Vaccine supply decisions and government interventions for recurring epidemics

Abstract

For infectious diseases that occurs recurringly or periodically, e.g., influenza, humans have tried to develop vaccines to effectively prevent infection. However, vaccination coverage, which is the most effective way to prevent infections, is undesirably low. Existing epidemiology studies have consistently shown that there is association between the vaccination decisions in different flu seasons (epidemic periods), but related research in operations management mainly focuses on the single-period model. In this paper we construct a multi-period vaccine demand model to study multi-period vaccine supply decisions and government interventions. We consider that members of the public make vaccination decisions at the beginning of an epidemic period, given the information of the last epidemic period. Both the manufacturer and government make multi-period decisions in our model. The vaccination coverage is determined by the minimum between the supply and demand for the vaccine. We derive the multi-period profit-maximizing coverage and compare it with the socially optimal coverage. In addition, we show that, besides supply uncertainty, vaccine demand may decrease or increase with the vaccination coverage in the last epidemic period, depending on the vaccine effectiveness. Furthermore, the coverage convergence depends on the vaccine effectiveness and infection loss distribution. Accordingly, the multi-period profit-maximizing coverage and government intervention depend on the vaccine effectiveness and coverage convergence. We also conduct numerical studies to generate practical implications of the analytical findings. Our results provide management insights on vaccine supply decisions, government interventions, and vaccination coverage.

Appendix

Proof of Lemma 1

$u_{t+1}^{m}(f_t, \phi _{t})=\frac{w}{P(f_t,\phi _{t} )-H(f_t,\phi _{t} )}=\frac{w(1-f_t)}{r(f_t,\phi _{t} )-H(f_t,\phi _{t} )}$;

$d_{t+1}(f_t, \phi _{t})=(1-\alpha -\beta )\bar{G}(u_{t+1}^{m})+\alpha =(1-\alpha -\beta )\bar{G}(\frac{w(1-f_t)}{r(f_t,\phi _{t} )-H(f_t,\phi _{t} )})+\alpha$. $\square$

Proof of Proposition 1

$H(f_t, \phi _{t})=\eta (1-\phi _{t}) (1-\phi _{t} f_t-\frac{1}{R_0})$, $\frac{\partial H(f_t, \phi _{t})}{\partial \phi _{t}}=\eta (-(1-\phi _{t} f_t-\frac{1}{R_0})-f_t(1-\phi _{t})) \le 0$, $\frac{\partial ^2 H(f_t, \phi _{t})}{\partial \phi _{t}^2}=2\eta f_t \ge 0$.

$P(f_t, \phi _{t})=\frac{r(f_t, \phi _{t})-f_t H(f_t, \phi _{t})}{1-f_t}$, $\frac{\partial P(f_t, \phi _{t})}{\partial \phi _{t}}= \frac{f_t(\eta -1-\frac{\eta }{R_0}+\eta f_t(1-2\phi _{t}))}{1-f_t}$, $\frac{\partial ^2 P(f_t, \phi _{t})}{\partial \phi _t^2}=\frac{-2\eta f_t^2}{1-f_t} \le 0$.

$\frac{\partial ^2 (P(f_t, \phi _{t})-H(f_t, \phi _{t}))}{\partial \phi _{t}^2}=\frac{-2\eta f_t^2}{1-f_t}-2\eta f_t \le 0$. When $\phi _{t}\le \frac{1-R_0}{2 f_t R_0}+\frac{1-\eta }{2\eta }$, $\frac{\partial (P(f_t, \phi _{t})-H(f_t, \phi _{t}))}{\partial \phi _{t} } \le 0$. For any disease, $R_0 \ge 0$. According to the estimation of $\eta$ in Mamani et al. (2012), $\eta \ge 1$. So, for $\phi$ in [0, 1], $\frac{\partial (P(f_t, \phi _{t})-H(f_t, \phi _{t}))}{\partial \phi _{t} } \le 0$. $\square$

Proof of Proposition 2

$\frac{\partial (d_{t+1})}{\partial (f_t)}= [(r(f_t,\phi _{t} )-H(f_t,\phi _{t} ))+$

$(1-f_t)(\frac{\partial r(f_t,\phi _{t} )}{\partial f_t}-\frac{\partial H(f_t,\phi _{t} )}{\partial f_t})] \frac{-w(1-\alpha -\beta )}{(r(f_t,\phi _{t} )-H(f_t,\phi _{t} ))^2} \frac{-dG(u^m_t)}{du^m_t} \!=\!\frac{-w(1-\alpha -\beta )(\phi _{t}+\frac{1}{R_0}-1)}{(1-\eta (1-\phi _{t}))(1-\phi _{t} f_t-\frac{1}{R_0})^2} \frac{dG(u^m_t)}{du^m_t}$;

$\frac{\partial ^2(d_{t+1})}{\partial f_t^2}=-g(\frac{w(1-f_t)}{r(f_t,\phi _{t} )-H(f_t,\phi _{t} )})[\frac{-2w(1-\alpha -\beta )\phi _{t}(\phi _{t}+\frac{1}{R_0}-1)}{(1-\eta (1-\phi _{t}))(1-\phi _{t} f_t-\frac{1}{R_0})^3}]-$

$g^{'} (\frac{w(1-f_t)}{r(f_t,\phi _{t} )-H(f_t,\phi _{t} )})\frac{w(1-\alpha -\beta )(\phi _{t}+\frac{1}{R_0}-1)}{(1-\eta (1-\phi _{t}))(1-\phi _{t} f_t-\frac{1}{R_0})^2}$.

$\frac{\partial ^2(d_{t+1})}{\partial f_t^2}\!=\!\frac{-w(1-\alpha -\beta )(\phi _{t}+\frac{1}{R_0}-1)}{(1-\eta (1-\phi _{t}))(1-\phi _{t} f_t-\frac{1}{R_0})^2}(g(\frac{w(1-f_t)}{r(f_t,\phi _{t} )-H(f_t,\phi _{t} )})\frac{2\phi _{t}}{(1-\phi _{t} f_t-\frac{1}{R_0})}\!+\!g^{\prime } (\frac{w(1-f_t)}{r(f_t,\phi _{t} )-H(f_t,\phi _{t} )}))$.

Set $J(f_t, \phi _{t} )=\frac{w(1-f_t)}{r(f_t,\phi _{t})-H(f_t,\phi _{t})}$.

When $\phi _{t}>1-\frac{1}{R_0}$, $(\phi _{t}+\frac{1}{R_0}-1)>0$. The sufficient condition for $\frac{\partial ^2(d_{t+1})}{\partial f_t^2}< 0$ is that $\frac{2 \phi _{t} g(J(f_t, \phi _{t} ))}{1-\phi _{t} f_t-\frac{1}{R_0}}+g'(J(f_t, \phi _{t}))> 0$. It is easy to prove that $\frac{2 \phi _{t} g(J(f_t, \phi _{t} ))}{1-\phi _{t} f_t-\frac{1}{R_0}}+g'(J(f_t, \phi _{t}))> 0$ for a uniform distribution g(.). On the other hand, if g(.) is a normal distribution, for $R_0=3$ (approximate value for influenza), $0\le w\le 1$, $\phi _{t}>1-\frac{1}{R_0}$, and $0\le f_t \le 1-\frac{1}{R_0}$, and we can get $\frac{2 \phi _{t} g(J(f_t, \phi _{t} ))}{1-\phi _{t} f_t-\frac{1}{R_0}}+g'(J(f_t, \phi _{t}))> 0$. So $\frac{\partial ^2(d_{t+1})}{\partial f_t^2}< 0$ for $\phi _{t}>1-\frac{1}{R_0}$.

(1)$\phi _{t}>1-\frac{1}{R_0}$, $\frac{\partial d_{t+1}(f_t, \phi _{t})}{\partial f_t }<0$, $d_{t+1}(f_t, \phi _{t})$ is a strictly concave decreasing function of $f_t$.

(2)$\phi _{t}<1-\frac{1}{R_0}$, $\frac{\partial d_{t+1}(f_t, \phi _{t})}{\partial f_t }>0$, $d_{t+1}(f_t, \phi _{t})$ is an increasing function of $f_t$.

(3)$\phi _{t}=1-\frac{1}{R_0}$, $\frac{\partial d_{t+1}(f_t, \phi _{t})}{\partial f_t }=0$, $d_{t+1}(f_t, \phi _{t})$ is a constant function of $f_t$. $\square$

Proof of Lemma 2

When the vaccine effectiveness satisfies $\phi _{t+1}>1-\frac{1}{R_0}$, for $f_t<\alpha$, $d_{t+1}(f_t, \phi _{t})=1-\beta$. Then $d_{t+1}(1-\beta , \phi _{t})=\alpha$. And the coverage would never be outside $[\alpha , 1-\beta ]$. Besides, when $1-\beta >f_{cf}$, $r(1-\beta ,\phi _{t} )=H(1-\beta ,\phi _{t} )=P(1-\beta ,\phi _{t} )=r(f_{cf},\phi _{t+1} )=H(f_{cf},\phi _{t} )=P(f_{cf},\phi _{t} )=0$. At this time, $d_{t+1}(1-\beta , \phi _{t})=d_{t+1}(f_{cf}, \phi _{t})$. It contradicts Proposition 2 (1). $\square$

Proof of Lemma 3

For $\phi _{t}>1-\frac{1}{R_0}$, $d_{t+1}(f_0, \phi _{t})$ is a decreasing function in $[\alpha , 1-\beta ]$. There are $d_{t+1}(\alpha , \phi _{t})-\alpha \ge 0$ and $d_{t+1}(1-\beta , \phi _{t})-(1-\beta ) \le 0$. So there must exist an $f_0$ satisfying $d_{t+1}(f_0, \phi _{t})=f_0$. $\square$

Proof of Proposition 3

$\frac{\partial B}{\partial f_t}=-g(J(f_t, \phi _{t} ))[\frac{w(1-\alpha -\beta )(\phi _{t}+\frac{1}{R_0}-1)}{(1-\eta (1-\phi _{t}))(1-\phi _{t} f_t-\frac{1}{R_0})^2}]+k_t$;

$\frac{\partial ^2B}{\partial f_t^2}=\frac{\partial ^2(d_{t+1})}{\partial f_t^2}=\frac{-w(1-\alpha -\beta )(\phi _{t}+\frac{1}{R_0}-1)}{(1-\eta (1-\phi _{t}))(1-\phi _{t} f_t-\frac{1}{R_0})^2}(g(\frac{w(1-f_t)}{r(f_t,\phi _{t} )-H(f_t,\phi _{t} )})\frac{2\phi _{t}}{(1-\phi _{t} f_t-\frac{1}{R_0})}$ $\qquad \qquad +g^{\prime } (\frac{w(1-f_t)}{r(f_t,\phi _{t} )-H(f_t,\phi _{t} )}))$.

(1) $\phi _{t}>1-\frac{1}{R_0}$. By the Proof of Proposition 2, when $\frac{2 \phi _{t} g(J(f_t, \phi _{t} ))}{1-\phi _{t} f_t-\frac{1}{R_0}}+g'(J(f_t, \phi _{t} ))\ge 0$, where $J(f_t, \phi _{t} )=\frac{w(1-f_t)}{r(f_t)-H(f_t)}$, $\frac{\partial ^2B}{\partial f_t^2}\le 0$. Then it is easy to get the results.

(2) $\phi _{t}<1-\frac{1}{R_0}$. $k_t>0$, $\frac{\partial B}{\partial f_t}>0$, $f_{2P}^{M*}=1-\beta$. $\square$

Proof of Lemma 4

As shown in the Proof of Proposition 8, $J (J(f_t,\phi _t ), \phi _{t+1} ) < f_t$ for $f_t>f_0$ and $J (J(f_t,\phi _t ), \phi _{t+1} )> f_t$ for $f_t<f_0$. For a $f_t<f_0$, $f_t+f_{t-1}>f_t+f_{t+1}$; for a $f_t>f_0$, $f_t+f_{t+1}>f_t+f_{t-1}$. Therefore, $f_{VI}>f_0$. $\square$

Proof of Proposition 4

When $\phi _{t}>1-\frac{1}{R_0}$, for $f_t\le f_{IV}$, $\frac{\partial B}{\partial f_t}\ge 0$; when $\phi _{t}<1-\frac{1}{R_0}$, for all $f_t\in [0,1]$, $\frac{\partial B}{\partial f_t}\ge 0$. For both $\phi _{t}>1-\frac{1}{R_0}$ and $\phi _{t}<1-\frac{1}{R_0}$, $\frac{\partial B}{\partial f_t}$ is an increasing function of $f_t$ in $[0, f_{IV}]$.

Proof of Lemma 5

By Lemma 2 and Proposition 3, when $\phi _{t}>1-\frac{1}{R_0}$, $B(\alpha , \phi _{t})=B(1-\beta , \phi _{t})$ and both are the minimum value for $f_t\in [\alpha , 1-\beta ]$. When $\phi _{t}<1-\frac{1}{R_0}$, $B(1-\beta , \phi _{t})$ is the maximum for $f_t\in [\alpha , 1-\beta ]$. So if $(w_a-\frac{c_a}{Y_a})(1+\alpha -\beta ) \ge 2(w_b-\frac{c_b}{Y_b})(1-\beta )$, $(w_a-\frac{c_a}{Y_a}) B(f_{t}, \phi _{t})\mid _{\phi _{t}>1-\frac{1}{R_0}}\ge (w_b-\frac{c_b}{Y_b})(1-\beta )B(f_{t}, \phi _{t})\mid _{\phi _{t}<1-\frac{1}{R_0}}$ for $f_t\in [\alpha , 1-\beta ]$. If $(w_a-\frac{c_a}{Y_a})(1+\alpha -\beta ) < 2(w_b-\frac{c_b}{Y_b})(1-\beta )$, there exists an $f_I$ satisfying $\pi _a(f_I,\phi _t)=\pi _b(f_I,\phi _t)$. $\square$

Proof of Proposition 5

$u_t$ and $f_t$ are for the same period, so $u_t (P(f_t,\phi _t )-H(f_t,\phi _t ))=w$ and $G (u_{t} ) =1-f_{t}$.

$\int _{u_{t}}^{1} v d G(v) =v G(v)|_{u_{t}} ^{1}-\int _{u_{t}}^{1} G(v) d v =1-u_{t} G (u_{t} )-\int _{u_{t}}^{1} G(v) d v$.

$\int _{0}^{u_{t}} v d G(v) =v G(v)|_{0}^{u_{t}}-\int _{0}^{u_{t}} G(v) d v =u_{t} G(u_{t})-\int _{0}^{u_{t}} G(v) d v$.

$\frac{\partial \int _{u_{t}}^{1} v d G(v)}{\partial f_{t}}=-\frac{\partial u_{t}}{\partial f_{t}} G\left( u_{t}\right) +u_{t}+G(u_{t}) \frac{\partial u_{t}}{\partial f_{t}}=u_{t}$, and $\frac{\partial \int _{0}^{u_{t}} v d G(v)}{\partial f_{t}}=-u_{t}$.

$\frac{\partial TC}{\partial f_t}=m[\frac{\partial H(f_t,\phi _t )}{\partial f_t}\int ^1_{u_t}vdG(v)+H(f_t,\phi _t )u_t+\frac{\partial P(f_t,\phi _t )}{\partial f_t}\int _0^{u_t}vdG(v)-P(f_t,\phi _t )u_t]+(m-1)w+\frac{c}{Y_t}=m[\frac{\partial H(f_t,\phi _t )}{\partial f_t}\int ^1_{u_t}vdG(v)+\frac{\partial P(f_t,\phi _t )}{\partial f_t}\int _0^{u_t}vdG(v)]-w+\frac{c}{Y_t}$.

It is easy to get $\frac{\partial H(f_t,\phi _t )}{\partial f_t}\le 0$, $\frac{\partial P(f_t,\phi _t )}{\partial f_t}\le 0$, and $-w+\frac{c}{Y_t}<0$. So $\frac{\partial TC}{\partial f_t}\le 0$.

$\frac{\partial \left( P\left( f, \phi _{t}\right) -H\left( f_{t}, \phi _{t}\right) \right) }{\partial f_{t}}=\frac{\left( 1-\eta \left( 1-\phi _{t}\right) \right) \left[ \left( -\phi _{t}\right) \left( 1-f_{t}\right) +1-\phi _{t} f_{t}-\frac{1}{R_{0}}\right] }{\left( 1-f_{t}\right) ^{2}} =\frac{\left( 1-\eta \left( 1-\phi _{t}\right) \right) \left( 1-\frac{1}{R_{0}}-\phi _{t}\right) }{\left( 1-f_{t}\right) ^{2}}$.

$\frac{\partial ^{2} P\left( f_{t}, \phi _{t}\right) }{\partial f_{t}^{2}}=\frac{\partial ^{2}\left( P\left( f_{t}, \phi _{t}\right) -H\left( f_{t}, \phi _{t}\right) \right) }{\partial f_{t}^{2}} =\frac{4\left( 1-\eta \left( 1-\phi _{t}\right) \right) \left( 1-\frac{1}{R_{0}}-\phi _{t}\right) }{\left( 1-f_{t}\right) ^{2}}$.

$\frac{\partial ^{2} T C_{1 P}}{\partial f_{t}^{2}}=m [\frac{\partial ^{2} H (f_{t}, \phi _{t} )}{\partial f_{t}^{2}} \int _{u_{t}}^{1} v d G(v)+\frac{\partial ^{2} P (f_{t}, \phi _{t} )}{\partial f_{t}^{2}} \int _{0}^{u_{t}} v d G(v) +\frac{\partial H (f_{t}, \phi _{t} )}{\partial f_{t}}{u_{t}}-\frac{\partial P (f_{t}, \phi _{t} )}{\partial f_{t}}{u_{t}} ]$

$=m \frac{(1-\eta (1-\phi _t)) (1-\frac{1}{R_{0}}-\phi _{t} )}{\left( 1-f_{t}\right) ^{2}} (4 \int _{0}^{u_{t}} v d G(v) -1 )$.

Therefore, for $\phi _{t}>1-\frac{1}{R_0}$, when $\int _{0}^{u_{t}} v d G(v) \ge \frac{1}{4}$, TC is concave decreasing function of $f_t$. When $\int _{0}^{u_{t}} v d G(v) <\frac{1}{4}$, TC is convex decreasing function of $f_t$.

For $\phi _{t}<1-\frac{1}{R_0}$, when $\int _{0}^{u_{t}} v d G(v) \ge \frac{1}{4}$, TC is convex decreasing function of $f_t$. When $\int _{0}^{u_{t}} v d G(v) <\frac{1}{4}$, TC is concave decreasing function of $f_t$. $\square$

Proof of Lemma 6

$CC^*_{1P}=((m-1) w_{a} +\frac{c_{a}}{Y_{a}}) \frac{R_{0}-1}{\phi R_{0}}-(m-1) w_{b}-\frac{c_{b}}{Y_{b}}-m r\left( 1, \phi _{t}\right) \int _{0}^{1} v d G(v)$.

$\frac{\partial CC^*_{1P}}{\partial R_0}=(((m-1) w_{a} +\frac{c_{a}}{Y_{a}}) \frac{1}{\phi } - m \int _{0}^{1} v d G(v))\frac{1}{R_0^2}$.

$\frac{\partial ^2 CC^*_{1P}}{\partial R_0^2}=-2 (((m-1) w_{a} +\frac{c_{a}}{Y_{a}}) \frac{1}{\phi } - m \int _{0}^{1} v d G(v))\frac{1}{R_0^3}$.

If $((m-1) w_{a} +\frac{c_{a}}{Y_{a}}) \frac{1}{\phi } - m \int _{0}^{1} v d G(v)>0$, $CC^*_{1P}$ is a concave increasing function of $R_0$.

If $((m-1) w_{a} +\frac{c_{a}}{Y_{a}}) \frac{1}{\phi } - m \int _{0}^{1} v d G(v)<0$, $CC^*_{1P}$ is a convex decreasing function of $R_0$. $\square$

Proof of Proposition 7

Set $f_{III}$ as the coverage achieving the minimum total subsidy. We use the contradiction to show that $f_{III}< f^{M*}_{2P}$.

If $f_{III}\ge f^{M*}_{2P}$, $TS_{2P}(f_{2P}^{T})\mid _{d_t}=S_M (f_{2P}^{T} - f^{M*}_{2P}) + S_C (f_{2P}^{T}-d_t)+ S_M (f_{2P}^{T} - f^{M*}_{2P})+ S_C (f_{2P}^{T}-d_{t+1}(f_{2P}^{T}))$. $\frac{\partial TS_{2P}(f_{2P}^{T})}{\partial f_{2P}^{T}}= 2(S_M+S_C)- 2S_C\frac{\partial d_{t+1}(f_{2P}^{T})}{\partial f_{2P}^{T}}$, and $\frac{\partial ^2 TS_{2P}(f_{2P}^{T})}{\partial (f_{2P}^{T})^2}= - 2 S_C \frac{\partial ^2 d_{t+1}(f_{2P}^{T})}{\partial (f_{2P}^{T})^2} \ge 0$. So $f_{III}$ satisfies $\frac{\partial d_{t+1}(f_{t})}{\partial f_{t}}=1+\frac{S_M}{S_C}$. And $f_{VI}$ satisfies $1+ \frac{\partial d_{t+1}(f_{t})}{\partial f_{t}}=0$. Since $\frac{\partial ^2 d_{t+1}(f_{t})}{\partial (f_{t})^2}\le 0$, $\frac{\partial d_{t+1}(f_{t})}{\partial f_{t}}$ is a decreasing function of $f_t$. So $f_{III}< f^{M*}_{2P}$. It contradict with the assumption.

If $f_{III}< f^{M*}_{2P}$, $TS_{2P}(f_{2P}^{T})\mid _{d_t}=S_C (f_{2P}^{T}-d_t) + S_C (f_{2P}^{T}-d_{t+1}(f_{2P}^{T}))$. $\frac{\partial TS_{2P}(f_{2P}^{T})}{\partial f_{2P}^{T}}= S_C (2- \frac{\partial d_{t+1}(f_{2P}^{T})}{\partial f_{2P}^{T}})$, $\frac{\partial ^2 TS_{2P}(f_{2P}^{T})}{\partial (f_{2P}^{T})^2}= - S_C \frac{\partial ^2 d_{t+1}(f_{2P}^{T})}{\partial (f_{2P}^{T})^2} \ge 0$. So $f_{III}$ satisfies $2- \frac{\partial d_{t+1}(f_{t})}{\partial f_{t}}=0$. And $f_{VI}$ satisfies $1+ \frac{\partial d_{t+1}(f_{t})}{\partial f_{t}}=0$. Since $\frac{\partial ^2 d_{t+1}(f_{t})}{\partial (f_{t})^2}\le 0$, $\frac{\partial d_{t+1}(f_{t})}{\partial f_{t}}$ is a decreasing function of $f_t$. So $f_{III}< f^{M*}_{2P}$. $\square$

Proof of Proposition 8

(1) Considering $f_t>f_0$, following that $d_{t+1}(f_t, \phi _{t})$ is a strictly concave decreasing function of $f_t$, we get

$\left\{ \begin{array}{l}f_{t}-f_{0}>f_{0}-f_{t+1} \\ f_{0}-f_{t+1}>f_{t+2}-f_{0}\end{array}\right.$.

So we get $J (J(f_t,\phi _t ), \phi _{t+1} ) < f_t$ for $f_t>f_0$, and it is similar to get $J (J(f_t,\phi _t ), \phi _{t+1} )> f_t$ for $f_t<f_0$. Therefore, the coverage converges to $f_0$.

(2) a. It means $d_{t+1}<f_t$. b. It means $d_{t+1}>f_t$. c. It means $d_{t+1}=f_t$. $\square$

Proof of Proposition 9

$\frac{\partial \pi _{MP}}{\partial f_t}=\frac{1}{e}[\frac{\partial (f_t+f_{t+1})}{\partial f_t}+\frac{\partial (f_{t+2}+f_{t+3})}{\partial f_{t+2}}\frac{\partial f_{t+2}}{\partial f_{t+1}}\frac{\partial f_{t+1}}{\partial f_{t}}+\ldots ]$.

(1) Regarding Proposition 3, we get $\frac{\partial ^2B}{\partial f_t^2}\le 0$. Besides, $\frac{\partial f_{t+2}}{\partial f_{t+1}}\le 0$ and $\frac{\partial f_{t+1}}{\partial f_{t}}\le 0$. a. We first consider the situation where $f_{t+2} < f_t$ for $f_t>f_0$ or $f_{t+2}> f_t$ for $f_t<f_0$. If $f_0 > f_{VI}$, for all $f_t>f_{VI}$, we have $f_{t+2}>f_{VI}$, $f_{t+4}>f_{VI}$, ...By Proposition 3, we can get $f^{M*}_{MP}<f_{VI}$. If $f_0 < f_{VI}$, for all $f_t<f_{VI}$, we will have $f_{t+2}<f_{VI}$, $f_{t+4}<f_{VI}$, ...By Proposition 3, we get $f^{M*}_{MP}>f_{VI}$. b. The situation where $f_{t+2}>f_t$ for $f_t>f_0$ or $f_{t+2}<f_t$ for $f_t<f_0$ is as follows. For all $f_t>max\{f_0, f_{VI}\}$, we have $f_{t+2}>max\{f_0, f_{VI}\}$, $f_{t+4}>max\{f_0, f_{VI}\}$, ...For all $f_t<min\{f_0, f_{VI}\}$, we have $f_{t+2}<min\{f_0, f_{VI}\}$, $f_{t+4}<min\{f_0, f_{VI}\}$, ...By Proposition 3, we get $f^{M*}_{MP}$ in the interval $[f_0, f_{VI}]$.

(2) When $\phi _{t+1}<1-\frac{1}{R_0}$, $\frac{\partial B}{\partial f_t}>0$ and $\frac{\partial f_{t+1}}{\partial f_{t}}\le 0$ for all $f_t$. So $\frac{\partial \pi _{MP}}{\partial f_t} \ge 0$. $\square$

Proof of Proposition 10

By Lemma 4, for a given $d_t$, the optimal production is $min\{f_{2P}^{M*}, d_t\}$. So for $d_t\ge f_{2P}^{M*}$, $f_t=f_{2P}^{M*}$; otherwise, $f_t=d_t$.

By Proposition 9, (1) When $\phi _{i}>1-\frac{1}{R_0}$ for $i=t, t+1, \ldots , M$, $J (J(f_t,\phi _t ), \phi _{t+1} ) < f_t$ for $f_t>f_0$ and $J (J(f_t,\phi _t ), \phi _{t+1} ) > f_t$ for $f_t<f_0$. $f_{2P}^{M*} > f_0$. If $f_t=f_{2P}^{M*}$, we get $f_{t+2}<f_{2P}^{M*}$, $f_{t+4}<f_{t+2}<f_{2P}^{M*}$, ...By Proposition 3, $B(f_t)< B(f_{t+2})<B(f_{t+4})\ldots$. So $\frac{2 }{M} B_{MP}$ decreases with M. (2) When $\phi _t, \phi _{t+1}, \ldots , \phi _{M}<1-\frac{1}{R_0}$, the profit-maximizing coverage is $1-\beta$ in every period. $\square$

Proof of Proposition 11

If $f_{VII}>f^{M*}_{2P}$, $\frac{\partial TS_{MP}}{\partial f_{MP}^{T}}=(S_M+S_C)-S_C \frac{\partial d_{t+1}}{\partial f_{MP}^{T}}$, $\frac{\partial ^2 TS_{MP}}{\partial (f_{MP}^{T})^2}=- S_C \frac{\partial ^2 d_{t+1}}{\partial (f_{MP}^{T})^2}\ge 0$. Set $f_{VII}$ satisfying $\frac{\partial d_{t+1}}{\partial f_{MP}^{T}}=\frac{S_M+S_C}{S_C}$. We get $f_{VII}<f^{M*}_{2P}$, which contradicts with the assumption.

Then considering $f_{VII}<f^{M*}_{2P}$, $\frac{\partial TS_{MP}}{\partial f_{MP}^{T}}=(S_M+S_C)(1- \frac{\partial d_{t+1}}{\partial f_{MP}^{T}})$, $\frac{\partial ^2 TS_{MP}}{\partial (f_{MP}^{T})^2}=- (S_M+S_C)\frac{\partial ^2 d_{t+1}}{\partial (f_{MP}^{T})^2}\ge 0$, we set $f_{VII}$ satisfying $1-\frac{\partial d_{t+1}}{\partial f_{VII}}=0$. $\square$