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May 2017, 22(3): 977-1000. doi: 10.3934/dcdsb.2017049
Management strategies in a malaria model combining human and transmission-blocking vaccines
Jemal Mohammed-Awel 1, , Ruijun Zhao 2, , Eric Numfor 3, and Suzanne Lenhart 4,
Department of Mathematics, Valdosta State University, Valdosta, GA 31698, USA
Department of Mathematics and Statistics, Minnesota State University, Mankato, Mankato, MN, 56001, USA
Department of Mathematics, Augusta University, Augusta, GA 30912, USA
Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA
Received August 2015 Revised September 2016 Published January 2017
Figure(9) / Table(4)
We propose a new mathematical model studying control strategies of malaria transmission. The control is a combination of human and transmission-blocking vaccines and vector control (larvacide). When the disease induced death rate is large enough, we show the existence of a backward bifurcation analytically if vaccination control is not used, and numerically if vaccination is used. The basic reproduction number is a decreasing function of the vaccination controls as well as the vector control parameters, which means that any effort on these controls will reduce the burden of the disease. Numerical simulation suggests that the combination of the vaccinations and vector control may help to eradicate the disease. We investigate optimal strategies using the vaccinations and vector controls to gain qualitative understanding on how the combinations of these controls should be used to reduce disease prevalence in malaria endemic setting. Our results show that the combination of the two vaccination controls integrated with vector control has the highest impact on reducing the number of infected humans and mosquitoes.
Keywords: Stability, malaria, transmission-blocking vaccine, optimal control, differential equations.
Mathematics Subject Classification: Primary: 92D30; Secondary: 49J1.
Citation: Jemal Mohammed-Awel, Ruijun Zhao, Eric Numfor, Suzanne Lenhart. Management strategies in a malaria model combining human and transmission-blocking vaccines. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 977-1000. doi: 10.3934/dcdsb.2017049
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Figure 1. The schematic diagram of the mathematical model
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Figure 2. The bifurcation diagram for the special case where $\xi_h = 0$. The basic reproduction number is calculated with $u$ varying. The two panels in the first row show a backward bifurcation for $\delta_h = 4\mu_h$, which might represent severe malaria version. The two panels in the second row show a forward bifurcation for $\delta_h = \mu_h$, which might represent a mild version of malaria. The other parameters are listed in Table 1. On the graphs, solid red lines represent the stable endemic equilibrium, dashed blue lines represent the unstable endemic equilibrium, solid green lines represent the stable disease-free equilibrium and dashed green lines represent the unstable disease-free equilibrium
Figure 3. The bifurcation diagram for the complete model. The basic reproduction number is calculated with $u = 0$ and $\xi_h$ varying from $0$ to $0.2$. The three panels in the first row show a backward bifurcation when $\delta_h = 4\mu_h$, which might represent severe malaria version. The three panels in the second row show a forward bifurcation when $\delta_h = 0.5\mu_h$, which might represent a mild version of malaria. On the graphs, solid red lines represent the stable endemic equilibrium, dashed blue lines represent the unstable endemic equilibrium, solid green lines represent the stable disease-free equilibrium and dashed green lines represent the unstable disease-free equilibrium
Figure 4. Simulation for different initial data when $\delta_h = 4\mu_h$, $u = 0$, and $\xi_h = 0.03$. The basic reproduction number is $\mathcal{R}_{vac} = 0.95907$. The three panels in the first row show the disease persists for some initial data. The three panels in the second row show the disease asymptotically dies out for some other initial data
Figure 5. Simulation for different vaccination effort when $\delta_h = 4\mu_h$ and $u = 0$. The three panels in the first row are generated when $\xi_h = 0.001$, in which $\mathcal{R}_{vac} = 1.62127$, and the disease persists. The three panels in the second row are generated when $\xi_h = 0.1$, in which $\mathcal{R}_{vac} = 0.45893$, and the disease asymptotically dies out
Figure 6. Optimal control solution for Case 5 in which the initial values are set to be the endemic equilibrium when no control is applied
Figure 7. Optimal solution of human and mosquito population for Case 5
Figure 8. Optimal control solution for Case 4
Figure 9. Optimal control solution for Case 5 control strategy with low level of infections as initial values for the state variables: $S_h^0=4000$, $I_h^0=10$, $R_h^0=2$, $V_B^0=0$, $J_h^0=0$, $M_h^0=0$, $R_h^0=0$, $S_v^0=10000$, $I_v^0=10$, and $V_v^0=0$
Table 1. Description of parameters of the basic malaria model, the detailed reference for each value can be found in [7,34]
Parameter Description Baseline values and range
$\Lambda_{h}$ Recruitment rate of $10^4/55 \in[10^{4}/72, 10^{4}/35]$
$L$ environmental carrying capacity per year $10^5 \in[10^{4}/72, 10^{4}/35]\times\frac{40}{3}$
$r_v$ mosquito growth constant per year $4\times365/21$
$\mu_{h}$ Natural death rate of host per year $1/55\in[1/72, 1/35]$
$\mu_{v}$ Natural death rate of vector per year $365/21\in[365/28, 365/14]$
$C_{hv}$ The effective transmission rate from humans to mosquitoes per year per bite $9\in[2.6, 32\times365]$
$C_{vh}$ The effective transmission rate from mosquitoes to humans per year per bite $0.8\in[0.001\times365, 0.27\times365]$
$\xi_h$ Vaccination rate of humans with mixture dose per year $0.01\in[0, \ln 5]$
$\rho_{h}$ Rate of loss of immunity per year $2\in[1/50, 4]$
$\eta_{h}$ Rate of development of temporal immunity per year $1\in [1/2, 6]$
$\delta_h$ Disease-induced death rate per year $0.1/55\in[0, 4.1\times10^{-4}]\times365$
$\nu_{h}$ Vaccination rate of humans per year with HV dose only $0.1 \in[0, \ln 5]$
$\omega_{h}$ Rate of loss of HV acquired-immunity per year in vaccinated group of humans $1/4\in[1/5, 1]$
$u$ $1-u$ represents mosquitoes birth due to control such asreduction factor of larvacide $0\in[0, 0.72]$
Table 2. Values for the parameters for our numerical scenarios
Parameter Value Parameter Value Parameter Value Parameter Value
A1 7.3 B1 A1 C1 100 $\xi _h^{\max }$ ln(4)/4
A2 A1 B2 B1/2 C2 100 $\nu _h^{\max }$ ln(4)/4
A3 A1/10 B3 B1/100 C3 100 ${u^{\max }}$ 0.4
Table 3. Values of the objective functional at the optimal control solution (column two) and at the upper bound values of the nonzero controls (column three); the fourth column is the percentage decrease in cost for each strategy at optimal control compared to strategy with maximum control
Strategy/Control Cost with optimal control Cost with constant maximum control Percentage decrease in cost at Optimal Control compared to maximum control
Case 1 181, 345 191, 066 5.09%
Case 2 199, 938 199, 943 0.0025%
Case 5 157, 635 175, 596 10.23%
Table 4. Values of the cost functional at the optimal control solution(column two); percentage cost decrease of each strategy at optimal control compared to strategy without control, Case 0 (column three); and percentage decrease in cost for each strategy at optimal control compared to strategy in Case 2 at optimal control, (column four).
Strategy/Control Cost with optimal control Percentage cost decrease compared with Case 0 Percentage decrease in cost compared with Case 2
Case 1 181, 345 27.26% 10.25%
Case 2 199, 938 15.43% 0.0%
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Jemal Mohammed-Awel Ruijun Zhao Eric Numfor Suzanne Lenhart