I think this could be the solution to your problem
% Parameters
L = 80;
gamma = 1e-2;
R_0 = 0.7;
beta = R_0*gamma;
epsilon = 0.0001;
sigma = -log(epsilon)/L;
% Time discretization
days = 10;
tspan = [1 days];
% Space discretization
y = linspace(0,L,100);
% Intial conditions
z0 = 0.2*ones(size(y));
[t,Z] = ode45(@(t,z) odefun(t,z,y,sigma,beta,gamma),tspan,z0);
plot(y,Z) % plot of Z vs space at various times
xlabel('y')
ylabel('Z')
legend([repmat('t = ',length(t),1),num2str(t)],'Location','EastOutside')
function dzdt = odefun(t,z,y,sigma,beta,gamma)
xmy = abs(y-y');
dzdt = beta*(1-z).*trapz(y,exp(-sigma*xmy).*z',2)-gamma*z;
end
