A Dynamic Model to Assess Human Papillomavirus Vaccination Strategies in a Heterosexual Population Combined with Men Who have Sex with Men

Abstract

Vaccination is effective in preventing human papillomavirus (HPV) infection. It is imperative to investigate who should be vaccinated and what the best vaccine distribution strategy is. In this paper, we use a dynamic model to assess HPV vaccination strategies in a heterosexual population combined with gay, bisexual, and other men who have sex with men (MSM). The basic reproduction numbers for heterosexual females, heterosexual males and MSM as well as their average for the total population are obtained. We also derive a threshold parameter, based on basic reproduction numbers, for model analysis. From the analysis and numerical investigations, we have several conclusions. (1) To eliminate HPV infection, the priority of vaccination should be given to MSM, especially in countries that have already achieved high coverage in females. The heterosexual population gets great benefit but MSM only get minor benefit from vaccinating heterosexual females or males. (2) The best vaccination strategy is to vaccinate MSM firstly as many as possible, then heterosexual females, lastly heterosexual males. (3) Given a fixed vaccination coverage of MSM, distributing the remaining vaccines to only heterosexual females or males leads to a similar prevalence in the total population. This prevalence is lower than that when vaccines are distributed to both genders. The evener the distribution, the higher the prevalence in the total population. (4) Vaccination becomes less effective in reducing the prevalence as more vaccines are given. It is more effective to allocate vaccines to a region with lower vaccination coverage. This study provides information that may help policymakers formulate guidelines for vaccine distribution to reduce HPV prevalence on the basis of vaccine availability and prior vaccination coverage. Whether these guidelines are affected when the objective is to reduce HPV-associated cancer incidence remains to be further studied.

Appendices

Appendix A: Proof of the Well-Posedness of System (1)

For reference, we introduce a theorem by Thieme (2018). Let $\mathfrak {R}_{+}^{n}=[0, +\infty )^n$ be the cone of nonnegative vector in $\mathfrak {R}^{n}$. Let $F: \mathfrak {R}_{+}^{n+1}\rightarrow \mathfrak {R}^{n}$

$$\begin{aligned} F(t,x)=(F_1(t,x), \cdot \cdot \cdot , F_n(t,x)),\quad x=(x_1, \cdot \cdot \cdot , x_n),\end{aligned}$$

be locally Lipschitz and $F_j(t,x)\ge 0$ when $t\ge 0$, $x\in \mathfrak {R}_{+}^{n}$, $x_j=0$. For every $x^0\in \mathfrak {R}_{+}^{n}$, there exists a unique solution of $x^{\prime }=F(t,x)$ and $x(0)=x^0$, which is defined on some interval [0, b) where $b>0$. If $b<\infty$, then

$$\begin{aligned} \limsup _{t\nearrow b}\sum _{j=1}^{n}x_j(t)=\infty .\end{aligned}$$

We use this theorem to prove that system (1) is well-posed. First, we show that the region

$$\begin{aligned} D&=\{(S_f,V_f,I_f,S_m,V_m,I_m,S_M,V_M,I_M)\in \mathfrak {R}_{+}^{9}:S_k+V_k+I_k\\&\qquad \le \Lambda _k/\mu _k, k=f, m, M\} \end{aligned}$$

is positively invariant and attracts all solutions of system (1). From $N_k^{\prime }=\Lambda _k-\mu _k N_k, \ k=f,m,M,$ we have

$$\begin{aligned} N_k(t)= \frac{\Lambda _k}{\mu _k}+\left[ N_k(0)-\frac{\Lambda _k}{\mu _k}\right] e^{-\mu _k t}.\end{aligned}$$

Therefore, we have $N_k(t)\le \Lambda _k/\mu _k$ for any $t\ge 0$ if $N_k(0)\le \Lambda _k/\mu _k$. This shows that D is positively invariant. Furthermore, if $N_k(0)> \Lambda _k/\mu _k$, then $N_k(t)$ approaches $\Lambda _k/\mu _k$ asymptotically. This shows that the region D attracts all solutions in $\mathfrak {R}_{+}^{9}$. Therefore, D is epidemiologically well-posed.

Next, we use Thieme’s theorem to prove that for any initial value in D, there exists a unique solution of system (1) with values in D for $t\in [0, +\infty )$. It follows from $\lim \nolimits _{t\rightarrow \infty }N_k(t)=N_k^*=\Lambda _k/\mu _k$ that there exists $T>0$ such that $N_k(t)>N_k^*/2$ for all $t>T$. By shifting, we can assume for $t\ge 0$, $N_k^*/2<N_k(t)\le N_k^*.$ Let $x=(S_f,V_f,I_f,S_m,V_m,I_m,S_M,V_M,I_M)\in \mathfrak {R}_{+}^{9}$ and $F(x)=(F_1(x), F_2(x), F_3(x), F_4(x), F_5(x), F_6(x), F_7(x), F_8(x), F_9(x)),$ where

$$\begin{aligned} F_1(x)&=(1-\phi _{f})\Lambda _{f}-\lambda _{f}S_{f}+\delta _{f}I_{f}-\mu _{f}S_{f},\\ F_2(x)&=\phi _{f}\Lambda _{f}-(1-\tau )\lambda _{f}V_{f}-\mu _{f}V_{f},\\ F_3(x)&=\lambda _{f}S_{f}+(1-\tau )\lambda _{f}V_{f}-(\delta _{f}+\mu _{f})I_{f}, \\ F_4(x)&=(1-\phi _{m})\Lambda _{m}-\lambda _{m}S_{m}+\delta _{m}I_{m}-\mu _{m}S_{m}, \\ F_5(x)&=\phi _{m}\Lambda _{m}-(1-\tau )\lambda _{m}V_{m}-\mu _{m}V_{m}, \\ F_6(x)&=\lambda _{m}S_{m}+(1-\tau )\lambda _{m}V_{m}-(\delta _{m}+\mu _{m})I_{m}, \\ F_7(x)&=(1-\phi _{M})\Lambda _{M}-\lambda _{M}S_{M}+\delta _{M}I_{M}-\mu _{M}S_{M}, \\ F_8(x)&=\phi _{M}\Lambda _{M}-(1-\tau )\lambda _{M}V_{M}-\mu _{M}V_{M}, \\ F_9(x)&=\lambda _{M}S_{M}+(1-\tau )\lambda _{M}V_{M}-(\delta _{M}+\mu _{M})I_{M}. \end{aligned}$$

For any $x,\overline{x}\in D$,

$$\begin{aligned} |F_1(x)-F_1(\overline{x})|&=|-\lambda _{f}(S_{f}-\overline{S}_{f})-(\lambda _f-\overline{\lambda }_f)\overline{S}_f\\&\quad +\delta _{f}(I_{f}-\overline{I}_{f})-\mu _{f}(S_{f}-\overline{S}_{f})|\\&\le K_1|S_f-\overline{S}_f|+\frac{c_{mf}\beta _{mf}\overline{S}_f}{N_f\overline{N}_f}|I_m \overline{N}_f -\overline{I}_m N_f|\\&\quad +\frac{c_{Mf}\beta _{Mf}\overline{S}_f}{N_f\overline{N}_f}|I_M \overline{N}_f-\overline{I}_M N_f|+\mu _f|S_{f}-\overline{S}_{f}|+\delta _{f}|I_{f}-\overline{I}_{f}|\\&\le K_1|S_f-\overline{S}_f|+\mu _f|S_{f}-\overline{S}_{f}|+\delta _{f}|I_{f}-\overline{I}_{f}|\\&\quad +\frac{c_{mf}\beta _{mf}\overline{S}_f}{N_f\overline{N}_f}(|I_m -\overline{I}_m|\overline{N}_f +\overline{I}_m|N_f-\overline{N}_f|)\\&\quad +\frac{c_{Mf}\beta _{Mf}\overline{S}_f}{N_f\overline{N}_f}(|I_M -\overline{I}_M|\overline{N}_f +\overline{I}_M|N_f-\overline{N}_f|)\\&\le K_1|S_f-\overline{S}_f|+\mu _f|S_{f}-\overline{S}_{f}|+\delta _{f}|I_{f}-\overline{I}_{f}|+K_1|N_f-\overline{N}_f|\\&\quad +c_{mf}\beta _{mf}|I_m-\overline{I}_m|+c_{Mf}\beta _{Mf}|I_M-\overline{I}_M|\\&\le (2K_1+\mu _f+\delta _f)(|S_{f}-\overline{S}_{f}|+|I_{f}-\overline{I}_{f}|+|V_f-\overline{V}_f|)\\&\quad +c_{mf}\beta _{mf}|I_m-\overline{I}_m|+c_{Mf}\beta _{Mf}|I_M-\overline{I}_M|\\&\le K_2(|S_{f}-\overline{S}_{f}|+|I_{f}-\overline{I}_{f}|+|V_f-\overline{V}_f|\\&\quad +|I_m-\overline{I}_m|+|I_M-\overline{I}_M|), \end{aligned}$$

where

$$\begin{aligned} K_1&=\frac{2(c_{mf}\beta _{mf}N_m^*+c_{Mf}\beta _{Mf}N_M^*)}{N_f^*},\\ K_2&=\mathrm{max}\{2K_1+\mu _f+\delta _f, c_{mf}\beta _{mf}, c_{Mf}\beta _{Mf}\}. \end{aligned}$$

By similar arguments, we can show that the other components of F(x) are locally Lipschitz in x. It’s easy to check that $||F(x)-F(\overline{x})||\le M||x-\overline{x}||$ for some $M>0$. Thus, F(x) is locally Lipschitz.

If $S_f=0$, then $F_1(x)=(1-\phi _{f})\Lambda _{f}+\delta _{f}I_{f}\ge 0$ whenever $t\ge 0$, $x\in \mathfrak {R}_+^9$. Similarly, we have $F_k(x)\ge 0$ whenever $t\ge 0$, $x\in \mathfrak {R}_+^9$, $k=2,\cdot \cdot \cdot ,9$. Therefore, by Thieme’s Theorem, for every $x^0\in D$, there exists a unique solution of $x^{\prime }=F(x)$ and $x(0)=x^0$, with values in D. The solution is defined on some interval [0, b), $b>0$. Because $\displaystyle \limsup \nolimits _{t\nearrow b}\sum \nolimits _{j=1}^{n}x_j(t)=N_f^*+N_m^*+N_M^*<\infty ,$ again by Thieme’s Theorem, we have $b=\infty$. Thus, our model is both epidemiologically and mathematically well posed.

Appendix B: Proof of Proposition 1

Assume the contrary. Then for every $\varepsilon >0$, $\displaystyle \limsup \nolimits _{t\rightarrow \infty } I_M(t)<\frac{\varepsilon }{2}.$ Thus, there exists $T_1(\varepsilon )>0$ such that $I_M(t)<\varepsilon$ for all $t>T_1(\varepsilon )$. Because $\displaystyle \lim \nolimits _{t\rightarrow \infty }N_M(t)=\frac{\Lambda _M}{\mu _M},$ there exists $T_2>0$ such that $\displaystyle N_M(t)>\frac{\Lambda _M}{2\mu _M}$ for all $t>T_2$. In particular, for

$$\begin{aligned} \varepsilon _0=\frac{[c_{MM}\beta _{MM}-(\delta _M+\mu _M)]\Lambda _M}{4\mu _Mc_{MM}\beta _{MM}}>0\end{aligned}$$

and $t>T=\mathrm{max}\{T_1(\varepsilon _0),T_2\}$, we have

$$\begin{aligned} I^{\prime }_{M}(t)&=\frac{c_{MM}\beta _{MM}I_M+c_{fM}\beta _{fM}I_f}{N_M}S_{M}-(\delta _{M}+\mu _{M})I_{M}\\&\ge \frac{c_{MM}\beta _{MM}I_M}{N_M}S_{M}-(\delta _{M}+\mu _{M})I_{M}\\&=\bigg [\frac{c_{MM}\beta _{MM}}{N_M}(N_{M}-I_{M})-(\delta _{M}+\mu _{M})\bigg ]I_{M}\\&>\bigg [c_{MM}\beta _{MM}-(\delta _{M}+\mu _{M})-c_{MM}\beta _{MM}\frac{2\mu _M\varepsilon _0}{\Lambda _M}\bigg ]I_{M}\\&=\frac{1}{2}(\delta _{M}+\mu _{M})(R_{0,MM}-1)I_{M}. \end{aligned}$$

Thus, $\displaystyle I_M(t)\ge I_M(0)e^{\frac{1}{2}(\delta _{M}+\mu _{M})(R_{0,MM}-1)t}.$ Because $R_{0,MM}>1$, we conclude that for $I_M(0)>0$ $\lim \nolimits _{t\rightarrow \infty }I_M(t)=+\infty .$ This contradicts the assumption. Thus, if $R_{0,MM}>1$, $I_M$ is uniformly weakly endemic.

Appendix C: Proof of Proposition 3

If $R_{0,MM}<1$, then $\displaystyle R_{0,f}=R_{0,fm}R_{0,mf}+R_{0,fM}\sum \nolimits _{n=0}^{\infty }R_{0,MM}^{n}R_{0,Mf}=R_{0,fm}R_{0,mf}+\frac{R_{0,fM}R_{0,Mf}}{1-R_{0,MM}}.$ From $R_{0,f}<1$, we have $\displaystyle R_{0,fm}R_{0,mf}<1$ and $\displaystyle \frac{R_{0,fM}R_{0,Mf}}{1-R_{0,MM}}<1.$ Thus,

$$\begin{aligned} R_{0,m}&=R_{0,mf}\bigg (1+R_{0,fM}\sum _{n=0}^{\infty }R_{0,MM}^{n}R_{0,Mf}\bigg )R_{0,fm}\\&=R_{0,fm}R_{0,mf}+R_{0,fm}R_{0,mf}\frac{R_{0,fM}R_{0,Mf}}{1-R_{0,MM}}\\&\le R_{0,fm}R_{0,mf}+\frac{R_{0,fM}R_{0,Mf}}{1-R_{0,MM}}\\&=R_{0,f}<1. \end{aligned}$$

We also have

$$\begin{aligned} R_{0,M}-1&=R_{0,MM}+R_{0,Mf}(1+R_{0,fm}R_{0,mf})R_{0,fM}-1\\&=(R_{0,MM}-1)\bigg (1-\frac{R_{0,fM}R_{0,Mf}}{1-R_{0,MM}}-\frac{R_{0,fM}R_{0,Mf}}{1-R_{0,MM}}R_{0,fm}R_{0,mf}\bigg ). \end{aligned}$$

From $R_{0,f}<1$ and $\displaystyle \frac{R_{0,fM}R_{0,Mf}}{1-R_{0,MM}}<1,$ we know

$$\begin{aligned}&\frac{R_{0,fM}R_{0,Mf}}{1-R_{0,MM}}+\frac{R_{0,fM}R_{0,Mf}}{1-R_{0,MM}}R_{0,fm}R_{0,mf}<\frac{R_{0,fM}R_{0,Mf}}{1-R_{0,MM}}\\&\quad +R_{0,fm}R_{0,mf} =R_{0,f}<1. \end{aligned}$$

It follows from $R_{0,MM}<1$ that $R_{0,M}-1<0$, i.e., $R_{0,M}<1$.

Appendix D: Proof of Theorem 2

Define the following Lyapunov function

$$\begin{aligned} L=&\bigg (S_f-S_f^0-S_f^0\ln \frac{S_f}{S_f^0}+I_f\bigg )+R_{0,mf}\bigg (S_m-S_m^0-S_m^0\ln \frac{S_m}{S_m^0}+I_m\bigg )\\&+\frac{R_{0,Mf}}{1-R_{0,MM}}\bigg (S_M-S_M^0-S_M^0\ln \frac{S_M}{S_M^0}+I_M\bigg ). \end{aligned}$$

It’s clear that when $R_0<1$, L is radially unbounded and positive definite in the entire space D. The derivative of L along the trajectories of system (2) yields

$$\begin{aligned} \dot{L}=&\bigg [(1-\frac{S_f^0}{S_f})S_f^{\prime }+I_f^{\prime }\bigg ]+R_{0,mf}\bigg [(1-\frac{S_m^0}{S_m})S_m^{\prime }+I_m^{\prime }\bigg ]+\frac{R_{0,Mf}}{1-R_{0,MM}}\bigg [(1-\frac{S_M^0}{S_M})S_M^{\prime }+I_M^{\prime }\bigg ]\\ =&\bigg (1-\frac{S_f^0}{S_f}\bigg )(\Lambda _{f}-\lambda _{f}S_{f}+\delta _{f}I_{f}-\mu _{f}S_{f})+[\lambda _{f}S_{f}-(\delta _{f}+\mu _{f})I_{f}]\\&+R_{0,mf}\bigg \{\bigg (1-\frac{S_m^0}{S_m}\bigg )(\Lambda _{m}-\lambda _{m}S_{m}+\delta _{m}I_{m}-\mu _{m}S_{m}) +[\lambda _{m}S_{m}-(\delta _{m}+\mu _{m})I_{m}]\bigg \}\\&+\frac{R_{0,Mf}}{1-R_{0,MM}}\bigg \{\bigg (1-\frac{S_M^0}{S_M}\bigg )(\Lambda _{M}-\lambda _{M}S_{M}+\delta _{M}I_{M}-\mu _{M}S_{M}) +[\lambda _{M}S_{M}-(\delta _{M}+\mu _{M})I_{M}]\bigg \}. \end{aligned}$$

Using the equilibrium conditions $\Lambda _k=\mu _kS_k^0,\ N_k=S_k^0,\ \frac{S_k^0}{S_k}\ge 1,\ k=f,m,M,$ and collecting terms, we obtain

$$\begin{aligned} \dot{L}=&\bigg [\bigg (1-\frac{S_f^0}{S_f}\bigg )\mu _f(S_f^0-S_f)+S_f^0\lambda _f-\bigg (\frac{S_f^0}{S_f}\delta _{f}+\mu _{f}\bigg )I_f\bigg ]\\&+R_{0,mf}\bigg [\bigg (1-\frac{S_m^0}{S_m}\bigg )\mu _m(S_m^0-S_m)+S_m^0\lambda _m-\bigg (\frac{S_m^0}{S_m}\delta _{m}+\mu _{m}\bigg )I_m\bigg ]\\&+\frac{R_{0,Mf}}{1-R_{0,MM}}\bigg [\bigg (1-\frac{S_M^0}{S_M}\bigg )\mu _M(S_M^0-S_M)+S_M^0\lambda _M-\bigg (\frac{S_M^0}{S_M}\delta _{M}+\mu _{M}\bigg )I_M\bigg ]\\ \le&-\frac{\mu _f}{S_f}(S_f^0-S_f)^2-R_{0,mf}\frac{\mu _m}{S_m}(S_m^0-S_m)^2-\frac{R_{0,Mf}}{1-R_{0,MM}}\frac{\mu _M}{S_M}(S_M^0-S_M)^2\\&+[c_{mf}\beta _{mf}I_m+c_{Mf}\beta _{Mf}I_M-(\delta _{f}+\mu _{f})I_f]\\&+R_{0,mf}[c_{fm}\beta _{fm}I_f-(\delta _{m}+\mu _{m})I_m]\\&+\frac{R_{0,Mf}}{1-R_{0,MM}}[c_{MM}\beta _{MM}I_M+c_{fM}\beta _{fM}I_f-(\delta _{M}+\mu _{M})I_M]\\ \le&-\frac{\mu _f}{S_f}(S_f^0-S_f)^2-R_{0,mf}\frac{\mu _m}{S_m}(S_m^0-S_m)^2-\frac{R_{0,Mf}}{1-R_{0,MM}}\frac{\mu _M}{S_M}(S_M^0-S_M)^2\\&+(\delta _{f}+\mu _{f})\bigg (R_{0,fm}R_{0,mf}+\frac{R_{0,fM}R_{0,Mf}}{1-R_{0,MM}}-1\bigg )I_{f}\le 0. \end{aligned}$$

$\dot{L}$ is 0 only at DFE. Therefore, the DFE $E_0$ is globally asymptotically stable when $R_0\le 1$.

Appendix E: Proof of Proposition 7

For reference, we introduce a theorem by Castillo-Chavez and Song (2004) [also in Martcheva (2015)]. Consider the following general system of ODEs with a parameter $\phi$:

$$\begin{aligned} \frac{dx}{dt}=f(x, \phi ), \quad f: \mathbb {R}^{n}\times \mathbb {R} \rightarrow \mathbb {R}^{n}, \quad f\in \mathbb {C}^{2}(\mathbb {R}^{n}\times \mathbb {R}), \end{aligned}$$

(3)

where $x=0$ is an equilibrium point of the system, i.e., $f(0, \phi )\equiv 0$ for all $\phi$. We assume that

A1 $\mathscr {A}=D_{x}f(0,0)=(\frac{\partial f_{i}}{\partial x_{j}}(0,0))$ is the linearized matrix of the system around the equilibrium 0 with $\phi$ evaluated at 0. Zero is a simple eigenvalue of $\mathscr {A}$, and other eigenvalues have negative real parts.

A2 The matrix $\mathscr {A}$ has a nonnegative right eigenvector w and a left eigenvector v each corresponding to the zero eigenvalue.

Let $f_{k}$ be the kth component of f, $\displaystyle a=\sum \nolimits _{k,i,j=1}^{n}v_{k}w_{i}w_{j}\frac{\partial ^2f_{k}}{\partial x_{i}\partial x_{j}}(0,0)$ and $\displaystyle b=\sum \nolimits _{k,i=1}^{n}v_{k}w_{i}\frac{\partial ^2f_{k}}{\partial x_{i}\partial \phi }(0,0).$ The local dynamics of the system around 0 are completely determined by the signs of a and b:

(i) $a>0$, $b>0$. When $\phi <0$ with $|\phi |\ll 1$, 0 is locally asymptotically stable and there exists a positive unstable positive equilibrium; when $0<\phi \ll 1$, 0 is locally asymptotically stable and there exists a positive unstable equilibrium;

(ii) $a<0$, $<0$. When $\phi <0$ with $|\phi |\ll 1$, 0 is unstable; when $0<\phi \ll 1$, 0 is unstable and there exists a negative and locally asymptotically stable equilibrium;

(iii) $a>0$, $b<0$. When $\phi <0$ with $|\phi |\ll 1$, 0 is unstable and there exists a locally asymptotically stable negative equilibrium; when $0<\phi \ll 1$, 0 is unstable and a positive unstable equilibrium appears;

(iv) $a<0$, $b>0$. When $\phi <0$ changes from negative to positive, 0 changes its stability from stable to unstable. Correspondingly, a negative unstable equilibrium becomes positive and locally asymptotically stable.

We notice the following when using the above theorem.

(1) The equilibrium 0 is the DFE in our model. The parameter $\phi$ is one of the parameters in the reproduction number and the critical value of $\phi$ is the value of the parameter that makes the reproduction number equal to one.

(2) Since the DFE has positive entries, the right eigenvector w doesn’t need to be nonnegative. The components of the right eigenvector w that correspond to positive entries in the DFE could be negative. However, the components that correspond to zero entries in the DFE should be nonnegative.

Now we use this theorem to prove Proposition 7. Choose $\beta _{fm}$ as the bifurcation parameter and let $\beta _{fm}^*$ be the critical value such that $R_0(\phi _f, \phi _m, \phi _M)=1$. Thus, $R_{0,MM}(\phi _M)<1$ and $\beta _{fm}^*$ satisfies $\displaystyle R_{0,fm}(\phi _m)R_{0,mf}(\phi _f)+R_{0,fM}(\phi _M)\frac{1}{1-R_{0,MM}(\phi _M)}R_{0,Mf}(\phi _f)=1.$

Reordering variables as $x=(I_f,I_m,I_M,S_f,V_f,S_m,V_m,S_M,V_m)^T$, the Jacobian matrix of system (1) evaluated at the $\overline{E}_0$ and $\beta _{fm}^*$ is $\mathscr {A}=\begin{pmatrix} J_{11} &{} 0 \\ J_{21} &{} J_{22} \end{pmatrix},$ where

$$\begin{aligned} J_{11}&=\left( \begin{matrix} -(\delta _f+\mu _f) &{} c_{mf}\beta _{mf}[(1-\phi _f)+(1-\tau )\phi _f] \\ c_{fm}\beta _{fm}^*[(1-\phi _m)+(1-\tau )\phi _m] &{} -(\delta _m+\mu _m) \\ c_{fM}\beta _{fM}[(1-\phi _M)+(1-\tau )\phi _M] &{} 0\\ \end{matrix}\right. \\&\qquad \qquad \quad \left. \begin{matrix} c_{Mf}\beta _{Mf}[(1-\phi _f)+(1-\tau )\phi _f]\\ 0\\ -(\delta _M+\mu _M)+c_{MM}\beta _{MM}[(1-\phi _M)+(1-\tau )\phi _M] \end{matrix}\right) ,\\ J_{21}&=\begin{pmatrix} \delta _f &{} -c_{mf}\beta _{mf}(1-\phi _f) &{} -c_{Mf}\beta _{Mf}(1-\phi _f)\\ 0 &{} -c_{mf}\beta _{mf}(1-\tau )\phi _f &{} -c_{Mf}\beta _{Mf}(1-\tau )\phi _f\\ -c_{fm}\beta _{fm}^*(1-\phi _m) &{} \delta _m &{} 0\\ -c_{fm}\beta _{fm}^*(1-\tau )\phi _m &{} 0 &{} 0\\ -c_{fM}\beta _{fM}(1-\phi _M) &{} 0 &{} \delta _M-c_{MM}\beta _{MM}(1-\phi _M)\\ -c_{fM}\beta _{fM}(1-\tau )\phi _M &{} 0 &{} -c_{MM}\beta _{MM}(1-\tau )\phi _M \end{pmatrix}, \end{aligned}$$

and

$$\begin{aligned} J_{22}&=\begin{pmatrix} -\mu _f &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} -\mu _f &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} -\mu _m &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} -\mu _m &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} -\mu _M &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -\mu _M \end{pmatrix}. \end{aligned}$$

It’s easy to check that zero is a simple eigenvalue of $\mathscr {A}$ and the other eigenvalues have negative real parts. Thus, condition A1 is satisfied. Moreover, it can be shown that $\mathscr {A}$ has a right eigenvector (corresponding to the zero eigenvalue), given by $w=(w_1, w_2, w_3, w_4, w_5, w_6, w_7, w_8, w_9)^T$, where

$$\begin{aligned} w_1&=1,\ \ \ w_2=\frac{\delta _f+\mu _f}{\delta _m+\mu _m}R_{0,fm}(\phi _m)|_{\beta _{fm}=\beta _{fm}^*}=:p>0, \\ w_3&=\frac{\delta _f+\mu _f}{\delta _M+\mu _M}\frac{R_{0,fM}(\phi _M)}{1-R_{0,MM}(\phi _M)}=:q>0,\\ w_4&=-\frac{1}{\mu _f}[c_{mf}\beta _{mf}(1-\phi _f)p+c_{Mf}\beta _{Mf}(1-\phi _f)q-\delta _f],\\ w_5&=-\frac{1}{\mu _f}[c_{mf}\beta _{mf}(1-\tau )\phi _fp+c_{Mf}\beta _{Mf}(1-\tau )\phi _fq], \ \\ w_6&=-\frac{1}{\mu _m}[c_{fm}\beta _{fm}^*(1-\phi _m)-\delta _{m}p],\\ w_7&=-\frac{1}{\mu _m}c_{fm}\beta _{fm}^*(1-\tau )\phi _m, \ \\ w_8&=-\frac{1}{\mu _M}\{c_{fM}\beta _{fM}(1-\phi _M)p+[c_{MM}\beta _{MM}(1-\phi _M)-\delta _M]q\},\\ w_9&=-\frac{1}{\mu _f}[c_{fM}\beta _{fM}(1-\tau )\phi _M+c_{MM}\beta _{MM}(1-\tau )\phi _{M}q]. \end{aligned}$$

Besides, $\mathscr {A}$ also has a left eigenvector (corresponding to the zero eigenvalue), given by $v=(v_1, v_2, v_3, v_4, v_5, v_6, v_7, v_8, v_9)^T$, where

$$\begin{aligned} v_1= & {} 1,\quad v_2=R_{0,mf}(\phi _f),\quad v_3=\frac{R_{0,Mf}(\phi _f)}{1-R_{0,MM}(\phi _M)},\\ v_4= & {} v_5=v_6=v_7=v_8=v_9=0. \end{aligned}$$

We denote the right-hand side functions of system (1) as $f_i$, $i=1,\cdots ,9$. Because the last six components of v are zeros, we only need the derivatives of $f_1$, $f_2$ and $f_3$. At the DFE and $\beta _{fm}=\beta _{fm}^*$, the associated nonzero secondary partial derivatives are

$$\begin{aligned} \frac{\partial ^2f_1}{\partial I_f\partial I_m}&=-\frac{c_{mf}\beta _{mf}\mu _{f}}{\Lambda _{f}},\quad \frac{\partial ^2f_1}{\partial I_f\partial I_M}=-\frac{c_{Mf}\beta _{Mf}\mu _{f}}{\Lambda _{f}},\\ \frac{\partial ^2f_2}{\partial I_f\partial I_m}&=-\frac{c_{fm}\beta _{fm}^*\mu _{m}}{\Lambda _{m}},\quad \frac{\partial ^2f_3}{\partial I_f\partial I_M}=-\frac{(c_{MM}\beta _{MM}+c_{fM}\beta _{fM})\mu _{M}}{\Lambda _{M}},\\ \frac{\partial ^2f_3}{\partial I_m\partial I_M}&=-\frac{c_{MM}\beta _{MM}\mu _{M}}{\Lambda _{M}},\quad \frac{\partial ^2f_3}{\partial I_M^2}=-\frac{2c_{MM}\beta _{MM}\mu _{M}}{\Lambda _{M}},\\ \frac{\partial ^2f_3}{\partial I_M\partial S_f}&=\frac{\partial ^2f_3}{\partial I_M\partial V_f}=\frac{\partial ^2f_3}{\partial I_M\partial S_m}=\frac{\partial ^2f_3}{\partial I_M\partial V_m}=\frac{\partial ^2f_3}{\partial I_M\partial S_M}=\frac{\partial ^2f_3}{\partial I_M\partial V_M}=-\frac{c_{MM}\beta _{MM}\mu _{M}}{\Lambda _{M}},\\ \frac{\partial ^2f_2}{\partial I_f\partial \beta _{fm}}&=c_{fm}. \end{aligned}$$

Therefore, $\displaystyle b=v_2w_1\frac{\partial ^2f_2}{\partial I_f\partial \beta _{fm}}=R_{0,mf}(\phi _f)c_{fm}>0$ and

$$\begin{aligned} a=&2v_1w_1w_2\frac{\partial ^2f_1}{\partial I_f\partial I_m}+2v_1w_1w_3\frac{\partial ^2f_1}{\partial I_f\partial I_M}+2v_2w_1w_2\frac{\partial ^2f_2}{\partial I_f\partial I_m}\\&+2v_3w_1w_3\frac{\partial ^2f_3}{\partial I_f\partial I_M}+2v_3w_2w_3\frac{\partial ^2f_3}{\partial I_m\partial I_M}+v_3w_3^2\frac{\partial ^2f_3}{\partial I_M^2}+2v_3w_3w_4\frac{\partial ^2f_3}{\partial I_M\partial S_f}\\&+2v_3w_3w_5\frac{\partial ^2f_3}{\partial I_M\partial V_f}+2v_3w_3w_6\frac{\partial ^2f_3}{\partial I_M\partial S_m}+2v_3w_3w_7\frac{\partial ^2f_3}{\partial I_M\partial V_m}\\&+2v_3w_3w_8\frac{\partial ^2f_3}{\partial I_M\partial S_M}+2v_3w_3w_9\frac{\partial ^2f_3}{\partial I_M\partial V_M}\\ =&2p\left( -\frac{c_{mf}\beta _{mf}\mu _{f}}{\Lambda _{f}})+2q(-\frac{c_{Mf}\beta _{Mf}\mu _{f}}{\Lambda _{f}}\right) +2R_{0,mf}(\phi _f)p\left( -\frac{c_{fm}\beta _{fm}^*\mu _{m}}{\Lambda _{m}}\right) \\&+2\frac{R_{0,Mf}(\phi _f)}{1-R_{0,MM}(\phi _M)}q\bigg \{\bigg [-\frac{(c_{MM}\beta _{MM}+c_{fM}\beta _{fM})\mu _{M}}{\Lambda _{M}}\bigg ]\\&+p\left( -\frac{c_{MM}\beta _{MM}\mu _{M}}{\Lambda _{M}}\right) +q\left( -\frac{c_{MM}\beta _{MM}\mu _{M}}{\Lambda _{M}}\right) \bigg \}\\&+2\frac{R_{0,Mf}(\phi _f)}{1-R_{0,MM}(\phi _M)}q\left( -\frac{c_{MM}\beta _{MM}\mu _{M}}{\Lambda _{M}}\right) (w_4+w_5+w_6+w_7+w_8+w_9)\\ =&-2p\frac{c_{mf}\beta _{mf}\mu _{f}}{\Lambda _{f}}-2q\frac{c_{Mf}\beta _{Mf}\mu _{f}}{\Lambda _{f}} -2R_{0,mf}(\phi _f)p\frac{c_{fm}\beta _{fm}^*\mu _{m}}{\Lambda _{m}}\\&-2\frac{R_{0,Mf}(\phi _f)}{1-R_{0,MM}(\phi _M)}q\frac{\mu _{M}}{\Lambda _{M}}[c_{fM}\beta _{fM}+c_{MM}\beta _{MM}(1+p+q)]\\&+2\frac{R_{0,Mf}(\phi _f)}{1-R_{0,MM}(\phi _M)}q\frac{c_{MM}\beta _{MM}\mu _{M}}{\Lambda _{M}}(1+p+q)\\ =&-2p\frac{c_{mf}\beta _{mf}\mu _{f}}{\Lambda _{f}}-2q\frac{c_{Mf}\beta _{Mf}\mu _{f}}{\Lambda _{f}} -2R_{0,mf}(\phi _f)p\frac{c_{fm}\beta _{fm}^*\mu _{m}}{\Lambda _{m}}\\&-2\frac{R_{0,Mf}(\phi _f)}{1-R_{0,MM}(\phi _M)}q\frac{\mu _{M}}{\Lambda _{M}}c_{fM}\beta _{fM}<0. \end{aligned}$$

According to the theorem by Castillo-Chavez and Song (2004), system (1) only has forward bifurcation.