A qualitative analysis of a Mycoplasma genitalium epidemiological model

Theorem 1.For any initial condition $(s_0,i_{s0},i_{r0})\in {({ℝ}_0^{+})}^3$, problems (1) and (2) admit a unique solution.

Proof.System (1) can be written in the form $X^{\prime }(t)=F(X(t))$, where for $t\in [0,T]$,

Theorem 2.The solution of system (1) and (2) is nonnegative.

Proof.Since

Theorem 3.The disease free equilibrium point $P_F=(1,0,0)$ is

• 1.

  locally asymptotically stable if $\beta /\gamma <1$;

• 2.

  unstable if $\beta /\gamma >1$;

If $\beta /\gamma =1$, then a critical case occurs.

Proof.The Jacobian matrix (5), evaluated at the disease free equilibrium point, is the following:

Since the spectrum of matrix (6) is

Theorem 4.If $R_0>1$, then model (1) has an endemic equilibrium point given by $P_E=(1/R_0,0,1-1/R_0)$, and is locally asymptotically stable.

Proof.In this case, the Jacobian matrix is given by

Theorem 5.If $R_0<1$, then the disease free equilibrium point $P_F=(1,0,0)$ of model (1) is globally asymptotically stable.

Proof.Consider the following Lyapunov function $V(i_s,i_r)=2\mu i_s+i_r.$ Differentiating V with respect to time gives

Thus, $\dot{V}\le 0$ for all $s,i_s,i_r\ge 0$. Moreover, $\dot{V}=0$ if and only if $(s,i_s,i_r)=P_F$. Consequently, by LaSalle's Invariance Principle,²¹ the disease free equilibrium point $P_F$ is globally asymptotically stable.

Theorem 6.Let $R_0>1$. Then, the endemic equilibrium point $P_E=(s^{\star },i_s^{\star },i_r^{\star })$ of model (1) is globally asymptotically stable.

Proof.Consider the following Lyapunov function

Observe that, for all $t\in [0,T]$, $1-\mu (i_r-i_r^{\star })/i_r>0$. In fact, if $(i_r-i_r^{\star })/i_r>0$, and since $\mu \in (0,1)$, we get that

Theorem 7. ([24])There exists an optimal control $\bar{\chi }$ and a set of corresponding solutions $(\overline{i_s},\overline{i_r})$ that minimize the objective functional (7) subject to (8) and (9). Moreover, there exists an adjoint function $\bar{\lambda }:[0,T]\to {ℝ}^2$,$\bar{\lambda }=(\overline{{\lambda }_1},\overline{{\lambda }_2})$, such that $(\overline{i_s},\overline{i_r},\bar{\chi },\bar{\lambda })$ is a solution of the following equations:

• 1.

  The control system

  $$\left\{\begin{array}{c}{i}_s^{\prime }(t)=\beta (1-i_s(t)-i_r(t))i_s(t)-(\gamma +\chi (t))i_s(t)\\ i_r^{\prime }(t)=\beta (1-i_s(t)-i_r(t))i_r(t)-\gamma i_r(t)+\chi (t)\mu i_s(t),\end{array}\right.$$
  that is, system (8);

• 2.

  The adjoint system:

  $$\left\{\begin{array}{c}{\lambda }_1^{\prime }=-w_s+{\lambda }_1(\beta i_s-\beta (1-i_s-i_r)+\gamma +\chi )+{\lambda }_2(\beta i_r-\chi \mu )\\ {\lambda }_2^{\prime }=-w_r+{\lambda }_1\beta i_s+{\lambda }_2(\beta i_r-\beta (1-i_s-i_r)+\gamma );\end{array}\right.$$ ()

• 3.

  The transversality condition

  $${\lambda }_1(T)=0={\lambda }_2(T).$$ ()

Furthermore, the following minimization condition is fulfilled:

Proof.The existence of an optimal control $\bar{\chi }$ is an immediate consequence of Reference 25 (corollary 4.1). In fact, it follows due to:

1. the convexity of integrand of ${𝒥}_1$ with respect to the control $\chi$;
2. a priori boundedness of the state solutions;
3. the state system (8) satisfies the Lipschitz property with respect to the state variables since state solutions are bounded.

The deduction of the necessary optimality conditions is a standard result from the Pontryagin's minimum principle. If $\bar{\chi }$ is an optimal control, there exists a nontrivial mapping $\bar{\lambda }:[0,T]\to {ℝ}^2$, such that the following equations are satisfied for $(\overline{i_s},\overline{i_r},\bar{\chi },\bar{\lambda })$:

1. The control system
   $$i_s^{\prime }=\frac{\partial H}{\partial {\lambda }_1}\text{and}{i}_r^{\prime }=\frac{\partial H}{\partial {\lambda }_2};$$
2. The adjoint system:
   $${\lambda }_1^{\prime }=-\frac{\partial H}{\partial i_s}\text{and}{\lambda }_2^{\prime }=-\frac{\partial H}{\partial i_r};$$
3. the transversality condition ${\lambda }_1(T)=0={\lambda }_2(T)$;

and also the minimization condition:

Remark 1.Although at the endemic equilibrium, $i_s=0$ and the equilibrium is globally stable, we can not ignore the class $i_s$ in the optimal control model, since in this case the results will be different. For example, using the same data as the ones used in Figure 4, but considering ${\omega }_s=0$, the results are displayed in Figure 6. As it can be seen, the evolution of infected individuals is distinct when ${\omega }_s=0$.