varepsilon _{v}\varphi _\mathrm{{f}}^\mathrm{{q}}\right) ^{2}}\left[ \mu _\mathrm{{f}}\mu _\mathrm{{m}}\sigma _\mathrm{{m}}^{\ell }\psi _\mathrm{{m}}^{\ell }\left( 1-\varepsilon _{v}\varphi _\mathrm{{f}}^\mathrm{{q}}\right) +\mu _\mathrm{{f}}\sigma _\mathrm{{m}}^{\ell ^{2}}\psi _\mathrm{{m}}^{\ell }\left( 1-\varepsilon _{v}\varphi _\mathrm{{f}}^\mathrm{{q}}\right) \right] \nonumber \\&\quad -\, \frac{2\left( \sigma _\mathrm{{m}}^{\ell }+\mu _\mathrm{{m}}\right) }{\mu _\mathrm{{f}}\pi _\mathrm{{m}}\left( \psi _\mathrm{{m}}^{\ell }+\mu _\mathrm{{m}}\right) \left( 1-\varepsilon _{v}\varphi _\mathrm{{f}}^\mathrm{{q}}\right) ^{2}}\left[ \beta _\mathrm{{m}}^{\ell }\mu _\mathrm{{m}}\sigma _\mathrm{{m}}^{\ell ^{2}}c_\mathrm{{f}}\left( 1-\varepsilon _{v}\varphi _\mathrm{{f}}^\mathrm{{q}}\right) +\mu _\mathrm{{f}}\mu _\mathrm{{m}}^{2}\sigma _\mathrm{{m}}^{\ell }\left( 1-\varepsilon _{v}\varphi _\mathrm{{f}}^\mathrm{{q}}\right) ^{2}\right] \nonumber \\&\quad -\,\frac{2\sigma _\mathrm{{m}}^{\ell }\varphi _\mathrm{{m}}^\mathrm{{q}}\varepsilon _{v}\left( \sigma _\mathrm{{m}}^{\ell }+\mu _\mathrm{{m}}\right) ^{2}\left( 1-\varphi _\mathrm{{m}}^\mathrm{{q}}\right) }{\pi _\mathrm{{m}}\mu _\mathrm{{f}}^{2}\left( 1-\varepsilon _{v}\varphi _\mathrm{{m}}^\mathrm{{q}}\right) ^{2}}-\frac{2\mu _\mathrm{{m}}\sigma _\mathrm{{m}}^{\ell }\left( \sigma _\mathrm{{m}}^{\ell }+\mu _\mathrm{{m}}\right) }{\pi _\mathrm{{m}}\mu _\mathrm{{f}}}\left[ 1+\frac{1}{\left( \psi _\mathrm{{m}}^{\ell }+\mu _\mathrm{{m}}\right) }\right] \nonumber \\&\quad -\,\frac{2\mu _\mathrm{{m}}\sigma _\mathrm{{m}}^{\ell }\left( \sigma _\mathrm{{m}}^{\ell }+\mu _\mathrm{{m}}\right) \left( 1-r_\mathrm{{m}}^{\ell }\right) \psi _\mathrm{{m}}^{\ell }}{\pi _\mathrm{{m}}\mu _\mathrm{{f}}\left( \psi _\mathrm{{m}}^{\ell }+\mu _\mathrm{{m}}\right) \left( \alpha _\mathrm{{m}}^{\ell }+\mu _\mathrm{{m}}\right) }\left[ 1+\frac{\left( 1-k_\mathrm{{m}}^{\ell }\right) \alpha _\mathrm{{m}}^{\ell }}{\left( n_\mathrm{{m}}+\mu _\mathrm{{m}}\right) }\right] <0. \end{aligned}$$

(51)

It should be recalled that, in (51), $0\le \varepsilon _v, \varphi _\mathrm{{f}}^\mathrm{{q}}, \varphi _\mathrm{{m}}^\mathrm{{q}} \le 1$. Hence, it follows from (51) that the bifurcation coefficient, $a0$, is negative for the low-risk-only model (8) with $\rho _\mathrm{{f}}^{\ell }=\rho _\mathrm{{m}}^{\ell }=0$. Thus, it follows from Item (iv) of Theorem 4.1 of Castillo-Chavez and Song (2004) that the low-risk-only model (8) does not undergo backward bifurcation in the absence of re-infection of recovered individuals.

6. Proof of Theorem 3.5

Proof

Consider the risk-structured model (3) with $\rho _\mathrm{{f}}^{\ell }=\rho _\mathrm{{m}}^{\ell }=\rho _\mathrm{{f}}^\mathrm{{h}}=\rho _\mathrm{{m}}^\mathrm{{h}}=\delta _\mathrm{{m}}=0$. The proof is based on using a Comparison Theorem. The equations for the infected components of the model (3) can be written as (it should be mentioned that system (52) satisfies the Type K condition discussed in Smith (1995)):

$$\begin{aligned} \frac{\mathrm{{d}}}{\mathrm{{d}}t}\left( \begin{array}{c} E_\mathrm{{f}}^{\ell }(t)\\ I_\mathrm{{f}}^{\ell }(t)\\ P_\mathrm{{f}}^{\ell }(t)\\ W_\mathrm{{f}}^{\ell }(t)\\ E_\mathrm{{f}}^\mathrm{{h}}(t)\\ I_\mathrm{{f}}^\mathrm{{h}}(t)\\ P_\mathrm{{f}}^\mathrm{{h}}(t)\\ G_\mathrm{{f}\ell }(t)\\ G_\mathrm{{fh}}(t)\\ C_\mathrm{{f}}^\mathrm{{c}}(t)\\ E_\mathrm{{m}}^{\ell }(t)\\ I_\mathrm{{m}}^{\ell }(t)\\ P_\mathrm{{m}}^{\ell }(t)\\ W_\mathrm{{m}}(t)\\ E_\mathrm{{m}}^\mathrm{{h}}(t)\\ I_\mathrm{{m}}^\mathrm{{h}}(t)\\ P_\mathrm{{m}}^\mathrm{{h}}(t)\\ G_\mathrm{{m}\ell }(t)\\ G_\mathrm{{mh}}(t)\\ C_\mathrm{{m}}^\mathrm{{r}}(t) \end{array}\right) = \left( \mathcal {F}_\mathrm{{r}}-\mathcal {H}_\mathrm{{r}}\right) \left( \begin{array}{c} E_\mathrm{{f}}^{\ell }(t)\\ I_\mathrm{{f}}^{\ell }(t)\\ P_\mathrm{{f}}^{\ell }(t)\\ W_\mathrm{{f}}^{\ell }(t)\\ E_\mathrm{{f}}^\mathrm{{h}}(t)\\ I_\mathrm{{f}}^\mathrm{{h}}(t)\\ P_\mathrm{{f}}^\mathrm{{h}}(t)\\ G_\mathrm{{f}\ell }(t)\\ G_\mathrm{{fh}}(t)\\ C_\mathrm{{f}}^\mathrm{{c}}(t)\\ E_\mathrm{{m}}^{\ell }(t)\\ I_\mathrm{{m}}^{\ell }(t)\\ P_\mathrm{{m}}^{\ell }(t)\\ W_\mathrm{{m}}(t)\\ E_\mathrm{{m}}^\mathrm{{h}}(t)\\ I_\mathrm{{m}}^\mathrm{{h}}(t)\\ P_\mathrm{{m}}^\mathrm{{h}}(t)\\ G_\mathrm{{m}\ell }(t)\\ G_\mathrm{{mh}}(t)\\ C_\mathrm{{m}}^\mathrm{{r}}(t) \end{array}\right) -\mathcal {J}_\mathrm{{r}}\left( \begin{array}{c} E_\mathrm{{f}}^{\ell }(t)\\ I_\mathrm{{f}}^{\ell }(t)\\ P_\mathrm{{f}}^{\ell }(t)\\ W_\mathrm{{f}}^{\ell }(t)\\ E_\mathrm{{f}}^\mathrm{{h}}(t)\\ I_\mathrm{{f}}^\mathrm{{h}}(t)\\ P_\mathrm{{f}}^\mathrm{{h}}(t)\\ G_\mathrm{{f}\ell }(t)\\ G_\mathrm{{fh}}(t)\\ C_\mathrm{{f}}^\mathrm{{c}}(t)\\ E_\mathrm{{m}}^{\ell }(t)\\ I_\mathrm{{m}}^{\ell }(t)\\ P_\mathrm{{m}}^{\ell }(t)\\ W_\mathrm{{m}}(t)\\ E_\mathrm{{m}}^\mathrm{{h}}(t)\\ I_\mathrm{{m}}^\mathrm{{h}}(t)\\ P_\mathrm{{m}}^\mathrm{{h}}(t)\\ G_\mathrm{{m}\ell }(t)\\ G_\mathrm{{mh}}(t)\\ C_\mathrm{{m}}^\mathrm{{r}}(t) \end{array}\right) , \end{aligned}$$

(52)

where the matrices $\mathcal {F}_\mathrm{{r}}$ and $\mathcal {H}_\mathrm{{r}}$ are as defined in Sect. 3.1, and

$$\begin{aligned} \mathcal {J}_\mathrm{{r}}&= \left[ \frac{S_\mathrm{{f}}^{*}+V_\mathrm{{f}}^{b^{*}}+\left( 1-\varepsilon _{v}\right) V_\mathrm{{f}}^{q^*}}{N_\mathrm{{m}}^{*}}-\frac{S_\mathrm{{f}}+V_\mathrm{{f}}^\mathrm{{q}}+\left( 1-\varepsilon _{v}\right) V_\mathrm{{f}}^\mathrm{{q}}}{N_\mathrm{{m}}^{*}}\right] \mathcal {J}_{1r}\\&+\left[ \frac{S_\mathrm{{f}}^{*}+\left( 1 -\varepsilon _{v}\right) V_\mathrm{{f}}^{b^{*}} +\left( 1-\varepsilon _{v}\right) V_\mathrm{{f}}^{q^*}}{N_\mathrm{{m}}^{*}} -\frac{S_\mathrm{{f}}+\left( 1-\varepsilon _{v}\right) V_\mathrm{{f}}^\mathrm{{q}} +\left( 1-\varepsilon _{v}\right) V_\mathrm{{f}}^\mathrm{{q}}}{N_\mathrm{{m}}^{*}}\right] \mathcal {J}_{2r}\\&+\left[ \frac{S_\mathrm{{m}}^{*}+\left( 1-\varepsilon _{v}\right) V_\mathrm{{m}}^{q^*}}{N_\mathrm{{m}}^{*}}-\frac{S_\mathrm{{m}}+\left( 1-\varepsilon _{v}\right) V_\mathrm{{m}}^\mathrm{{q}}}{N_\mathrm{{m}}^{*}}\right] \left( \mathcal {J}_{3r}+\mathcal {J}_{4r}\right) , \end{aligned}$$

where

$$\begin{aligned} \mathcal {J}_{1r}&= \left( \begin{array}{c@{\quad }c} \mathbf{0}_{10 \times 10} &{} \mathcal {J}_{1}\\ \mathbf{0}_{10 \times 10}&{} \mathbf{0}_{10 \times 10} \end{array}\right) , \quad \mathcal {J}_{2r}=\left( \begin{array}{c@{\quad }c} \mathbf{0}_{10 \times 10} &{}\mathcal {J}_{2}\\ \mathbf{0}_{10 \times 10} &{}\mathbf{0}_{10 \times 10}\end{array}\right) ,\\ \mathcal {J}_{3r}&= \left( \begin{array}{c@{\quad }c} \mathbf{0}_{10 \times 10} &{} \mathbf{0}_{10 \times 10} \\ \mathcal {J}_{3}&{} \mathbf{0}_{10 \times 10} \end{array}\right) \quad \mathrm{and} \quad \mathcal {J}_{4r}=\left( \begin{array}{c@{\quad }c} \mathbf{0}_{10 \times 10} &{}\mathbf{0}_{10 \times 10} \\ \mathcal {J}_{4}&{}\mathbf{0}_{10 \times 10}\end{array}\right) , \end{aligned}$$

with

$$\begin{aligned} \mathcal {J}_{1}&= \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \beta _\mathrm{{m}}^{\ell }c_\mathrm{{f}}\eta _\mathrm{{m}}^{\ell }&{} \beta _\mathrm{{m}}^{\ell }c_\mathrm{{f}}&{} \beta _\mathrm{{m}}^{\ell }c_\mathrm{{f}}\theta _\mathrm{{m}}^{\ell }&{}0 &{}0&{}0&{}0&{}0&{}0&{}0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0&{}0&{}0&{}0&{}0&{} 0&{} 0&{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\end{array}\right) ,\\ \mathcal {J}_{2}&= \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 0&{}0 &{} 0&{}0 &{}0&{}0&{}0&{}0&{}0&{}0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0&{}0&{}0&{}0&{}\beta _\mathrm{{m}}^\mathrm{{h}}c_\mathrm{{f}}\eta _\mathrm{{m}}^\mathrm{{h}}&{} \beta _\mathrm{{m}}^\mathrm{{h}}c_\mathrm{{f}}&{} \beta _\mathrm{{m}}^\mathrm{{h}}c_\mathrm{{f}}\theta _\mathrm{{m}}^\mathrm{{h}}&{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\end{array}\right) , \end{aligned}$$

$$\begin{aligned} \mathcal {J}_{3}&= \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \beta _\mathrm{{f}}^{\ell }c_\mathrm{{f}}\eta _\mathrm{{f}}^{\ell }&{} \beta _\mathrm{{f}}^{\ell }c_\mathrm{{f}} &{} \beta _\mathrm{{f}}^{\ell }c_\mathrm{{f}}\theta _\mathrm{{f}}^{\ell }&{}0 &{}0&{}0&{}0&{}0&{}0&{}0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0&{}0&{}0&{}0&{}0&{} 0&{} 0&{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\end{array}\right) , \end{aligned}$$

and

$$\begin{aligned} \mathcal {J}_{4}=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 0&{} 0 &{} 0&{}0 &{}0&{}0&{}0&{}0&{}0&{}0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0&{}0&{}0&{}0&{}\beta _\mathrm{{f}}^\mathrm{{h}}c_\mathrm{{f}}\eta _\mathrm{{f}}^\mathrm{{h}}&{} \beta _\mathrm{{f}}^\mathrm{{h}}c_\mathrm{{f}}&{} \beta _\mathrm{{f}}^\mathrm{{h}}c_\mathrm{{f}}\theta _\mathrm{{f}}^\mathrm{{h}}&{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\\ 0 &{} 0 &{} 0&{} 0 &{} 0 &{} 0&{} 0&{} 0&{} 0&{} 0\end{array}\right) . \end{aligned}$$

It should be noted that $\mathcal {J}_{1r}, \mathcal {J}_{2r}, \mathcal {J}_{3r}$ and $\mathcal {J}_{4r}$ are non-negative matrices. Furthermore, since

$$\begin{aligned} S_\mathrm{{f}}(t)\le S_\mathrm{{f}}^*(t),\;\; V_\mathrm{{f}}^\mathrm{{b}}(t)\le V_\mathrm{{f}}^{b^{*}}(t), \;\; V_\mathrm{{f}}^\mathrm{{q}}(t)\le V_\mathrm{{f}}^{q^{*}}(t), \end{aligned}$$

and

$$\begin{aligned} S_\mathrm{{m}}(t)\le S_\mathrm{{m}}^*(t), \;\; V_\mathrm{{m}}^\mathrm{{q}}(t)\le V_\mathrm{{m}}^{q^{*}}(t), \end{aligned}$$

$\mathrm{(for\; all\; t\ge 0\; in\; \mathcal {D}_\mathrm{{r}}^{*})},$ the matrix $\mathcal {J}_\mathrm{{r}}$ is non-negative. Thus, it follows from (52) that

$$\begin{aligned} \frac{\mathrm{{d}}}{\mathrm{{d}}t}\left( \begin{array}{c} E_\mathrm{{f}}^{\ell }(t)\\ I_\mathrm{{f}}^{\ell }(t)\\ P_\mathrm{{f}}^{\ell }(t)\\ W_\mathrm{{f}}^{\ell }(t)\\ E_\mathrm{{f}}^\mathrm{{h}}(t)\\ I_\mathrm{{f}}^\mathrm{{h}}(t)\\ P_\mathrm{{f}}^\mathrm{{h}}(t)\\ G_\mathrm{{f}\ell }(t)\\ G_\mathrm{{fh}}(t)\\ C_\mathrm{{f}}^\mathrm{{c}}(t)\\ E_\mathrm{{m}}^{\ell }(t)\\ I_\mathrm{{m}}^{\ell }(t)\\ P_\mathrm{{m}}^{\ell }(t)\\ W_\mathrm{{m}}(t)\\ E_\mathrm{{m}}^\mathrm{{h}}(t)\\ I_\mathrm{{m}}^\mathrm{{h}}(t)\\ P_\mathrm{{m}}^\mathrm{{h}}(t)\\ G_\mathrm{{m}\ell }(t)\\ G_\mathrm{{mh}}(t)\\ C_\mathrm{{m}}^\mathrm{{r}}(t) \end{array}\right) \le \left( \mathcal {F}_\mathrm{{r}}-\mathcal {H}_\mathrm{{r}}\right) \left( \begin{array}{c} E_\mathrm{{f}}^{\ell }(t)\\ I_\mathrm{{f}}^{\ell }(t)\\ P_\mathrm{{f}}^{\ell }(t)\\ W_\mathrm{{f}}^{\ell }(t)\\ E_\mathrm{{f}}^\mathrm{{h}}(t)\\ I_\mathrm{{f}}^\mathrm{{h}}(t)\\ P_\mathrm{{f}}^\mathrm{{h}}(t)\\ G_\mathrm{{f}\ell }(t)\\ G_\mathrm{{fh}}(t)\\ C_\mathrm{{f}}^\mathrm{{c}}(t)\\ E_\mathrm{{m}}^{\ell }(t)\\ I_\mathrm{{m}}^{\ell }(t)\\ P_\mathrm{{m}}^{\ell }(t)\\ W_\mathrm{{m}}(t)\\ E_\mathrm{{m}}^\mathrm{{h}}(t)\\ I_\mathrm{{m}}^\mathrm{{h}}(t)\\ P_\mathrm{{m}}^\mathrm{{h}}(t)\\ G_\mathrm{{m}\ell }(t)\\ G_\mathrm{{mh}}(t)\\ C_\mathrm{{m}}^\mathrm{{r}}(t) \end{array}\right) . \end{aligned}$$

(53)

Using the fact that the eigenvalues of the matrix $\mathcal {F}_\mathrm{{r}}-\mathcal {H}_\mathrm{{r}}$ all have negative real parts (see local stability result in Sect. 3.1, where $\rho (\mathcal {F}_\mathrm{{r}}\mathcal {H}_\mathrm{{r}}^{-1})<1$ if $\mathcal {R}_{01}^\mathrm{{r}}<1$, which is equivalent to $\mathcal {F}_\mathrm{{r}}-\mathcal {H}_\mathrm{{r}}$ having eigenvalues with negative real parts when $\mathcal {R}_{01}^\mathrm{{r}}<1$van den Driessche and Watmough 2002), it follows that the linearized differential inequality system (53) is stable whenever $\mathcal {R}_{01}^\mathrm{{r}}<1$. Thus, it follows, by Comparison Theorem (Lakshmikantham et al. 1989), that

$$\begin{aligned}&\lim _{t\rightarrow \infty }\;\;(E_\mathrm{{f}}^{\ell }(t),I_\mathrm{{f}}^{\ell }(t), P_\mathrm{{f}}^{\ell }(t),W_\mathrm{{f}}^{\ell }(t),E_\mathrm{{f}}^\mathrm{{h}}(t),I_\mathrm{{f}}^\mathrm{{h}}(t),P_\mathrm{{f}}^\mathrm{{h}}(t),G_\mathrm{{f}\ell }(t), G_\mathrm{{fh}}(t),C_\mathrm{{f}}^\mathrm{{c}}(t),E_\mathrm{{m}}^{\ell }(t),\\&\quad I_\mathrm{{m}}^{\ell }(t), P_\mathrm{{m}}^{\ell }(t),W_\mathrm{{m}}(t), E_\mathrm{{m}}^\mathrm{{h}}(t), I_\mathrm{{m}}^\mathrm{{h}}(t),P_\mathrm{{m}}^\mathrm{{h}}(t),G_\mathrm{{m}\ell }(t),G_\mathrm{{mh}}(t),C_\mathrm{{m}}^\mathrm{{r}}(t))\\&\quad =(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0). \end{aligned}$$

Substituting $E_\mathrm{{f}}^{\ell }=I_\mathrm{{f}}^{\ell }=P_\mathrm{{f}}^{\ell }=W_\mathrm{{f}}^{\ell }=E_\mathrm{{f}}^\mathrm{{h}}=I_\mathrm{{f}}^\mathrm{{h}}=P_\mathrm{{f}}^\mathrm{{h}}=G_\mathrm{{f}\ell }=G_\mathrm{{fh}}=C_\mathrm{{f}}^\mathrm{{c}}=E_\mathrm{{m}}^{\ell }$=$I_\mathrm{{m}}^{\ell }=P_\mathrm{{m}}^{\ell }=W_\mathrm{{m}}=E_\mathrm{{m}}^\mathrm{{h}}= I_\mathrm{{m}}^\mathrm{{h}}=P_\mathrm{{m}}^\mathrm{{h}}=G_\mathrm{{m}\ell }=G_\mathrm{{mh}}=C_\mathrm{{m}}^\mathrm{{r}}=0$ into the equations of the model (3) gives $S_\mathrm{{f}}(t)\rightarrow S_\mathrm{{f}}^{*}, V_\mathrm{{f}}^\mathrm{{b}}(t)\rightarrow V_\mathrm{{f}}^{b^{*}}, V_\mathrm{{f}}^\mathrm{{q}}(t)\rightarrow V_\mathrm{{f}}^{q^{*}},$ $S_\mathrm{{m}}(t)\rightarrow S_\mathrm{{m}}^{*}$ and $V_\mathrm{{m}}^\mathrm{{q}}(t)\rightarrow V_\mathrm{{m}}^{q^{*}}$, as $t\rightarrow \infty$ for $\mathcal {R}_{01}^\mathrm{{r}}<1$. Thus,

$$\begin{aligned}&\lim _{t\rightarrow \infty }\;\;(S_\mathrm{{f}}(t), V_\mathrm{{f}}^\mathrm{{b}}(t), V_\mathrm{{f}}^\mathrm{{q}}(t),E_\mathrm{{f}}^{\ell }(t),I_\mathrm{{f}}^{\ell }(t),P_\mathrm{{f}}^{\ell }(t),W_\mathrm{{f}}^{\ell }(t),E_\mathrm{{f}}^\mathrm{{h}}(t),I_\mathrm{{f}}^\mathrm{{h}}(t),P_\mathrm{{f}}^\mathrm{{h}}(t),G_\mathrm{{f}\ell }(t),\\&G_\mathrm{{fh}}(t),C_\mathrm{{f}}^\mathrm{{c}}(t),S_\mathrm{{m}}(t), V_\mathrm{{m}}^\mathrm{{q}}(t),E_\mathrm{{m}}^{\ell }(t),I_\mathrm{{m}}^{\ell }(t), P_\mathrm{{m}}^{\ell }(t),W_\mathrm{{m}}(t),E_\mathrm{{m}}^\mathrm{{h}}(t),\\&I_\mathrm{{m}}^\mathrm{{h}}(t),P_\mathrm{{m}}^\mathrm{{h}}(t),G_\mathrm{{m}\ell }(t),G_\mathrm{{mh}}(t),C_\mathrm{{m}}^\mathrm{{r}}(t))=\mathcal {E}_{0}^\mathrm{{r}}, \end{aligned}$$

so that the DFE, $\mathcal {E}_{0}^\mathrm{{r}}$, of the model (3) is GAS in $\mathcal {D}_\mathrm{{r}}^{*}$ whenever $\mathcal {R}_{01}^\mathrm{{r}}<1$.$\square$